Recent zbMATH articles in MSC 60https://zbmath.org/atom/cc/602021-01-08T12:24:00+00:00WerkzeugStaircase tableaux and an alternative matrix formula for steady state probabilities in the asymmetric exclusion process \((q=1)\).https://zbmath.org/1449.050092021-01-08T12:24:00+00:00"Gonzales, Ken Joffaniel M."https://zbmath.org/authors/?q=ai:gonzales.ken-joffaniel-m"Celeste, Richell O."https://zbmath.org/authors/?q=ai:celeste.richell-o"Corcino, Roberto B."https://zbmath.org/authors/?q=ai:corcino.roberto-bagsarsaSummary: We derive an alternative matrix formula for steady state probabilities in the asymmetric exclusion process where particles hop at equal rates inside a one-dimensional finite lattice. The result is derived using the combinatorial properties of staircase tableaux and alternative tableaux.The nonlocal conjugation problem for one-dimensional parabolic equation with discontinuous coefficients and associated Feller semigroup.https://zbmath.org/1449.601242021-01-08T12:24:00+00:00"Kopytko, B. I."https://zbmath.org/authors/?q=ai:kopytko.bogdan-i"Shevchuk, R. V."https://zbmath.org/authors/?q=ai:shevchuk.r-vSummary: By the boundary integral equations method we establish the classical solvability of the conjugation problem for one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) with nonlocal Feller-Wentzell conjugation condition. Using the solution of this problem, we construct the two-parameter Feller semigroup associated with the inhomogeneous diffusion process in bounded domain with moving membrane.Stabilization of nonlinear stochastic systems without unforced dynamics via time-varying feedback.https://zbmath.org/1449.601002021-01-08T12:24:00+00:00"Florchinger, Patrick"https://zbmath.org/authors/?q=ai:florchinger.patrickSummary: In this paper we give sufficient conditions under which a nonlinear stochastic differential system without unforced dynamics is globally asymptotically stabilizable in probability via time-varying smooth feedback laws. The technique developed to design explicitly the time-varying stabilizers is based on the stochastic Lyapunov technique combined with the strategy used to construct bounded smooth stabilizing feedback laws for passive nonlinear stochastic differential systems. The interest of this work is that the class of stochastic systems considered in this paper contains a lot of systems which cannot be stabilized via time-invariant feedback laws.Analysis of stability of the equilibria for stochastic SIQR epidemic models with vaccination.https://zbmath.org/1449.341632021-01-08T12:24:00+00:00"Wang, Suxia"https://zbmath.org/authors/?q=ai:wang.suxia"Dong, Lingzhen"https://zbmath.org/authors/?q=ai:dong.lingzhen"Wang, Xiaoyan"https://zbmath.org/authors/?q=ai:wang.xiaoyanSummary: Based on the deterministic SIQR model with vaccination, we introduce the random perturbations, and establish the stochastic SIQR epidemic models with vaccination. By constructing suitable Lyapunov functions and using Itô's formula, it is proven that equilibria of the stochastic SIQR epidemic models under certain conditions are stochastically asymptotically stable. Further, we conjecture that the stability of equilibria is destroyed when the intensity of random perturbations is more larger. Finally, numerical simulations are presented to illustrate our conclusions and conjectures.Homogenization of a degenerate PDE with a nonlinear Neumann boundary condition.https://zbmath.org/1449.601062021-01-08T12:24:00+00:00"Coulibaly, A."https://zbmath.org/authors/?q=ai:coulibaly.alioune"Diedhiou, A."https://zbmath.org/authors/?q=ai:diedhiou.alassane"Sane, I."https://zbmath.org/authors/?q=ai:sane.ibrahimaSummary: We establish homogenization results of a degenerate semilinear PDE with a nonlinear Neumann boundary condition. Our approach is entirely probabilistic, and extends the result of \textit{Y. Ouknine} and \textit{É. Pardoux} [Prog. Probab. 52, 229--242 (2002; Zbl 1038.60049)].A strong deviation theorem for nonhomogeneous Markov chains indexed by a tree.https://zbmath.org/1449.600442021-01-08T12:24:00+00:00"Jin, Shaohua"https://zbmath.org/authors/?q=ai:jin.shaohua"Ma, Leiyu"https://zbmath.org/authors/?q=ai:ma.leiyu"Yu, Kaili"https://zbmath.org/authors/?q=ai:yu.kailiSummary: In this paper, by constructing a non-negative martingale on a nonhomogeneous tree, a strong deviation theorem for \(m\)-ordered nonhomogeneous Markov chains indexed by a tree is established.Intersection local times in \(L_2\) for Markov processes.https://zbmath.org/1449.601222021-01-08T12:24:00+00:00"Rudenko, Alexey"https://zbmath.org/authors/?q=ai:rudenko.alexey-vSummary: We provide sufficient conditions for the existence of intersection and self-intersection local times with additional weight in the space of square integrable random variables for Markov processes under specific local upper bounds for their transition density. We determine when this condition is satisfied for standard Brownian motion, symmetric stable processes and Brownian motions on Carnot group.The boundedness of maximal dyadic derivative operator on dyadic martingale Hardy space with variable exponents.https://zbmath.org/1449.420362021-01-08T12:24:00+00:00"Zhang, Chuanzhou"https://zbmath.org/authors/?q=ai:zhang.chuanzhou"Xia, Qi"https://zbmath.org/authors/?q=ai:xia.qi"Zhang, Xueying"https://zbmath.org/authors/?q=ai:zhang.xueyingSummary: In this paper, we research dyadic martingale Hardy spaces with variable exponents. By the characterization of log-Hölder continuity, the Doob's inequality is derived. Moreover, we prove the boundedness of the maximal dyadic derivative operator by the atomic decomposition of the variable exponent martingale space, which generalizes the conclusion in the classical case.The generalized Poisson count technique and its statistical inference.https://zbmath.org/1449.620492021-01-08T12:24:00+00:00"Wu, Qin"https://zbmath.org/authors/?q=ai:wu.qin"Liu, Yin"https://zbmath.org/authors/?q=ai:liu.yin"Ruan, Jian"https://zbmath.org/authors/?q=ai:ruan.jianSummary: Based on the generalized Poisson distribution, the generalized Poisson count technique is introduced to solve the over-dispersion and under-dispersion in the Poisson item count technique. For the statistical inference, the iterative algorithm using EM algorithm and MM algorithm is studied to calculate the maximum likelihood estimate in the model by introducing the missing data and constructing the substitution function. Furthermore, in the simulation, the bias of the estimate is presented and the simulation results are discussed to find effective information.Almost sure asymptotic expansions for profiles of simply generated random trees.https://zbmath.org/1449.600862021-01-08T12:24:00+00:00"Bogun, Vladyslav"https://zbmath.org/authors/?q=ai:bogun.vladyslavSummary: This paper is a continuation of the analysis of Edgeworth expansions for one-split branching random walk and its application to random trees. We provide new results for profile, mode and width for several simply generated random trees, in particular for random recursive trees, \(p\)-oriented recursive trees and \(D\)-ary random trees. Our results are corollaries of a general Edgeworth expansion for a one-split branching random walk proved by \textit{Z. Kabluchko} et al. [Ann. Appl. Probab. 27, No. 6, 3478--3524 (2017; Zbl 1382.60068)]. We derive an additional characterization of the random variables appearing in the coefficients of the asymptotic expansions by calculating explicitly corresponding fixed-point equations of a branching type. We further provide numerical simulations justifying our theoretical findings.Inverse problem of heat source identification based on Bayesian differential evolution algorithm.https://zbmath.org/1449.652312021-01-08T12:24:00+00:00"Yin, Weishi"https://zbmath.org/authors/?q=ai:yin.weishi"Li, Jiaqi"https://zbmath.org/authors/?q=ai:li.jiaqiSummary: Using the Bayesian differential evolution algorithm, we discussed the two-dimensional heat conduction equation. The inversion estimation of the heat source position was given through the observation temperature at different time of an observation point. The numerical experiment results show that, with the increase of the number of iterations, the error of the position parameter of the heat source decreases. When the number of iterations reaches 120, the relative error of the parameter inversion is controlled within 2\%. When 5\% and 10\% white noise are added to the observed data, the relative error changes little, which indicates that the algorithm has good stability.A note on local law of the iterated logarithm for increments of a Brownian motion.https://zbmath.org/1449.600702021-01-08T12:24:00+00:00"Mo, Yongxiang"https://zbmath.org/authors/?q=ai:mo.yongxiangSummary: Using large deviation of Brownian motion, a local law of the iterated logarithm for increments of a Brownian motion under the sup-norm is investigated.The transient queue length distribution of \(M/G/1\) queueing system with delayed vacation and Min\( (N,V)\)-policy.https://zbmath.org/1449.900762021-01-08T12:24:00+00:00"Hu, Rong"https://zbmath.org/authors/?q=ai:hu.rong"Tang, Yinghui"https://zbmath.org/authors/?q=ai:tang.yinghuiSummary: In this paper the delayed vacation is introduced in the \(M/G/1\) queueing system with Min\( (N,V)\)-policy based on multiple server vacations and the transient property of the queue length is studied, in which \(N\) is a predefined threshold that the server immediately interrupts his (her) vacation. Applying the method of the total probability decomposition technique and the Laplace transform tool, the transient queue length distribution from the beginning of any initial state is discussed. We obtain the expressions of the Laplace transformation of the transient queue length distribution.Approximation of centre manifolds for multiplicative noise driven stochastic dynamical systems.https://zbmath.org/1449.370512021-01-08T12:24:00+00:00"Li, Qin"https://zbmath.org/authors/?q=ai:li.qin"Chen, Guanggan"https://zbmath.org/authors/?q=ai:chen.guanggan"Yang, Min"https://zbmath.org/authors/?q=ai:yang.minSummary: In this paper, we study the Wong-Zakai type approximation of the centre manifold for a class of stochastic evolution equations driven by multiplicative noise. Based on the convergence of solutions on invariant manifolds, the centre manifold of a stochastic system with smooth noise is used to approximate the centre manifold of the original system, so that the dynamic behavior of the original stochastic system is more clear.Complete convergence and complete integral convergence of negatively dependent sequences under sub-linear expectations.https://zbmath.org/1449.600532021-01-08T12:24:00+00:00"Li, Jie"https://zbmath.org/authors/?q=ai:li.jie.2|li.jie|li.jie.1"Wu, Qunying"https://zbmath.org/authors/?q=ai:wu.qunyingSummary: By using the Markov inequality, under an exponential moment condition, we give the complete convergence and complete integral convergence of sequence of negatively dependent (ND) random variables with the same distribution in the sub-linear expectation space. Thus, the complete convergence and complete moment convergence in probability spaces are extended to the sub-linear expectation space, and similar results are obtained.Precise asymptotics in the law of iterated logarithm for moment convergence of ANA random variables.https://zbmath.org/1449.600602021-01-08T12:24:00+00:00"Tan, Xili"https://zbmath.org/authors/?q=ai:tan.xili"Guo, Shuang"https://zbmath.org/authors/?q=ai:guo.shuangSummary: Let \(\{{X_n}, n \ge 1\}\) be a strictly stationary sequence of ANA random variables. Using the central limit theorem and the moment inequality of ANA random sequences, the precise asymptotics in the law of iterated logarithm for the moment convergence of ANA random variables are obtained under suitable conditions.Limit theorems for one statistic of FBM in the model of real observations.https://zbmath.org/1449.600312021-01-08T12:24:00+00:00"Aiubova, N. S."https://zbmath.org/authors/?q=ai:aiubova.n-sSummary: In this article the central limit theorem as Hurst index \(H\in (0,\frac{3}{4}]\) and the non-central limit theorem as Hurst index \(H\in (\frac{3}{4},1]\) for statistics of fraction Brownian motion in the model of real observations are obtained.Valuation on Quanto options in jump-diffusion model with stochastic volatility.https://zbmath.org/1449.911632021-01-08T12:24:00+00:00"Wei, Zhu'e"https://zbmath.org/authors/?q=ai:wei.zhue"Xi, Huan"https://zbmath.org/authors/?q=ai:xi.huan"He, Jiawen"https://zbmath.org/authors/?q=ai:he.jiawenSummary: As the dynamic stock price and exchange rate satisfy the combined jump-diffusion model with stochastic volatility, by using stochastic analysis approaches including the semi-martingale Itô formula, multivariate characteristic functions, Girsanov theorem and Fourier inverse transform technique, the explicit formulas for the Quanto European call options are obtained. Numerical results show that the volatility factors have significant impacts on options values.Inequalities on generalized normalized \(\delta\)-Casorati curvatures for submanifolds in statistical manifolds of constant curvatures.https://zbmath.org/1449.530132021-01-08T12:24:00+00:00"Cai, Dandan"https://zbmath.org/authors/?q=ai:cai.dandan"Liu, Xudong"https://zbmath.org/authors/?q=ai:liu.xudong"Zhang, Liang"https://zbmath.org/authors/?q=ai:zhang.liang.1Summary: We considered submanifolds in statistical manifolds of constant curvatures by using Oprea's optimization method, and obtained some geometric inequalities involving the generalized normalized \(\delta\)-Casorati curvatures. We gave the upper bound and the lower bound of the normalized scalar curvature of the submanifolds, respectively, and the properties of submanifolds satisfying the equality cases.Some limit theorems for \(m\)-pairwise negative quadrant dependent random variables.https://zbmath.org/1449.600612021-01-08T12:24:00+00:00"Wu, Yongfeng"https://zbmath.org/authors/?q=ai:wu.yongfeng"Peng, Jiangyan"https://zbmath.org/authors/?q=ai:peng.jiangyanSummary: The authors first establish the Marcinkiewicz-Zygmund inequalities with exponent \(p\) (\(1\leq p\leq2\)) for \(m\)-pairwise negatively quadrant dependent (\(m\)-PNQD) random variables. By means of the inequalities, the authors obtain some limit theorems for arrays of rowwise \(m\)-PNQD random variables, which extend and improve the corresponding results in [\textit{Y. Meng} and \textit{Z. Lin}, Stat. Probab. Lett. 79, No. 23, 2405--2414 (2009; Zbl 1179.60012); \textit{S. H. Sung}, Appl. Math. Lett. 26, No. 1, 18--24 (2013; Zbl 1256.60016)]. It is worthy to point out that the open problem of [\textit{S. H. Sung, S. Lisawadi}, and \textit{A. Volodin}, J. Korean Math. Soc. 45, No. 1, 289--300 (2008; Zbl 1136.60319)] can be solved easily by using the obtained inequality in this paper.The Lasso estimator: distributional properties.https://zbmath.org/1449.621622021-01-08T12:24:00+00:00"Jagannath, Rakshith"https://zbmath.org/authors/?q=ai:jagannath.rakshith"Upadhye, Neelesh S."https://zbmath.org/authors/?q=ai:upadhye.neelesh-sSummary: The least absolute shrinkage and selection operator (LASSO) is a popular technique for simultaneous estimation and model selection. There have been a lot of studies on the large sample asymptotic distributional properties of the LASSO estimator, but it is also well-known that the asymptotic results can give a wrong picture of the LASSO estimator's actual finite-sample behaviour. The finite sample distribution of the LASSO estimator has been previously studied for the special case of orthogonal models. The aim in this work is to generalize the finite sample distribution properties of LASSO estimator for a real and linear measurement model in Gaussian noise.
In this work, we derive an expression for the finite sample characteristic function of the LASSO estimator, we then use the Fourier slice theorem to obtain an approximate expression for the marginal probability density functions of the one-dimensional components of a linear transformation of the LASSO estimator.Gaussian approximation for functionals of Gibbs particle processes.https://zbmath.org/1449.600122021-01-08T12:24:00+00:00"Flimmel, Daniela"https://zbmath.org/authors/?q=ai:flimmel.daniela"Beneš, Viktor"https://zbmath.org/authors/?q=ai:benes.viktorSummary: In the paper asymptotic properties of functionals of stationary Gibbs particle processes are derived. Two known techniques from the point process theory in the Euclidean space \(\mathbb{R}^d\) are extended to the space of compact sets on \(\mathbb{R}^d\) equipped with the Hausdorff metric. First, conditions for the existence of the stationary Gibbs point process with given conditional intensity have been simplified recently. Secondly, the Malliavin-Stein method was applied to the estimation of Wasserstein distance between the Gibbs input and standard Gaussian distribution. We transform these theories to the space of compact sets and use them to derive a Gaussian approximation for functionals of a planar Gibbs segment process.On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy.https://zbmath.org/1449.911072021-01-08T12:24:00+00:00"Peng, Xuanhua"https://zbmath.org/authors/?q=ai:peng.xuanhua"Su, Wen"https://zbmath.org/authors/?q=ai:su.wen"Zhang, Zhimin"https://zbmath.org/authors/?q=ai:zhang.zhimin.1Summary: In this paper, we model the insurance company's surplus flow by a perturbed compound Poisson model. Suppose that at a sequence of random time points, the insurance company observes the surplus to decide dividend payments. If the observed surplus level is larger than the maximum of a threshold \(b>0\) and the last observed level (after dividends payment if possible), then a fraction \(0<\theta<1\) of the excess amount is paid out as a lump sum dividend. We assume that the solvency is also discretely monitored at these observation times, so that the surplus process stops when the observed value becomes negative. Integro-differential equations for the expected discounted dividend payments before ruin and the Gerber-Shiu expected discounted penalty function are derived, and solutions are also analyzed by Laplace transform method. Numerical examples are given to illustrate the applicability of our results.Complete convergence for arrays of rowwise WOD random variables and its application.https://zbmath.org/1449.600622021-01-08T12:24:00+00:00"Ye, Qingyuan"https://zbmath.org/authors/?q=ai:ye.qingyuan"Wang, Lili"https://zbmath.org/authors/?q=ai:wang.lili.1|wang.lili.5|wang.lili|wang.lili.3|wang.lili.7|wang.lili.2|wang.lili.4|wang.lili.6|wang.lili.8"Chen, Kan"https://zbmath.org/authors/?q=ai:chen.kan"Li, Xingchen"https://zbmath.org/authors/?q=ai:li.xingchen"Wang, Xuejun"https://zbmath.org/authors/?q=ai:wang.xuejun|wang.xuejun.1Summary: We establish the convergence rate for arrays of rowwise WOD random variables under some general conditions. The result is applied to the nonparametric regression model and the complete consistency for the weighted estimator is provided, which improves and extends some corresponding ones in the literature. The main contribution of the research is greatly relaxing the limit for the dominating coefficients.Complete convergence of maximal partial sum with \(\tilde\varphi\)-mixing random variable series.https://zbmath.org/1449.600502021-01-08T12:24:00+00:00"Huang, Min"https://zbmath.org/authors/?q=ai:huang.min.1|huang.minSummary: We investigate the complete convergence of maximal partial sum with \(\tilde\varphi\)-mixing random variable series. As an application, we obtain the convergence rates in Marcinkiewicz-Zygmund-type strong law of large numbers for \(\tilde\varphi\)-mixing random variable series. The results include Baum-Katz-type theorem and Hsu-Bobbins-type Theorem as special cases, and generalize the results for the partial sum of Stocia to the case of maximal partial sum.Moments of the extended Erlang (2) model on multi-point processes at ruin \(T\)-time.https://zbmath.org/1449.910382021-01-08T12:24:00+00:00"Xue, Ying"https://zbmath.org/authors/?q=ai:xue.ying"Niu, Yaoming"https://zbmath.org/authors/?q=ai:niu.yaomingSummary: The classical risk model is transformed and generalized. It is assumed that a ``jump'' in the model corresponds to multiple claims, and the time interval of claims obeys Erlang (2) distribution. The expression of the moment of ruin \(T\) is given.A class of the stochastic predator-prey model with delay and Lévy jump.https://zbmath.org/1449.342942021-01-08T12:24:00+00:00"Shi, Lili"https://zbmath.org/authors/?q=ai:shi.lili"Liu, Guirong"https://zbmath.org/authors/?q=ai:liu.gui-rong|liu.gui-rong.1Summary: This research focuses on a class of stochastic predator-prey models with delay and Lévy jump. Firstly, the Lyapunov method and Itô formula are used to give the existence and uniqueness of the global positive solution of this model. Then according to Chebyshev's inequality, exponential martingale inequality and Borel-Cantelli lemma, etc., the stochastic ultimate boundedness and extinction are obtained. Finally, the theoretical results are illustrated by numerical simulations.A numerical solution of the pricing model of Asian options under sub-fractional jump-diffusion process.https://zbmath.org/1449.651832021-01-08T12:24:00+00:00"Hu, Pan"https://zbmath.org/authors/?q=ai:hu.panSummary: Under the assumption of the sub-fractional Ho-Lee stochastic interest rate model, this research firstly uses the delta hedging principle and establishes the partial differential equation of geometric average Asian options under the sub-fractional jump-diffusion process with transaction costs and dividends. Secondly, the pricing model is simplified to the Cauchy problem by using the variable substitution. Finally, a numerical solution of the pricing model is given by using the finite difference method and the composite trapezoid method. An example is also given to verify the effectiveness of the algorithm design.Necessary condition for near-optimal control of a stochastic SIRS epidemic model.https://zbmath.org/1449.490252021-01-08T12:24:00+00:00"Mu, Xiaojie"https://zbmath.org/authors/?q=ai:mu.xiaojie"Zhang, Qimin"https://zbmath.org/authors/?q=ai:zhang.qimin"Wang, Zong"https://zbmath.org/authors/?q=ai:wang.zongSummary: The stochastic SIRS model with imprecise parameters and white noise is established. We obtain priori estimates of the susceptible, infected and recovered populations. Necessary condition for the near optimality of the SIRS model is established with Ekeland's principle and a nearly maximum condition. A numerical example is provided for verifying the theoretical results.Stochastic differential equations driven by multi-fractional Brownian motion and Poisson point process.https://zbmath.org/1449.601072021-01-08T12:24:00+00:00"Liu, Hailing"https://zbmath.org/authors/?q=ai:liu.hailing"Xu, Liping"https://zbmath.org/authors/?q=ai:xu.liping"Li, Zhi"https://zbmath.org/authors/?q=ai:li.zhiSummary: In this paper, we study a class of stochastic differential equations with additive noise that contains a non-stationary multi-fractional Brownian motion (mBm) with a Hurst parameter as a function of time and a Poisson point process of class (QL). The differential equation of this kind is motivated by the reserve processes in a general insurance model, in which there is the long term dependence between the claim payment and the past history of liability. By using the variable order fractional calculus on the fractional Wiener-Poisson space and a multi-fractional derivative operator, and employing Girsanov theorem for multi-fractional Brownian motion, we prove the existence of weak solutions to the SDEs under consideration. As a consequence, we deduce the uniqueness in law and the pathwise uniqueness.Dynamics of a nonlinear stochastic viscoelastic equation with multiplicative noise.https://zbmath.org/1449.350462021-01-08T12:24:00+00:00"Caraballo, Tomas"https://zbmath.org/authors/?q=ai:caraballo.tomas"Pina, Nicolas"https://zbmath.org/authors/?q=ai:pina.nicolas"Munoz, Jaime"https://zbmath.org/authors/?q=ai:munoz.jaimeSummary: The well-posedness and stability properties of a stochastic viscoelastic equation with multiplicative noise, Lipschitz and locally Lipschitz nonlinear terms are investigated. The method of Lyapunov functions is used to investigate the asymptotic dynamics when zero is not a solution of the equation by using an appropriate cocycle and random dynamical system. The stability of mild solutions is proved in both cases of Lipschitz and locally Lipschitz nonlinear terms. Furthermore, we investigate the existence of a non-trivial stationary solution which is exponentially stable, by using a general random fixed point theorem for general cocycles. In this case, the stationary solution is generated by the composition of random variable and Wiener shift. In addition, the theory of random dynamical system is used to construct another cocycle and prove the existence of a random fixed point exponentially attracting every path.Fast calibration of the libor market model with stochastic volatility and displaced diffusion.https://zbmath.org/1449.601252021-01-08T12:24:00+00:00"Devineau, Laurent"https://zbmath.org/authors/?q=ai:devineau.laurent"Arrouy, Pierre-Edouard"https://zbmath.org/authors/?q=ai:arrouy.pierre-edouard"Bonnefoy, Paul"https://zbmath.org/authors/?q=ai:bonnefoy.paul"Boumezoued, Alexandre"https://zbmath.org/authors/?q=ai:boumezoued.alexandreSummary: This paper demonstrates the efficiency of using Edgeworth and Gram-Charlier expansions in the calibration of the libor market model with stochastic volatility and displaced diffusion (DD-SV-LMM). Our approach brings together two research areas; first, the results regarding the SV-LMM since the work of \textit{L. Wu} and \textit{F. Zhang} [ibid. 2, No. 2, 199--227 (2006; Zbl 1135.91363)], especially on the moment generating function, and second the approximation of density distributions based on Edgeworth or Gram-Charlier expansions. By exploring the analytical tractability of moments up to fourth order, we are able to perform an adjustment of the reference Bachelier model with normal volatilities for skewness and kurtosis, and as a by-product to derive a smile formula relating the volatility to the moneyness with interpretable parameters. As a main conclusion, our numerical results show a 98\% reduction in computational time for the DD-SV-LMM calibration process compared to the classical numerical integration method developed by \textit{S. L. Heston} [Rev. Financ. Stud. 6, No. 2, 327--343 (1993; Zbl 1384.35131)].A review on stochastic multi-symplectic methods for stochastic Maxwell equations.https://zbmath.org/1449.601112021-01-08T12:24:00+00:00"Zhang, Liying"https://zbmath.org/authors/?q=ai:zhang.liying"Chen, Chuchu"https://zbmath.org/authors/?q=ai:chen.chuchu"Hong, Jialin"https://zbmath.org/authors/?q=ai:hong.jialin"Ji, Lihai"https://zbmath.org/authors/?q=ai:ji.lihaiSummary: Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by \textit{J. Hong} et al. [J. Comput. Phys. 268, 255--268 (2014; Zbl 1349.65536)]. Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.Large deviations estimation for some random variable sequences.https://zbmath.org/1449.600432021-01-08T12:24:00+00:00"Feng, Decheng"https://zbmath.org/authors/?q=ai:feng.decheng"Zheng, Rui"https://zbmath.org/authors/?q=ai:zheng.rui"Xie, Jingfang"https://zbmath.org/authors/?q=ai:xie.jingfangSummary: Let \(\{{X_n}, n \ge 1\}\) be a sequence of random variables defined on a probability space \( (\Omega, \mathcal{F}, P)\), \(\{{S_n}, n\geq 1\}\) be the sequence of partial sums of \(\{{X_n}, n \ge 1\}\). In this paper, some large deviation estimations on the partial sums sequence are obtained for martingale difference sequences, \(\varphi\)-mixing sequences, \(p\)-order M-Z type random variable sequences and NOD sequences under the condition of \(\sum\limits_{i=1}^n E|{X_i}|^p = O (n)\).Properties of sublinear \(g\)-expectations and their applications.https://zbmath.org/1449.601012021-01-08T12:24:00+00:00"Ji, Ronglin"https://zbmath.org/authors/?q=ai:ji.ronglin"Zhou, Jinming"https://zbmath.org/authors/?q=ai:zhou.jinmingSummary: Under the basic assumptions on generators of backward stochastic differential equations, the one-to-one correspondence between subadditivity (resp., homogeneity) of \(g\)-expectations and generators of backward stochastic differential equations is obtained. Thus it is proved that the definitions of dynamic coherent risk measures in previous work are completely the same under the framework of \(g\)-expectations. Furthermore, a relationship between time-consistent dynamic coherent risk measures via \(g\)-expectations and sublinearity of \(g\)-expectations is established.Complete integration convergence for array of rowwise END random variables under sub-linear expectations.https://zbmath.org/1449.600592021-01-08T12:24:00+00:00"Tang, Rongxiu"https://zbmath.org/authors/?q=ai:tang.rongxiu"Wu, Qunying"https://zbmath.org/authors/?q=ai:wu.qunyingSummary: We study the complete integration convergence for a sequence of random variables under the sub-linear expectations. By using the Rosenthal's inequality, we obtain the complete integration convergence for the array of rowwise END (extended negatively dependent) random variables for the sub-linear expectation space under the general moment condition.The setting and optimization of quick queue with customer loss.https://zbmath.org/1449.601312021-01-08T12:24:00+00:00"Li, Kai"https://zbmath.org/authors/?q=ai:li.kai"Pan, Yuqian"https://zbmath.org/authors/?q=ai:pan.yuqian"Liu, Bohai"https://zbmath.org/authors/?q=ai:liu.bohai"Cheng, Bayi"https://zbmath.org/authors/?q=ai:cheng.bayiSummary: At the peak of a service system, customers may hesitate and even leave in the face of unavoidable queuing. This phenomenon not only affects the customer's satisfaction, but also causes the loss of the company's revenue. This paper establishes a fluid model of customer queuing behavior for the customer losses. The goal is to reduce the customer losses, and the setting and optimization method of quick queue in random service systems is studied. We construct two queuing models, in which one includes only regular queues and the other includes both regular and quick queues. We analyze the queuing systems, and describe the different forms of the objective function based on the fluid model of customer behavior. Then we compare and analyze the impact of the adoption of quick queues on the performance of the service system during peak period, and design a calculation method to obtain the optimal value for setting the number of quick queues. Thus, the overall performance of the system is optimized. Finally, we take the setting and optimization of quick queue in the supermarket service system as an example, which verifies the validity of the proposed method, and shows the reference value of this method to management practice.Power Lindley-logarithmic distribution and parameter estimation.https://zbmath.org/1449.622252021-01-08T12:24:00+00:00"Wang, Zeqi"https://zbmath.org/authors/?q=ai:wang.zeqi"Liu, Luqin"https://zbmath.org/authors/?q=ai:liu.luqinSummary: This paper proposes a new lifetime distribution named Power Lindley-Logarithmic distribution (PLL) by compounding the Power Lindley distribution and the Logarithmic distribution. In the paper, its moment, quantile, hazard rate function, limiting distribution of the order statistics and the MLE (maximum likelihood estimation) of the parameters are discussed, the consistency and asymptotic normality of the MLE are verified, and EM (expectation-maximization) algorithm is used to get the MLE. In the end, the paper carries out Monte Carlo simulation which indicates that the MLE obtained by EM algorithm perfectly reflects the true value of the parameters, and the parameters' MLE of the PPL distribution has good asymptotic normality.Complete convergence of weighted sums for END random variables sequence under sub-linear expectation.https://zbmath.org/1449.600542021-01-08T12:24:00+00:00"Ma, Xiaochen"https://zbmath.org/authors/?q=ai:ma.xiaochen"Wu, Qunying"https://zbmath.org/authors/?q=ai:wu.qunyingSummary: We study the complete convergence of weighted sums for END random variables sequence under sub-linear expectation space. With the condition that the Choquet integral of the order \(2 + r/\alpha \) moment of the random variable is finite, we prove the complete convergence of weighted sums for END random variables sequence extending from the probability space to the sub-linear expectation space.Asymptotic behavior of stochastic predator-prey model with epidemic in the predator.https://zbmath.org/1449.341742021-01-08T12:24:00+00:00"Zhang, Qiumei"https://zbmath.org/authors/?q=ai:zhang.qiumei"Wu, Jiajie"https://zbmath.org/authors/?q=ai:wu.jiajieSummary: In this paper, a stochastic predator-prey model with epidemic in the predator under the influence of white noise is formulated. The existence and uniqueness of the positive solution of the model are discussed, the asymptotic behavior of this model is studied.Additive and multiplicative noise excitability of stochastic partial differential equations.https://zbmath.org/1449.354652021-01-08T12:24:00+00:00"Guo, Zhongkai"https://zbmath.org/authors/?q=ai:guo.zhongkai"Cheng, Shuilin"https://zbmath.org/authors/?q=ai:cheng.shuilin"Wang, Weifeng"https://zbmath.org/authors/?q=ai:wang.weifengSummary: In this paper, noise excitability of energy solution to stochastic partial differential equation with additive and multiplicative cases is considered. By using Itô's formula and energy estimate method, we obtain excitation indices of \(u\) at different cases with different results. Thus, from this point of view, the effects of additive noise and multiplicative noise on the system are different.Effect of information on the strategic behavior of customers in a discrete-time bulk service queue.https://zbmath.org/1449.601342021-01-08T12:24:00+00:00"Panda, Gopinath"https://zbmath.org/authors/?q=ai:panda.gopinath"Goswami, Veena"https://zbmath.org/authors/?q=ai:goswami.veenaSummary: We consider the equilibrium and socially optimal behavior of strategic customers in a discrete-time queue with bulk service. The service batch size varies from a single customer to a maximum of `b' customers. We study the equilibrium and socially optimal balking strategies under two information policies: observable and unobservable. In the former policy, a service provider discloses the queue length information to arriving customers and conceals it in the latter policy. The effect of service batch size and other queueing parameters on the equilibrium strategies under both information policies are compared and illustrated with numerical experiments.Closure property of random sum and its maximum of random variables from class \(\mathcal{D}\) based on precise large deviation principles.https://zbmath.org/1449.600872021-01-08T12:24:00+00:00"Guo, Duo"https://zbmath.org/authors/?q=ai:guo.duo"Hang, Min"https://zbmath.org/authors/?q=ai:hang.min"Wang, Shijie"https://zbmath.org/authors/?q=ai:wang.shijieSummary: Let \(X = \{{X_1}, {X_2}, \cdots \}\) be a sequence of independent but not necessarily identically distributed random variables, and let \(\eta \) be an integer-valued counting random variable independent of \(X\). The random sum \({S_\eta} = \sum\limits_{k = 1}^\eta {X_k}\) and its maximum \({S_{ (\eta)}} = {\mathrm{max}}\{ {{S_0}, \cdots, {S_\eta}} \}\) are studied. Assuming that each \({X_k}\) belongs to the class of \(\mathcal{D}\), by using the result of the precise large deviation principles on class \(\mathcal{D}\), it is proven that the distributions of \({{S_\eta}}\) and \({S_{(\eta)}}\) belong to the same class under some conditions. The obtained results expand the related ones of previous studies.Stabilized IMLS based element free Galerkin method for stochastic elliptic partial differential equations.https://zbmath.org/1449.653162021-01-08T12:24:00+00:00"Izadpanah, Komeil"https://zbmath.org/authors/?q=ai:izadpanah.komeil"Mesforush, Ali"https://zbmath.org/authors/?q=ai:mesforush.ali"Nazemi, Ali"https://zbmath.org/authors/?q=ai:nazemi.ali-rezaSummary: In this paper, we propose a numerical method to solve the elliptic stochastic partial differential equations (SPDEs) obtained by Gaussian noises using an element free Galerkin method based on stabilized interpolating moving least square shape functions. The error estimates of the method is presented. The method is tested via several problems. The numerical results show the usefulness and accuracy of the new method.Optimal investment and reinsurance under the influence of threshold dividend.https://zbmath.org/1449.911162021-01-08T12:24:00+00:00"Zhang, Xuefang"https://zbmath.org/authors/?q=ai:zhang.xuefang"Jin, Yansheng"https://zbmath.org/authors/?q=ai:jin.yanshengSummary: We study the optimal investment and optimal reinsurance problem to maximize the expected dividend of an insurance company in jump diffusion risk model. Under the expected premium principle and the threshold dividend policy, the HJB equation satisfying the model is obtained by using stochastic optimal control theory and diffusion approximation theory, and then explicit solutions of optimal policy, value function and dividend policy are obtained. Finally, we carry out numerical simulation and analyze the influence of some parameters on the optimal policy.On the moments of order statistics from the standard two-sided power distribution.https://zbmath.org/1449.621082021-01-08T12:24:00+00:00"Akhter, Zuber"https://zbmath.org/authors/?q=ai:akhter.zuber"Mirmostafaee, S. M. T. K."https://zbmath.org/authors/?q=ai:mirmostafaee.s-m-t-k"Athar, Haseeb"https://zbmath.org/authors/?q=ai:athar.haseebSummary: In this paper, we obtain new explicit expressions for the single and product moments of order statistics from the standard two-sided power (STSP) distribution. These expressions can be used to compute the means, variances and the covariances of order statistics from the STSP distribution. We also have a glance at the application of the results to the lifetimes of the coherent systems. Two real data examples are given to illustrate the flexibility of the STSP distribution.Option pricing method and parameter calibration for jump-diffusion model.https://zbmath.org/1449.911652021-01-08T12:24:00+00:00"Xu, Congcong"https://zbmath.org/authors/?q=ai:xu.congcong"Xu, Zuoliang"https://zbmath.org/authors/?q=ai:xu.zuoliangSummary: In this paper, the pricing method and parameter calibration of jump-diffusion model are investigated. First, the risk-neutral characteristic function of jump-diffusion model is derived under the mean correction equivalent martingale measure. The option under jump-diffusion model is priced by using the COS pricing method. Then, the pricing error of the COS algorithm is analyzed and the effectiveness of the COS pricing method is verified through numerical experiment. Subsequently, the parameters of the jump-diffusion model are calibrated by the relative entropy regularization method. Numerical experiments demonstrate the accuracy and reliability of the proposed method. Finally, the calibration method is tested by analyzing the S\&P500 market data. The results show that the values of calibrated parameter are qualitatively for each maturity. Moreover, the results indicate a better fitting to the market data for the Merton jump-diffusion model in comparison to the Black-Scholes model.On the Sin-G class of distributions: theory, model and application.https://zbmath.org/1449.600222021-01-08T12:24:00+00:00"Souza, Luciano"https://zbmath.org/authors/?q=ai:souza.luciano"Rosa de O. Júnior, Wilson"https://zbmath.org/authors/?q=ai:rosa-de-o.wilson-jun"de Brito, Cicero Carlos R."https://zbmath.org/authors/?q=ai:de-brito.cicero-carlos-r"Chesneau, Christophe"https://zbmath.org/authors/?q=ai:chesneau.christophe"Ferreira, Tiago A. E."https://zbmath.org/authors/?q=ai:ferreira.tiago-a-e"Soares, Lucas G. M."https://zbmath.org/authors/?q=ai:soares.lucas-g-mSummary: This paper is devoted to the study of the Sin-G class of distributions and one of its special member. We first explore the mathematical properties of the Sin-G class, giving the cumulative and probability density functions and their expansions, quantile function, moments, moment generating function, reliability parameter, Rényi entropy and order statistics. Then, we focus our attention on the special member defined with the inverse Weibull distribution as baseline, denoted by SinIW. The mathematical and practical aspects of the SinIW distribution are investigated. In order to illustrate the usefulness of the SinIW model, an application to a real life data set is carried out.Transmuted Dagum distribution with applications.https://zbmath.org/1449.600162021-01-08T12:24:00+00:00"Elbatal, Ibrahim"https://zbmath.org/authors/?q=ai:elbatal.ibrahim"Aryal, Gokarna"https://zbmath.org/authors/?q=ai:aryal.gokarna-rajSummary: In this article, we will use the quadratic rank transmutation map (QRTM) in order to generate a flexible family of probability distributions in which a three parameter Dagum distribution is embedded. Various structural properties of the new distribution including explicit expressions for the moments, random number generation and order statistics are derived. Estimation by maximum likelihood and inference for large samples are addressed. It will be shown that the analytical results are applicable to model the real world data.Hermite Ornstein-Uhlenbeck processes mixed with a gamma distribution.https://zbmath.org/1449.600332021-01-08T12:24:00+00:00"Douissi, Soukaina"https://zbmath.org/authors/?q=ai:douissi.soukaina"Es-Sebaiy, Khalifa"https://zbmath.org/authors/?q=ai:es-sebaiy.khalifa"Tudor, Ciprian A."https://zbmath.org/authors/?q=ai:tudor.ciprian-aAfter some preliminaries concerning the basic properties of multiple Wiener-Itô integrals and Hermite processes, the authors construct the gamma-mixed Hermite Ornstein-Uhlenbeck process as the limit of the aggregated sum of the solution to the Langevin equation with Hermite noise and random coefficients and analyze the properties of the solution to the Langevin equation. In the second part, the authors consider the asymptotic behavior of the aggregated sum with respect to its parameter.
Reviewer: Oleg K. Zakusilo (Kyïv)Letter to the editor on the paper ``Stochastic representations and a geometric parametrization of the two-dimensional Gaussian law''.https://zbmath.org/1449.600142021-01-08T12:24:00+00:00"Rau, Christian"https://zbmath.org/authors/?q=ai:rau.christianSummary: With regard to the above named paper [\textit{T. Dietrich} et al., ibid. 4, No. 2, 27--59 (2013; Zbl 1449.60011)] we clarify an issue regarding coupling, and make some further remarks on correlation, and cone measure.
A reply to the letter of Christian Rau is also given by Wolf-Dieter Richter.A coding theorem for nonhomogeneous Markov source.https://zbmath.org/1449.940542021-01-08T12:24:00+00:00"Zhou, Dan"https://zbmath.org/authors/?q=ai:zhou.dan"Wang, Zhongzhi"https://zbmath.org/authors/?q=ai:wang.zhongzhiSummary: The aim of this paper is to extend the memoryless discrete source coding theorem to the case of nonhomogeneous Markov chain which leads to a greater application range. Firstly, we establish a strong law of large numbers of a delayed nonhomogeneous Markov chain by the classical Borel-Cantelli lemma. Then, we apply an independent random source to approximate the nonhomogeneous Markov source in order to obtain the general coding theorem of the nonhomogeneous Markov source. Finally, with the help of the general coding theorem of nonhomogeneous Markov source, we propose a numerical method of the lowest fault tolerance rate in batch data hypothesis test.Performance analysis and optimization for cognitive radio networks with a finite primary user buffer and a probability returning scheme.https://zbmath.org/1449.680052021-01-08T12:24:00+00:00"Zhao, Yuan"https://zbmath.org/authors/?q=ai:zhao.yuan"Yue, Wuyi"https://zbmath.org/authors/?q=ai:yue.wuyiSummary: In this paper, in order to reduce possible packet loss of the primary users (PUs) in cognitive radio networks, we assume there is a buffer with a finite capacity for the PU packets. At the same time, focusing on the packet interruptions of the secondary users (SUs), we introduce a probability returning scheme for the interrupted SU packets. In order to evaluate the influence of the finite buffer setting and the probability returning scheme to the system performance, we construct and analyze a discrete-time Markov chain model. Accordingly, we determine the expressions of some important performance measures of the PU packets and the SU packets. Then, we show numerical results to evaluate how the buffer setting of the PU packets and the returning probability influence the system performance. Moreover, we optimize the system access actions of the SU packets. We determine their individually and the socially optimal strategies by considering different buffer settings for PU packets and different returning probabilities for SU packets. Finally, a pricing policy by introducing an admission fee is also provided to coincide the two optimal strategies.The gamma modified Weibull distribution.https://zbmath.org/1449.620272021-01-08T12:24:00+00:00"Cordeiro, Gauss M."https://zbmath.org/authors/?q=ai:cordeiro.gauss-moutinho"Aristizábal, William D."https://zbmath.org/authors/?q=ai:aristizabal.william-d"Suárez, Dora M."https://zbmath.org/authors/?q=ai:suarez.dora-m"Lozano, Sébastien"https://zbmath.org/authors/?q=ai:lozano.sebastienSummary: The Weibull distribution has been highlighted in recent decades for its wide use in important applied areas [\textit{D. N. Prabhakar Murthy} et al., Weibull models. Chichester: Wiley (2004; Zbl 1047.62095)]. The modified Weibull was proposed as a more flexible alternative for modeling data [\textit{C. D. Lai}, \textit{M. Xie} and \textit{D. N. Prabhakar Murthy}, ``A modified Weibull distribution'', IEEE Trans. Reliab. 52, No. 1, 33--37 (2003; \url{doi:10.1109/TR.2002.805788})]. \textit{K. Zografos} and \textit{N. Balakrishnan} [Stat. Methodol. 6, No. 4, 344--362 (2009; Zbl 05898144)] pioniered a new class of distributions called the gamma-G with the advantage of having only one parameter to transform an arbitrary distribution. This simple fact allows us to explore a large number of skewed and non-skewed behaviors. In this paper, we present the main properties of the gamma modified Weibull distribution. We provide the moments, quantile function and other important measures. In addition, an application to a real data set demonstrates the usefulness of the new model.The (functional) law of the iterated logarithm of the sojourn time for a multiclass queue.https://zbmath.org/1449.601292021-01-08T12:24:00+00:00"Guo, Yongjiang"https://zbmath.org/authors/?q=ai:guo.yongjiang"Song, Yuantao"https://zbmath.org/authors/?q=ai:song.yuantaoSummary: Two types of the law of iterated logarithm (LIL) and one functional LIL (FLIL) are established for the sojourn time process for a multiclass queueing model, having a priority service discipline, one server and \(K\) customer classes, with each class characterized by a batch renewal arrival process and independent and identically distributed (i.i.d.) service times. The LIL and FLIL limits quantify the magnitude of asymptotic stochastic fluctuations of the sojourn time process compensated by its deterministic fluid limits in two forms: the numerical and functional. The LIL and FLIL limits are established in three cases: underloaded, critically loaded and overloaded, defined by the traffic intensity. We prove the results by a approach based on strong approximation, which approximates discrete performance processes with reflected Brownian motions. We conduct numerical examples to provide insights on these LIL results.A wrapped flexible generalized skew-normal model for a bimodal circular distribution of wind directions.https://zbmath.org/1449.622582021-01-08T12:24:00+00:00"Hernández-Sánchez, Estefanía"https://zbmath.org/authors/?q=ai:hernandez-sanchez.estefania"Scarpa, Bruno"https://zbmath.org/authors/?q=ai:scarpa.brunoSummary: Motivated by the analysis of wind directions, in this paper we consider skew-symmetric circular distributions generated by perturbation of a symmetric circular distribution. This class of models is able to describe different distribution shapes, including symmetric, skewed, and bimodal, which are often observed in circular data, as in our motivating example of wind directions at a site in Spain. We propose a wrapped version of the flexible generalized skew-normal distribution to fit these data. The model is presented and the parameters of the proposed model are estimated by the maximum likelihood method. We also note that in the considered area, typical sea breeze and land breeze directions are observed. Thus, we consider a subdivision of the data to fit these directions separately. The likelihood ratio test shows that the proposed model outperforms the wrapped skew-normal, and the Akaike information criterion reveals that it fits our wind-direction data with comparable performances with respect to other well-known bimodal circular distributions.BSDENMs: enlargement of filtration and insider trading.https://zbmath.org/1449.600952021-01-08T12:24:00+00:00"Akdim, K."https://zbmath.org/authors/?q=ai:akdim.khadijaSummary: The aim of this paper is to study backward stochastic differential equations driven by general normal martingales \((M_t)_{t\in[0,T]}\) with deterministic bracket \(\langle M,M\rangle_t\) (BSDENM), having the chaos representation property. We prove the existence and uniqueness of the pertaining solution under the stochastic Lipschitz condition. In this work, the backward stochastic differential equations,
(BSDENM) are wealth equations, and we use Jacod's method of enlargement of filtration to model the asymmetrical information. We also compare the strategies of an insider trader and a non insider one.Optimal proportional reinsurance and pairs trading polices for insurer.https://zbmath.org/1449.910992021-01-08T12:24:00+00:00"Huang, Boqiang"https://zbmath.org/authors/?q=ai:huang.boqiang"Li, Qicai"https://zbmath.org/authors/?q=ai:li.qicaiSummary: This paper discusses optimization problem in which the insurer transfers the claims risk by proportional reinsurance and manages the wealth by pairs trading. The surplus of claims is modeled by compound Poisson process. The insurer can invest it's wealth into pairs portfolio which includes a long position on one stock and a short on another stock. The price spread of this pair follows a mean-reverting stochastic process. Under maximizing of expect exponent utility of the terminal wealth, the optimal proportional reinsurance and pairs trading polices and value function are solved by stochastic control theory.Two-factor Markov-modulated stochastic volatility models for option pricing.https://zbmath.org/1449.911542021-01-08T12:24:00+00:00"Liu, Xueru"https://zbmath.org/authors/?q=ai:liu.xueru"Li, Meihong"https://zbmath.org/authors/?q=ai:li.meihong"Tian, Fan"https://zbmath.org/authors/?q=ai:tian.fan"Liu, Guoxiang"https://zbmath.org/authors/?q=ai:liu.guoxiangSummary: We consider the option pricing problem when the risky underlying assets are driven by a two-factor Markov-modulated stochastic volatility model, with the first volatility factor driven by the Cox-Ingersoll-Ross process and the second volatility factor driven by a continuous-time hidden Markov process. The states of the Markov process can be interpreted as the unobservable states of the economy. The market described by a two-factor Markov-modulated stochastic volatility model is incomplete in general and, hence, the martingale measure is not unique. We adopt the regime switching Esscher transform to determine an equivalent martingale pricing measure. We consider the valuation of the European and American options. A system of coupled partial differential integral equations satisfied by the European option prices is derived. We also derive a decomposition result for an American put option into its European counterpart and early exercise premium. Finally, numerical illustrations are given.Convergence analysis of an elitist non-homogeneous genetic algorithm with crossover/mutation probabilities adjusted by a fuzzy controller.https://zbmath.org/1449.601212021-01-08T12:24:00+00:00"Pereira, André"https://zbmath.org/authors/?q=ai:pereira.andre-g-c"Campos, Viviane"https://zbmath.org/authors/?q=ai:campos.viviane-s-m"Roveda, José"https://zbmath.org/authors/?q=ai:roveda.jose-arnaldo-f"Santana, Fágner"https://zbmath.org/authors/?q=ai:santana.fagner-l"de Medeiros, Francisco"https://zbmath.org/authors/?q=ai:de-medeiros.franciscoSummary: In recent years, several attempts to improve the efficiency of the canonical genetic algorithm have been presented. The advantage of the elitist non-homogeneous genetic algorithm is that, variations of the mutation probabilities permit the algorithm to broaden its search space at the start and restrict it later on, however the way in which the mutation probabilities vary is defined before the algorithm is initiated. To solve this problem various types of controllers can be used to adjust such changes. This work presents an elitist non-homogeneous genetic algorithm where the mutation probability is adjusted by a fuzzy controller. Many simulation studies have used fuzzy controllers to adjust the parameters in order to improve the performance of the genetic algorithm. However, no previous investigation has discussed the conditions that must be met by the controller in order to ensure convergence of the genetic algorithm. A generalized example will be used to illustrate how sufficient conditions for the algorithm convergence can be readily achieved. And finally, numerical simulations are used to compare the proposed algorithm with the canonical genetic algorithm.Asymptotic properties of the conditional hazard function and its maximum estimation under right-censoring and left-truncation.https://zbmath.org/1449.622082021-01-08T12:24:00+00:00"Agbokou, Komi"https://zbmath.org/authors/?q=ai:agbokou.komi"Gneyou, Kossi Essona"https://zbmath.org/authors/?q=ai:gneyou.kossi-essonaSummary: The second author [``A central limit theorem for a nonparametric maximum conditional
hazard rate in presence of right censoring'', Int. J. Stat. Probab. 2, No. 3, 110--124 (2013; \url{doi:10.5539/ijsp.v2n3p110}); J. Multivariate Anal. 128, 10--18 (2014; Zbl 1352.62066)] considered the estimation of the maximum hazard rate under random censorship with covariate random and established strong representation and strong uniform consistency with rate of the estimate. Then he studied the asymptotic normality of his estimator. The first author et al. [Far East J. Theor. Stat. 54, No. 2, 141--173 (2018; Zbl 1433.62279)] generalize this work to the case of right censored and left truncated data with covariate and established strong representation and strong uniform consistency with rate of the estimate of the said estimator and of a non-parametric estimator of its maximum value. The aim of this paper is to study the asymptotic normality result of the two nonparametric estimators.Precise asymptotics for partial sums of \(\rho\)-mixing sequences.https://zbmath.org/1449.600482021-01-08T12:24:00+00:00"Fu, Rui"https://zbmath.org/authors/?q=ai:fu.rui"Fei, Dandan"https://zbmath.org/authors/?q=ai:fei.dandan"Fu, Zongkui"https://zbmath.org/authors/?q=ai:fu.zongkui"Wang, Gaixia"https://zbmath.org/authors/?q=ai:wang.gaixiaSummary: Let \(\{{X_n}; n \geq 1\}\) be a sequence of strictly stationary \(\rho\)-mixing random variables with zero mean and finite variance. Using the weak convergence theorem and probability inequalities of \(\rho \)-mixing sequences, under some proper conditions, we obtained general laws of precise asymptotics for partial sums of \(\rho\)-mixing sequences.Almost sure stability for uncertain differential equation.https://zbmath.org/1449.601182021-01-08T12:24:00+00:00"Liu, Hongjian"https://zbmath.org/authors/?q=ai:liu.hongjian"Ke, Hua"https://zbmath.org/authors/?q=ai:ke.hua"Fei, Weiyin"https://zbmath.org/authors/?q=ai:fei.weiyinSummary: Uncertain differential equation is a type of differential equation driven by a Liu process. So far, concepts of stability and stability in mean for uncertain differential equations have been proposed. This paper aims at providing a concept of almost sure stability for uncertain differential equation. A sufficient condition is given for an uncertain differential equation being almost surely stable, and some examples are given to illustrate the effectiveness of the sufficient condition.A zero-truncated Poisson-Aradhana distribution with applications.https://zbmath.org/1449.620332021-01-08T12:24:00+00:00"Shanker, Rama"https://zbmath.org/authors/?q=ai:shanker.rama"Shukla, Kamlesh Kumar"https://zbmath.org/authors/?q=ai:shukla.kamlesh-kumarSummary: In this paper, a zero-truncation of Poisson-Aradhana distribution proposed by the first author [``The discrete Poisson-Aradhana distribution'', Turkiye Klinikleri J. Biostat. 9, No. 1, 12--22 (2017; \url{doi:10.5336/biostatic.2017-54834})] named `zero-truncated Poisson-Aradhana distribution' has been introduced and investigated. A general expression for the \(r\)th factorial moment about origin has been obtained and thus the first four moments about origin and the central moments have been given. Also, the expressions for coefficient of variation, skewness, kurtosis, and the index of dispersion of the distribution have been presented and their natures have been discussed graphically. The method of moments and the method of maximum likelihood estimation have also been discussed for estimating its parameter. Two examples of observed real datasets have been given to test the goodness of fit of of the proposed distribution over zero-truncated Poisson-Sujatha distribution, zero-truncated Poisson-Lindley distribution and zero-truncated Poisson distribution.Convergence for sums of asymptotically almost negatively associated random variables.https://zbmath.org/1449.600552021-01-08T12:24:00+00:00"Meng, Bing"https://zbmath.org/authors/?q=ai:meng.bing"Wang, Dingcheng"https://zbmath.org/authors/?q=ai:wang.dingcheng"Wu, Qunying"https://zbmath.org/authors/?q=ai:wu.qunyingSummary: In this paper, by applying the moment inequality for asymptotically almost negatively associated (AANA, in short) random sequence and truncated method, the equivalent conditions of complete moment convergence of the maximum partial for weighted sums of AANA random variables are obtained without assumptions of identical distributionwhich generalize and improve the corresponding ones of literatures.Properties and applications of alpha power gamma distribution.https://zbmath.org/1449.600182021-01-08T12:24:00+00:00"Niu, Yulin"https://zbmath.org/authors/?q=ai:niu.yulin"Yan, Zaizai"https://zbmath.org/authors/?q=ai:yan.zaizaiSummary: In this paper, the gamma distribution has been extended by adding an extra shape parameter; we refer to the new distribution as alpha power gamma distribution. It is found that the distribution has a relatively flexible hazard rate function. The properties of the new distribution are studied, including explicit expressions for the \(s^{\mathrm{th}}\) raw moments, moment generating function and distributions of order statistics are derived. Also, integral expressions for the entropy, mean residual life and mean waiting time are obtained. Maximum likelihood estimators of the distribution parameters under complete sample are discussed, the Fisher information matrix is derived. Then, the estimation of the parameters under the general progressive type-II censoring is studied. Finally, a real data set is used to illustrate the practicality of the proposed distribution.Complete convergence of WOD random variable sequences.https://zbmath.org/1449.600642021-01-08T12:24:00+00:00"Zhang, Qian"https://zbmath.org/authors/?q=ai:zhang.qian"Cai, Guanghui"https://zbmath.org/authors/?q=ai:cai.guanghui"Zheng, Yuyan"https://zbmath.org/authors/?q=ai:zheng.yuyanSummary: In this paper, we present a complete convergence result for widely orthant dependent (WOD) random variables by using the maximum Rosenthal's type moment inequality and the truncation methods (into three parts). The results generalize and improve the related known works in the literature.A finite horizon linear quadratic optimal stochastic control problem driven by both Brownian motion and Lévy processes.https://zbmath.org/1449.490182021-01-08T12:24:00+00:00"Hu, Shipei"https://zbmath.org/authors/?q=ai:hu.shipei"He, Zhimin"https://zbmath.org/authors/?q=ai:he.zhiminSummary: We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and Lévy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.Stochastic fractional non-autonomous Ginzburg-Landau equations with multiplicative noise in weighted space.https://zbmath.org/1449.354662021-01-08T12:24:00+00:00"Wang, Yunxiao"https://zbmath.org/authors/?q=ai:wang.yunxiao"Shu, Ji"https://zbmath.org/authors/?q=ai:shu.ji"Yang, Yuan"https://zbmath.org/authors/?q=ai:yang.yuan"Li, Qian"https://zbmath.org/authors/?q=ai:li.qian"Wang, Chunjiang"https://zbmath.org/authors/?q=ai:wang.chunjiangSummary: In this paper, we consider the asymptotic dynamic for random attractors of stochastic fractional non-autonomous Ginzburg-Landau equations with multiplicative noise in \(L_\rho^2 (\text bf{R}^n)\). Firstly, we transform the partial differential equation into the random equation that only induces the random parameters. Then, using estimates for far-field values of solutions and a cut-off technique, asymptotic compactness is proved. At last, the existence of a random attractor in \(L_\rho^2(\text bf{R}^n)\) for the random dynamical system is established.Comparison principle and stability analysis of a class of stochastic parabolic equations with delay and Markovian switching.https://zbmath.org/1449.350552021-01-08T12:24:00+00:00"Li, Zhao"https://zbmath.org/authors/?q=ai:li.zhao"Li, Shuyong"https://zbmath.org/authors/?q=ai:li.shuyongSummary: This paper investigates the mean square stability for stochastic parabolic equations with delay and Markovian switching. By establishing the comparison principle, using delay differential inequality and stochastic analysis techniques, the mean square stability, mean square uniform stability, mean square asymptotic stability and mean square exponential stability for the system are obtained. Finally, an example is given to illustrate the main theoretical result.Complete convergence for Sung's type weighted sums of dependent random variables with general moment conditions.https://zbmath.org/1449.600572021-01-08T12:24:00+00:00"Qiu, Dehua"https://zbmath.org/authors/?q=ai:qiu.dehua"Yi, Yanchun"https://zbmath.org/authors/?q=ai:yi.yanchun"Chen, Pingyan"https://zbmath.org/authors/?q=ai:chen.pingyanSummary: In this paper, the complete convergence theorems for Sung's type weighted sums of END random variables and PNQD random variables with general moment conditions are obtained. The theorems extend the related known works in the literature.A balance sheet optimal multi-modes switching problem.https://zbmath.org/1449.600822021-01-08T12:24:00+00:00"Eddahbi, M'hamed"https://zbmath.org/authors/?q=ai:eddahbi.mhamed"Fakhouri, Imade"https://zbmath.org/authors/?q=ai:fakhouri.imade"Ouknine, Youssef"https://zbmath.org/authors/?q=ai:ouknine.youssefSummary: We study a finite horizon balance sheet optimal multi-modes switching problem related to trade-off strategies between expected profit and cost cash flows. The problem is formulated in terms of Snell envelopes for the profit and the cost yields which act as obstacles to each other, moreover we fully characterize the optimal strategies. Then using the link between the Snell envelope of processes and reflected backward stochastic differential equations (RBSDEs for short), solving the problem turns out actually to solving the related system of RBSDEs, for which we prove the existence of a continuous minimal solution using an approximation scheme.Appointment scheduling with customer impatience based on operating cost model.https://zbmath.org/1449.901182021-01-08T12:24:00+00:00"Song, Minshan"https://zbmath.org/authors/?q=ai:song.minshan"Zhang, Yulin"https://zbmath.org/authors/?q=ai:zhang.yulin.1Summary: An appointment scheduling problem is studied with the consideration of customer impatience. On the assumption that both the time of leaving queue and the time of service are exponentially distributed, in order to minimize the joint cost, the optimal appointment schedule of the fixed number of customers is studied. The joint cost function is composed of customers' expected delay time and service availability time. The expected delay time of each customer in the queue is recursively computed in terms of customer interarrival time. Furthermore, the effect of impatience on the optimal schedule as well as the total operating cost is studied. The results show that as the impatience rate increases, the optimal interarrival time becomes shorter and the interarrival time of the last few customers gradually approaches that of the customers in the middle. In addition, impatient behaviors can increase the joint cost.A series expansion method for solving the boundary value problem connected with the Helmholtz equation.https://zbmath.org/1449.653052021-01-08T12:24:00+00:00"Du, Xinwei"https://zbmath.org/authors/?q=ai:du.xinweiSummary: A boundary value problem connected with the Helmholtz equation is studied in a smooth bounded domain. A series expansion method is proposed for obtaining an approximate solution to the problem. Tikhonov regularization is applied to the problem with noisy data. Numerical experiments are presented to show the effectiveness of the proposed method.Asymptotic properties of a stochastic mutualism model with a saturation term and Lévy jumps.https://zbmath.org/1449.341782021-01-08T12:24:00+00:00"Zhao, Xiaodan"https://zbmath.org/authors/?q=ai:zhao.xiaodan"Zhao, Aimin"https://zbmath.org/authors/?q=ai:zhao.aimin"Liu, Guirong"https://zbmath.org/authors/?q=ai:liu.gui-rong.1Summary: This paper investigates a stochastic mutualism model with a saturation term and Lévy jumps. It chooses a suitable Lyapunov function to demonstrate the existence and uniqueness of global positive solutions. Using Itô formula, sufficient conditions for the extinction of each species are established. The results in this paper extend results of the existing literature. Finally, some numerical simulations are given to illustrate the theoretical results.Anticipated time-dependent backward stochastic evolution equations.https://zbmath.org/1449.601102021-01-08T12:24:00+00:00"Zhan, Desheng"https://zbmath.org/authors/?q=ai:zhan.desheng"Chu, Jing"https://zbmath.org/authors/?q=ai:chu.jingSummary: A class of anticipated time-dependent backward stochastic evolution equation in a Hilbert space was discussed. The existence and uniqueness of the evolution solution were proved. As an application, the evolution solution for a class of anticipated backward stochastic partial differential equations was derived. Some well-known results were generalized and extended.The unit-Birnbaum-Saunders distribution with applications.https://zbmath.org/1449.620312021-01-08T12:24:00+00:00"Mazucheli, Josmar"https://zbmath.org/authors/?q=ai:mazucheli.josmar"Menezes, André F. B."https://zbmath.org/authors/?q=ai:menezes.andre-f-b"Dey, Sanku"https://zbmath.org/authors/?q=ai:dey.sankuSummary: In this paper a new probability distribution supported on unit interval is introduced. This distribution arises from the Birnbaum and Saunders distribution. It depends on two parameters and can be considered as an alternative to the classical Beta distribution and Kumaraswamy distribution. It presents the advantage of not including any additional parameter(s) or special function in its formulation and it has closed-form for the moments. The new transformed model, called the unit-Birnbaum-Saunders (UBS) distribution exhibits decreasing, upside down bathtub and then bathtub shaped density while the hazard rate function can be increaseing or bathtub shaped. The method of maximum likelihood and moments are used to estimate the model parameters. Monte Carlo simulation is carried out to examine the bias and root mean squared error of the maximum likelihood and moment estimators of the parameters. Finally, the potentiality of the model is studied using two real data sets.A goodness of fit test for the Pareto distribution.https://zbmath.org/1449.621112021-01-08T12:24:00+00:00"Suárez-Espinosa, Javier"https://zbmath.org/authors/?q=ai:suarez-espinosa.javier"Villaseñor-Alva, José A."https://zbmath.org/authors/?q=ai:villasenor-alva.jose-a"Hurtado-Jaramillo, Annel"https://zbmath.org/authors/?q=ai:hurtado-jaramillo.annel"Pérez-Rodríguez, Paulino"https://zbmath.org/authors/?q=ai:perez-rodriguez.paulinoSummary: The Pareto distribution has been studied by many authors for modeling phenomena such as: income distribution, flood levels of rivers, droughts, major insurance claims and financial issues among others. We propose a goodness of fit test for the type II Pareto distribution; the test does not require the estimation of the parameters and is based on the mean residual life function. The test statistic is the Kendall's correlation coefficient between the empirical mean residual life function and the sample order statistics, whose distribution has been obtained through simulation. A simulation study shows that the power of the test is very good when data come from light-tailed distributions and is reasonably good when data come from heavy-tailed distributions. The test will be specially useful for practitioners from the extreme events framework.The multiple vacations queueing system with impatient customers and working breakdowns.https://zbmath.org/1449.900782021-01-08T12:24:00+00:00"Ma, Zhanyou"https://zbmath.org/authors/?q=ai:ma.zhanyou"Cao, Jian"https://zbmath.org/authors/?q=ai:cao.jian"Yu, Xiangran"https://zbmath.org/authors/?q=ai:yu.xiangran"Guo, Shanshan"https://zbmath.org/authors/?q=ai:guo.shanshan"Chen, Li"https://zbmath.org/authors/?q=ai:chen.li.6|chen.li.4|chen.li.2|chen.li.7|chen.li.3|chen.li.5|chen.li.1Summary: In order to expand the random queueing theory, on the basis of the \(M/M/1\) queueing model with multiple vacations, the impatient customers strategy and working breakdowns strategy are introduced. Therefore, a new queueing system is constructed. The balance equations of the queueing system in steady state are constructed. The probability generating functions of the number of customers in the system when the server is in different states are derived by using the generating function approach. Furthermore, the explicit expressions of the performance measures are derived in steady state, such as the average queue length and the rate of abandonment of the customers etc. The relation between the parameters and performance measures is examined by numerical analysis. According to the game theory, the equilibrium strategy of the customer and the social optimal strategy are analyzed in detail by constructing the utility functions. The multiple vacations queueing system with impatient customers and working breakdowns is analyzed. Service providers and customers can make the risk prediction and evaluation decision according to the results in the realistic queue.The optimal thresholds of pairs trading with a stop-loss condition.https://zbmath.org/1449.911422021-01-08T12:24:00+00:00"Bi, Xiuchun"https://zbmath.org/authors/?q=ai:bi.xiuchun"Liu, Bo"https://zbmath.org/authors/?q=ai:liu.bo.3|liu.bo.4|liu.bo.1"Yuan, Lvning"https://zbmath.org/authors/?q=ai:yuan.lvning"Zhang, Shuguang"https://zbmath.org/authors/?q=ai:zhang.shuguangSummary: As a market-neutral trading strategy, pairs trading has been used in various investment practices. However, taking into account the volatility and uncertainty of the stock market, there may still be a great risk of loss in the pairs trading. At present, there is still less research on the optimal threshold problem with a stop-loss condition. This article assumes that stock prices are subject to geometric Brownian motions. We add a stop-loss curve to the original two threshold curves of buying and selling. By maximizing the reward function, the optimal threshold problem is transformed into a stochastic control problem, and the corresponding HJB equation is solved to obtain the optimal thresholds. Subsequently, this paper selects two stocks of the Bank of Beijing and the Bank of Huaxia from a shares to verify the optimal thresholds. The calculated annualized rate of return was 14.55\%, and the maximum retracement was 1.99\% lower than that before the stop loss. It was verified that the optimal thresholds after adding the stop-loss curve were effective.Discrimination method of mutual independence and non-correlation of two-dimensional random variables.https://zbmath.org/1449.600032021-01-08T12:24:00+00:00"Ma, Shuanghong"https://zbmath.org/authors/?q=ai:ma.shuanghong"Chen, Jinshu"https://zbmath.org/authors/?q=ai:chen.jinshuSummary: By analyzing the essence of independence and non-correlation of random variables, a new and simpler discrimination method is proposed. At the same time, an actual example is given to verify the validity and feasibility of this method. Further, the difference and relation of mutual independence and non-correlation are studied and it is proved that the attribute of their equivalence is not peculiar to the two-dimensional normal distribution.Infinite phase type representation for queue length distribution of a vacation queue.https://zbmath.org/1449.601382021-01-08T12:24:00+00:00"Zhang, Hongbo"https://zbmath.org/authors/?q=ai:zhang.hongbo|zhang.hongbo.1"Wang, Zheng"https://zbmath.org/authors/?q=ai:wang.zhengSummary: This paper considers an \(\mathrm{M}/\mathrm{M}/1\) vacation queue with T-SPH vacation time, where T-SPH denotes the discrete-time phase type distribution defined on a birth and death process with countable many states. The queue model can be described by a quasi-birth-and-death (QBD) process with infinite phases. Firstly, concrete forms of rate operator and operator-geometric solution of the QBD process are given. Secondly, the stochastic decomposition structure of the stationary queue length is presented. Finally, it is proved that the additional queue length follows discrete-time infinite phase type distribution.Coupon collector's problem with unlike probabilities.https://zbmath.org/1449.600092021-01-08T12:24:00+00:00"Nakata, Toshio"https://zbmath.org/authors/?q=ai:nakata.toshioSummary: In this note, we study the coupon collector's problem with unlike probabilities using majorization and a Schur concave function.Study on the influence of time to ruin and deficit at ruin on insurance company based on Erlang risk model.https://zbmath.org/1449.911122021-01-08T12:24:00+00:00"Yang, Liping"https://zbmath.org/authors/?q=ai:yang.liping"Zhou, Wenxin"https://zbmath.org/authors/?q=ai:zhou.wen-xinSummary: Although insurance companies play an important role in market economy, survival of the fittest is the inevitable result of free economy. In order to understand the real situation of insurance company survival, we study the Sparre Andersen risk model with the individual claim amount obeying Erlang (3) distribution and the claim time interval obeying Erlang (2) distribution when the initial surplus value is greater than zero. Applying the expected discounted penalty function, the join density of time to ruin and the deficit at ruin is obtained by using Lagrange implicit function theorem and Laplace transform. Finally, according to the actual data of Chinese listed insurance companies, an empirical study on the ruin probability of Chinese insurance companies is carried out, and the model is proved to be effective.The odd Nadarajah-Haghighi family of distributions: properties and applications.https://zbmath.org/1449.600172021-01-08T12:24:00+00:00"Nascimento, Abraão D. C."https://zbmath.org/authors/?q=ai:nascimento.abraao-d-c"Silva, Kássio F."https://zbmath.org/authors/?q=ai:silva.kassio-f"Cordeiro, Gauss M."https://zbmath.org/authors/?q=ai:cordeiro.gauss-moutinho"Alizadeh, Morad"https://zbmath.org/authors/?q=ai:alizadeh.morad"Yousof, Haitham M."https://zbmath.org/authors/?q=ai:yousof.haitham-m"Hamedani, G. G."https://zbmath.org/authors/?q=ai:hamedani.gholamhoss-g|hamedani.gholamhossein-gSummary: We study some mathematical properties of a new generator of continuous distributions called the odd Nadarajah-Haghighi (ONH) family. In particular, three special models in this family are investigated, namely the ONH gamma, beta and Weibull distributions. The family density function is given as a linear combination of exponentiated densities. Further, we propose a bivariate extension and various characterization results of the new family. We determine the maximum likelihood estimates of ONH parameters for complete and censored data. We provide a simulation study to verify the precision of these estimates. We illustrate the performance of the new family by means of a real data set.Dynamics of a stochastic predator-prey model with pulse input in a polluted environment.https://zbmath.org/1449.341312021-01-08T12:24:00+00:00"Fu, Yingjie"https://zbmath.org/authors/?q=ai:fu.yingjie"Lan, Guijie"https://zbmath.org/authors/?q=ai:lan.guijie"Zhang, Shuwen"https://zbmath.org/authors/?q=ai:zhang.shuwen"Wei, Chunjin"https://zbmath.org/authors/?q=ai:wei.chunjinSummary: In this paper, we show a stochastic predator-prey model with pulse input in a polluted environment. The existence and uniqueness of the positive global solution and the boundedness of expectation of the system are all proved. Sufficient conditions for the existence and boundedness of periodical solution are obtained, and it is globally attractive with probability 1. The threshold of population extinction and persistence in the mean are obtained too. Finally, some numerical simulations are carried out to illustrate the main results.Almost sure central limit theorems for weighted dependent sequences.https://zbmath.org/1449.600342021-01-08T12:24:00+00:00"Gonchigdanzan, Khurelbaatar"https://zbmath.org/authors/?q=ai:gonchigdanzan.khurelbaatarSummary: Let \(\{X_n : n\geq 1\}\) be a sequence of dependent random variables and let \(\{w_{nk} : 1\leq k \leq n, n\geq 1\}\) be a triangular array of real numbers. We prove the almost sure version of the CLT proved by \textit{M. Peligrad} and \textit{S. Utev} [Ann. Probab. 34, No. 4, 1608--1622 (2006; Zbl 1101.60014)] for weighted partial sums of mixing and associated sequences of random variables, i.e. \[\lim_{n\to\infty}\frac{1}{\log n}\sum_{k=1}^n\frac{1}{k} \;{\text{I}} \left(\sum_{i=1}^kw_{ki}X_i\leq x\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^xe^{-\frac{1}{2}t^2} dt \;\; {\text{a.s.}} \;\; .\]Performance evaluation and analysis of a discrete queue system with multiple working vacations and non-preemptive priority.https://zbmath.org/1449.601332021-01-08T12:24:00+00:00"Ma, Zhanyou"https://zbmath.org/authors/?q=ai:ma.zhanyou"Wang, Wenbo"https://zbmath.org/authors/?q=ai:wang.wenbo"Hu, Linmin"https://zbmath.org/authors/?q=ai:hu.linminSummary: In this paper, we introduce a discrete time \(\mathrm{Geo}/\mathrm{Geo}/1\) queue system with non-preemptive priority and multiple working vacations. We assume that there are two types of customers in this queue system named ``customers of type-I'' and ``customers of type-II''. customer of type-II has a higher priority with non-preemption than customer of type-I. By building a discrete time four-dimensional Markov chain which includes the numbers of customers with different priorities in the system, the state of the server and the service state, we obtain the state transition probability matrix. Using a birth-and-death chain and matrix-geometric method, we deduce the average queue length, the average waiting time of the two types of customers, and the average busy period of the system. Then, we provide some numerical results to evaluate the effect of the parameters on the system performance. Finally, we develop some benefit functions to analyse both the personal and social benefit, and obtain some optimization results within a certain range.On the collision local time of sub-bifractional Brownian motions.https://zbmath.org/1449.600792021-01-08T12:24:00+00:00"Kuang, Nenghui"https://zbmath.org/authors/?q=ai:kuang.nenghuiSummary: Let \(S^{{H_i}, {K_i}} = \{S_t^{{H_i}, {K_i}}, t \ge 0\}, i = 1, 2\) be two independent sub-bifractional Brownian motions of dimension 1, with indices \({H_i} \in (0, 1)\) and \({K_i} \in (0, 1]\). We consider the collision local time \[{\ell_T} = \int_0^T \delta (S_t^{{H_1}, {K_1}}-S_t^{{H_2}, {K_2}}){\mathrm{d}}t,\; 0 < T < \infty,\] where \(\delta\) denotes the Dirac delta function. We show that \({\ell_T}\) exists in \({L^2}\), and furthermore, it is also smooth in the sense of Meyer-Watanabe if \(\min\{{H_1}{K_1}, {H_2}{K_2}\} < \frac{1}{3}\).Moderate deviations of LS estimators in the linear EV regression model with martingale difference errors.https://zbmath.org/1449.600322021-01-08T12:24:00+00:00"Bai, Yanghua"https://zbmath.org/authors/?q=ai:bai.yanghua"Chen, Xia"https://zbmath.org/authors/?q=ai:chen.xia.1"Yan, Li"https://zbmath.org/authors/?q=ai:yan.liSummary: In this paper, we study the linear errors in variables (EV) model: \({\eta_i} = \theta + \beta {x_i} + {\varepsilon_i},\; {\xi_i} = {x_i} + {\delta_i} (1 \le i \le n)\). When the errors \( ({\varepsilon_i}, {\delta_i})\) are martingale difference sequences, moderate deviations of the least square estimators for the unknown parameters \(\beta\) and \(\theta\) are established.The odd exponentiated half-logistic-G family: properties, characterizations and applications.https://zbmath.org/1449.620232021-01-08T12:24:00+00:00"Afify, Ahmed Z."https://zbmath.org/authors/?q=ai:afify.ahmed-z"Altun, Emrah"https://zbmath.org/authors/?q=ai:altun.emrah"Alizadeh, Morad"https://zbmath.org/authors/?q=ai:alizadeh.morad"Ozel, Gamze"https://zbmath.org/authors/?q=ai:ozel.gamze"Hamedani, G. G."https://zbmath.org/authors/?q=ai:hamedani.gholamhossein-gSummary: We introduce a new class of continuous distributions called the odd exponentiated half-logistic-G family. Some special models of the new family are provided. These special models are capable of modeling various shapes of aging and failure criteria. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, order statistics, probability weighted moments and characterizations are obtained. The maximum likelihood method is used for estimating the model parameters. The flexibility of the generated family is illustrated by means of three applications to real data sets.A new extension of power Lindley distribution for analyzing bimodal data.https://zbmath.org/1449.620252021-01-08T12:24:00+00:00"Alizadeh, Morad"https://zbmath.org/authors/?q=ai:alizadeh.morad"MirMostafaee, S. M. T. K."https://zbmath.org/authors/?q=ai:mirmostafaee.s-m-t-k"Ghosh, Indranil"https://zbmath.org/authors/?q=ai:ghosh.indranilSummary: In this article, we introduce a new three-parameter odd log-logistic power Lindley distribution and discuss some of its properties. These include the shapes of the density and hazard rate functions, mixture representation, the moments, the quantile function, and order statistics. Maximum likelihood estimation of the parameters and their estimated asymptotic standard errors are derived. Three algorithms are proposed for generating random data from the proposed distribution. A simulation study is carried out to examine the bias and mean square error of the maximum likelihood estimators of the parameters. An application of the model to two real data sets is presented finally and compared with the fit attained by some other well-known two and three-parameter distributions for illustrative purposes. It is observed that the proposed model has some advantages in analyzing lifetime data as compared to other popular models in the sense that it exhibits varying shapes and shows more flexibility than many currently available distributions.Multistage multivariate nested distance: an empirical analysis.https://zbmath.org/1449.902722021-01-08T12:24:00+00:00"Vitali, Sebastiano"https://zbmath.org/authors/?q=ai:vitali.sebastianoSummary: Multistage stochastic optimization requires the definition and the generation of a discrete stochastic tree that represents the evolution of the uncertain parameters in time and space. The dimension of the tree is the result of a trade-off between the adaptability to the original probability distribution and the computational tractability. Moreover, the discrete approximation of a continuous random variable is not unique. The concept of the best discrete approximation has been widely explored and many enhancements to adjust and fix a stochastic tree in order to represent as well as possible the real distribution have been proposed. Yet, often, the same generation algorithm can produce multiple trees to represent the random variable. Therefore, the recent literature investigates the concept of distance between trees which are candidate to be adopted as stochastic framework for the multistage model optimization. The contribution of this paper is to compute the nested distance between a large set of multistage and multivariate trees and, for a sample of basic financial problems, to empirically show the positive relation between the tree distance and the distance of the corresponding optimal solutions, and between the tree distance and the optimal objective values. Moreover, we compute a lower bound for the Lipschitz constant that bounds the optimal value distance.Local times of linear multifractional stable sheets.https://zbmath.org/1449.600902021-01-08T12:24:00+00:00"Shen, Guang-jun"https://zbmath.org/authors/?q=ai:shen.guangjun"Yu, Qian"https://zbmath.org/authors/?q=ai:yu.qian"Li, Yun-meng"https://zbmath.org/authors/?q=ai:li.yunmengSummary: Let \(X^{H(u)}(u)=\{X^{H(u)}(u), u \in \mathbb{R}_+^N\}\) be linear multifractional stable sheets with index functional \(H(u)\), where \(H(u) = (H_1(u),\dots, H_N(u))\) is a function with values in \((0, 1)^N \). Based on some assumptions of \(H(u)\), we obtain the existence of the local times of \(X^{H(u)} (u)\) and establish its joint continuity and the Hölder regularity. These results generalize the corresponding results about fractional stable sheets to multifractional stable sheets.Equilibrium analysis of queueing system with two types of parallel customers and partial failures.https://zbmath.org/1449.601392021-01-08T12:24:00+00:00"Zhang, Songtai"https://zbmath.org/authors/?q=ai:zhang.songtai"Xu, Xiuli"https://zbmath.org/authors/?q=ai:xu.xiuliSummary: In this paper, we consider the equilibrium behavior of customers in a queueing system with two types of parallel customers, partial failures and delayed repairs, where two types of customers are independent and follow different negative exponential distribution, respectively, and the system server is not completely reliable. When a partial failure occurs, the server continues to serve the customers on spot at a low rate and does not admit a new arrival. Assuming that two types of customers both have the option of joining or balking in order to maximize their own benefits, we analyze the equilibrium balking strategies of the two types of customers and the average social benefits of the system in the fully observable case and the almost observable case, respectively. On this basis, the effects of information level on customers' strategic behavior are presented by some numerical examples.One non-sibling mating only bisexual branching processes.https://zbmath.org/1449.601262021-01-08T12:24:00+00:00"Xing, Yongsheng"https://zbmath.org/authors/?q=ai:xing.yongshengSummary: One class of bisexual branching is introduced which allows only non-sibling mating. Under some assumptions on mating function, the necessary condition for almost sure extinction is obtained.Representation of the number operator in the continuous-time Guichardet-Fock space.https://zbmath.org/1449.601132021-01-08T12:24:00+00:00"Zhou, Yulan"https://zbmath.org/authors/?q=ai:zhou.yulan"Li, Xiaohui"https://zbmath.org/authors/?q=ai:li.xiaohui"Cheng, Xiuqiang"https://zbmath.org/authors/?q=ai:cheng.xiuqiang"Xue, Rui"https://zbmath.org/authors/?q=ai:xue.ruiSummary: The paper considers the representation of the number operator \(N\) in the continuous-time Guichardet-Fock space \({L^2} (\Gamma;\eta)\). Firstly, the gradient-Skorohod integral representation of \(N\) is given by using modified stochastic gradient \({\tilde \nabla}\) and non-adaptive Skorohod integral \(\delta: N = \delta \circ \tilde \nabla\). Secondly, the representation of Bochner integral is given: \(N = \int_{\mathbb{R}_+}\tilde \nabla_s^*{\tilde \nabla_s}{\mathrm{d}}s\) in the sense of the inner product, by means of the family of isometric operator \(\{{\tilde \nabla_s}^*{\tilde \nabla_s}; s \in {\mathbb{R}_+}\}\). Meanwhile, the spectrum of \(N\) is just the nonnegative integral \(N\), and for any \(n \ge 0\), the closed subspace \({L^2} (\Gamma^{ (n)};\eta)\) of Guichardet-Fock space \({L^2} (\Gamma;\eta)\) is just the eigenspace corresponding to the eigenvalue \(n\), and \(N\) has the spectrum representation: \(N = \sum\limits_{n=1}^\infty n{J_n}\), where \({J_n}:{L^2} (\Gamma;\eta) \to {L^2} (\Gamma^{ (n)};\eta)\) is the orthogonal projection.On the largest part size and its multiplicity of a random integer partition.https://zbmath.org/1449.050242021-01-08T12:24:00+00:00"Mutafchiev, Ljuben"https://zbmath.org/authors/?q=ai:mutafchiev.lyuben-rSummary: Let \(\lambda\) be a partition of the positive integer \(n\) chosen uniformly at random among all such partitions. Let \(L_n = L_n(\lambda)\) and \(M_n = M_n(\lambda)\) be the largest part size and its multiplicity, respectively. For large \(n\), we focus on a comparison between the partition statistics \(L_n\) and \(L_n M_n\). In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of \(L_n M_n - L_n\) grows as fast as \(\frac{1}{2}\log n\). We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.On the laws of large numbers in possibilistic theory.https://zbmath.org/1449.600492021-01-08T12:24:00+00:00"Gal, Sorin G."https://zbmath.org/authors/?q=ai:gal.sorin-gheorgheSummary: In this paper we obtain some possibilistic variants of the probabilistic laws of large numbers, different from those obtained by other authors, but very natural extensions of the corresponding ones in probability theory. Our results are based on the possibility measure and on the ``maxitive'' definitions for possibility expectation and possibility variance. In the frame of this paper, we have only strong law of large numbers, because the weak form of the law of large numbers, will always imply the strong law of large numbers.On the existence of some skew-normal stationary processes.https://zbmath.org/1449.622052021-01-08T12:24:00+00:00"Minozzo, Marco"https://zbmath.org/authors/?q=ai:minozzo.marco"Ferracuti, Laura"https://zbmath.org/authors/?q=ai:ferracuti.lauraSummary: Recently, some authors have introduced in the literature stationary stochastic processes in the time and spatial domains, whose finite-dimensional marginal distributions are multivariate skew-normal. In this paper, we show with a counter-example that the characterizations of these processes are not valid and so these processes do not exist. In particular, we exhibit through a marginalization argument that the set of finite-dimensional marginal distributions of these processes is not self-coherent. In addition, we point our attention to some valid constructions of stationary stochastic processes, which can be used to model skewed data.The third-order random tensor Gaussian distribution.https://zbmath.org/1449.600232021-01-08T12:24:00+00:00"He, Lingling"https://zbmath.org/authors/?q=ai:he.lingling"Lin, Zerong"https://zbmath.org/authors/?q=ai:lin.zerong"Zhang, Ziming"https://zbmath.org/authors/?q=ai:zhang.ziming"Xu, Changqing"https://zbmath.org/authors/?q=ai:xu.changqingSummary: In this paper, we study the density function and the characteristic function of the random vector and random matrix Gaussian distribution. We firstly define the characteristic function of the random tensor, its Gaussian distribution and the density function of the standard random tensor. Then, we calculate the characteristic function of the standard random tensor Gaussian distribution through the inner product of the random tensor. Finally, we obtain the density function and the characteristic function of the third-order random tensor by tensor product and matrixing.Large deviation principle for the prototype model of the wind-driven ocean circulation.https://zbmath.org/1449.600452021-01-08T12:24:00+00:00"Pu, Xueke"https://zbmath.org/authors/?q=ai:pu.xueke"Lu, Yuting"https://zbmath.org/authors/?q=ai:lu.yutingSummary: Recently, the stochastic quasi-geostrophic equation has attracted the attention of many scholars' and it has been intensively studied. One reason is due to that it is similar with another more well-known model, the stochastic Navier-Stokes equation. The quasi-geostrophic equation is important in the geostrophic fluid dynamics. In this research, we discuss the prototype model of the wind-driven ocean circulation with multiplicative noise. First, we verify the existence of strong solutions and the uniqueness of mild solutions in dimension two and three. Then, we prove a large deviation principle based on the Laplace principle and the weak convergence methods.Convergence of the linear fractional self-repelling diffusion.https://zbmath.org/1449.600802021-01-08T12:24:00+00:00"Li, Hongwei"https://zbmath.org/authors/?q=ai:li.hongwei"Ge, Yong"https://zbmath.org/authors/?q=ai:ge.yong"Yan, Litan"https://zbmath.org/authors/?q=ai:yan.litanSummary: In this paper, we investigate the asymptotic behavior of the linear self-repelling diffusion driven by fractional Brownian motion. \(X_t^H = B_t^H + a\int_0^t \int_0^s (X_s^H - X_u^H){\mathrm{d}}u{\mathrm{d}}s + vt\), \(X_0^H = 0\) where \({B^H}\) is a fractional Brownian motion with Hurst index \(H > 1/2\), \(a > 0\), \(v \in \mathbb{R}\) are two constants. The authors prove that \(te^{-1/2at^2}X_t^H\) converges to a normal random variable as \(t\, (t\to +\infty)\) almost surely and in \({L^2} (\Omega)\).Apollonius ``circle'' in spherical geometry.https://zbmath.org/1449.510232021-01-08T12:24:00+00:00"Ionaşcu, Eugen J."https://zbmath.org/authors/?q=ai:ionascu.eugen-julienSummary: We investigate the analog of the circle of Apollonious in spherical geometry. This can be viewed as the pre-image through the stereographic projection of an algebraic curve of degree three. This curve consists of two connected components each being the ``reflection'' of the other through the center of the sphere. We give an equivalent equation for it, which is surprisingly, this time, of degree four.The change point test of Poisson process based on U-type statistics.https://zbmath.org/1449.620982021-01-08T12:24:00+00:00"Fan, Zimiao"https://zbmath.org/authors/?q=ai:fan.zimiao"Tian, Mengqin"https://zbmath.org/authors/?q=ai:tian.mengqinSummary: The change point of Poisson process is studied. With the assumption of only one change point at most, the model of Poisson process parameter change point is considered. By using U-type statistics, we construct the test statistics and show decision criteria to determine the existence of change point. In the case of change point, the estimation of the change point position is given.Bottom-up modeling of domestic appliances with Markov chains and semi-Markov processes.https://zbmath.org/1449.601202021-01-08T12:24:00+00:00"Drenyovszki, Rajmund"https://zbmath.org/authors/?q=ai:drenyovszki.rajmund"Kovács, Lóránt"https://zbmath.org/authors/?q=ai:kovacs.lorant"Tornai, Kálmán"https://zbmath.org/authors/?q=ai:tornai.kalman"Oláh, András"https://zbmath.org/authors/?q=ai:olah.andras"Pintér, István"https://zbmath.org/authors/?q=ai:pinter.istvanSummary: In our paper we investigate the applicability of independent and identically distributed random sequences, first order Markov and higher order Markov chains as well as semi-Markov processes for bottom-up electricity load modeling. We use appliance time series from publicly available data sets containing fine grained power measurements. The comparison of models are based on metrics which are supposed to be important in power systems like load factor, loss of load probability. Furthermore, we characterize the interdependence structure of the models with autocorrelation function as well. The aim of the investigation is to choose the most appropriate and the most parsimonious models for smart grid simulation purposes and applications like demand side management and load scheduling. According to our results most appliance types can be modeled adequately with two states (on/off model) and the semi-Markov process can reproduce the properties of an aggregate load well compared to the original time series. With the price of more parameters of the semi-Markov model compared to identically distributed random sequence and first order Markov chain, it gives better results when the autocorrelation function, loss of load probability and load factor are considered.A discrete Sparre Andersen risk model with general income rate.https://zbmath.org/1449.910352021-01-08T12:24:00+00:00"Győrfy-Bátori, András"https://zbmath.org/authors/?q=ai:gyorfy-batori.andras"Mihálykó, Csaba"https://zbmath.org/authors/?q=ai:mihalyko.csaba"Orbán-Mihálykó, Éva"https://zbmath.org/authors/?q=ai:orban-mihalyko.evaSummary: A discrete Sparre Andersen risk process with general income rate is investigated. A discrete version of the Gerber-Shiu function is introduced and a difference equation is set up for it. The existence and the uniqueness of its solution is investigated and an analytical solution is given in the case when the claim size has negative binomial distribution. An example is given for illustrating the computations.Intertemporal utility of tax-deferred pension insurance based on Markov chain.https://zbmath.org/1449.911052021-01-08T12:24:00+00:00"Liu, Hua"https://zbmath.org/authors/?q=ai:liu.hua"Guan, Xuepiao"https://zbmath.org/authors/?q=ai:guan.xuepiaoSummary: By buying a tax-deferred pension, the policyholder can lower his or her taxes and increase disposable income after retirement. Therefore, tax-deferred pension insurance can help to raise individual utility to some extent. However, in practice, health status and lifetime are important factors that affect the utility of purchasing tax-deferred pension insurance. Under the framework of intertemporal utility, the Markov chain model of health status is introduced here, and it is found that the deterioration in health and the short lifetime cause the reduction in individual utility, and thus drop people's desire of purchasing tax-deferred pension insurance. Therefore, the policies initiated by the government and products designed by the insurer should supply these population with a minimum guarantee under the development of tax-deferred pension insurance.An existence and uniqueness theorem of stochastic differential equations and the properties of their solution.https://zbmath.org/1449.600932021-01-08T12:24:00+00:00"Bae, Mun-Jin"https://zbmath.org/authors/?q=ai:bae.mun-jin"Park, Chan-Ho"https://zbmath.org/authors/?q=ai:park.chanho"Kim, Young-Ho"https://zbmath.org/authors/?q=ai:kim.youngho.1Summary: In this paper, we show the existence and uniqueness of solutions to stochastic differential equations under weakened Hölder's condition and a weakened linear growth condition. Furthermore, the properties of their solutions investigated and estimate for the error between Picard iterations \(x_n(t)\) and the unique solution \(x(t)\) of SDEs.Generalized dominoes tiling's Markov chain mixes fast.https://zbmath.org/1449.050462021-01-08T12:24:00+00:00"Kayibi, K. K."https://zbmath.org/authors/?q=ai:kayibi.koko-kalambay"Samee, U."https://zbmath.org/authors/?q=ai:samee.umatul|samee.uma"Merajuddin"https://zbmath.org/authors/?q=ai:merajuddin.pirzada|merajuddin.m"Pirzada, S."https://zbmath.org/authors/?q=ai:pirzada.shariefuddinSummary: A generalized tiling is defined as a generalization of the properties of tiling a region of \(\mathbb{Z}^2\) with dominoes, and comprises tiling with rhombus and any other tilings that admits height functions which can be ordered into a distributive lattice. By using properties of the distributive lattice, we prove that the Markov chain consisting of moving from one height function to the next by a flip is fast mixing and the mixing time \(\tau(\epsilon)\) is given by \(\tau(\epsilon)\leq(kmn)^3(mn\;\ln\;k+\ln\;\epsilon^{-1})\), where \(mn\) is the area of the grid \(\Gamma\) that is a \(k\)-regular polycell. This result generalizes the result of \textit{K. K. Kayibi} and \textit{S. Pirzada} [Theor. Comput. Sci. 714, 1--14 (2018; Zbl 1387.05043)] and improves on the mixing time obtained by using coupling arguments by \textit{N. Destainville} [in: Discrete models: combinatorics, computation, and geometry. Proceedings of the 1st international conference (DM-CCG), Paris, France, July 2--5, 2001. Paris: Maison de l'Informatique et des Mathématiques Discrètes (MIMD). 1--22 (2001; Zbl 1017.68148)] and by \textit{M. Luby} et al. [SIAM J. Comput. 31, No. 1, 167--192 (2001; Zbl 0992.82013)].Mean correcting martingale measure for exponential semimartingale market models.https://zbmath.org/1449.600852021-01-08T12:24:00+00:00"Yao, Luogen"https://zbmath.org/authors/?q=ai:yao.luogen"Yang, Gang"https://zbmath.org/authors/?q=ai:yang.gang"Yang, Xiangqun"https://zbmath.org/authors/?q=ai:yang.xiangqunSummary: A martingale measure is constructed by using a mean correcting transform in a general semimartingale market model. It is shown that this measure is the mean correcting martingale measure if there exists a continuous local martingale part in the semimartingale. Although this measure cannot be equivalent to the physical probability for a pure jump semimartingale process, we show that the option price of a European option with a convex payoff function under this measure is still arbitrage free if the arbitrage-free interval can reach universal bounds.Moderate deviations for Lotka-Nagaev estimator of a simple branching process.https://zbmath.org/1449.600412021-01-08T12:24:00+00:00"Zhu, Yanjiao"https://zbmath.org/authors/?q=ai:zhu.yanjiao"Gao, Zhenlong"https://zbmath.org/authors/?q=ai:gao.zhenlongSummary: In this paper, we consider moderate deviations for Lotka-Nagaev estimator of a simple branching process. We obtain that the moderate deviation probabilities for Lotka-Nagaev estimator of a simple branching process have exponential rates of decay. Particularly, we show that there is phase transition in the decay rates of moderate deviation probability when the offspring law has heavy tails.Further results on the generalized cumulative entropy.https://zbmath.org/1449.600252021-01-08T12:24:00+00:00"Di Crescenzo, Antonio"https://zbmath.org/authors/?q=ai:di-crescenzo.antonio"Toomaj, Abdolsaeed"https://zbmath.org/authors/?q=ai:toomaj.abdolsaeedSummary: Recently, a new concept of entropy called generalized cumulative entropy of order \(n\) was introduced and studied in the literature. It is related to the lower record values of a sequence of independent and identically distributed random variables and with the concept of reversed relevation transform. In this paper, we provide some further results for the generalized cumulative entropy such as stochastic orders, bounds and characterization results. Moreover, some characterization results are derived for the dynamic generalized cumulative entropy. Finally, it is shown that the empirical generalized cumulative entropy of an exponential distribution converges to normal distribution.The moderate deviation principle for a stochastic 3D LANS-\(\alpha\) model driven by multiplicative Lévy noise.https://zbmath.org/1449.600352021-01-08T12:24:00+00:00"Huang, Jianhua"https://zbmath.org/authors/?q=ai:huang.jianhua"Zhang, Zaiyun"https://zbmath.org/authors/?q=ai:zhang.zaiyun"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yong.1Summary: In this paper, we construct the moderate deviation principle for a stochastic 3D LANS-\(\alpha\) model driven by multiplicative Lévy noise by the weak convergence method.Pricing European lookback option in a special kind of mixed jump-diffusion Black-Scholes model.https://zbmath.org/1449.911672021-01-08T12:24:00+00:00"Yang, Zhaoqiang"https://zbmath.org/authors/?q=ai:yang.zhaoqiangSummary: This article considers the pricing problem of European fixed strike lookback options under the environment of mixed jump-diffusion fractional Brownian motion. Under the conditions of Merton assumptions, we analyze the Cauchy initial problem of stochastic parabolic partial differential equations for risk assets. By using the perturbation method of multiscale-parameter, the approximate pricing formulae of European lookback options are given by solving stochastic parabolic partial differential equations. Then the error estimates of the approximate solutions are given by using Feynman-Kac formula. Numerical simulation illustrate that the European lookback options have exact solutions when the volatilities are constant, and as the order of simulation increases, the approximate solutions gradually approximate the exact solutions.Consistency of least squares estimation to the parameter for stochastic differential equations under distribution uncertainty.https://zbmath.org/1449.620532021-01-08T12:24:00+00:00"Fei, Chen"https://zbmath.org/authors/?q=ai:fei.chen"Fei, Weiyin"https://zbmath.org/authors/?q=ai:fei.weiyinSummary: Under distribution uncertainty, on the basis of discrete observation data we investigate the consistency of the least squares estimator of the parameter for the stochastic differential equation where the noise is characterized by \(G\)-Brownian motion. In order to obtain the main result of consistency of parameter estimation, we provide some lemmas by the theory of stochastic calculus of sublinear expectation. The result shows that under some regularity conditions, the least squares estimator is strong consistent uniformly on the prior set. An illustrative example is discussed.Queue length distribution for \(\mathrm{Geo}/\mathrm{G}/1\) queue with \(\theta\)-entering discipline during multiple adaptive vacations.https://zbmath.org/1449.601372021-01-08T12:24:00+00:00"Wei, Yingyuan"https://zbmath.org/authors/?q=ai:wei.yingyuan"Tang, Yinghui"https://zbmath.org/authors/?q=ai:tang.yinghui"Yu, Miaomiao"https://zbmath.org/authors/?q=ai:yu.miaomiaoSummary: This paper considers a discrete time \(\mathrm{Geo}/\mathrm{G}/1\) queue with multiple adaptive vacations where the customers who arrive during server vacations enter the system with probability \(\theta\) (\(0 < \theta \le 1\)). Using renewal process theory and total probability decomposition technique, from the beginning of arbitrary initial state, the \(z\)-transform recursive expressions of the transient queue length distribution at time epoch \({n^+}\) are obtained. Based on the transient analysis, the explicit recursive formulae of the steady state queue length distribution at time epochs \({n^-}\); \(n\) and \({n^+}\) are derived, respectively. Furthermore, the results obtained in this paper indicate that the equilibrium queue length distribution no longer follows the stochastic decomposition structure. Finally, numerical results are offered to discuss the sensitivity of the steady state queue length distribution towards system parameters, and illustrate the significant application value of the recursive formulae for the steady state queue length distribution in the system capacity optimum design.Markov spectral clustering algorithm with DCBM for community detection.https://zbmath.org/1449.910852021-01-08T12:24:00+00:00"Ren, Shuxia"https://zbmath.org/authors/?q=ai:ren.shuxia"Zhang, Shubo"https://zbmath.org/authors/?q=ai:zhang.shubo"Wu, Tao"https://zbmath.org/authors/?q=ai:wu.taoSummary: Spectral clustering algorithm is one of the classical community detection algorithms. Due to the current constructed similarity graphs carry less community structure information, the actual clustering effect has a big gap with the ideal clustering effect. Therefore, based on degree corrected stochastic block model and Markov chain, a novel spectral clustering approach for community detection, called MSCD, is proposed. Firstly, probability matrix composed of the connection probability between nodes is introduced based on DCBM, and the mapping relationship is established between probability matrix and similar matrix. Then, Markov chain is utilized to reconstruct the similar graph of spectral clustering. Finally, the reconstructed similar graph is used to partition the networks into clusters. Three typical algorithms of SC, MRW-KNN and FluidC are performed on synthetic networks and real networks. Comparative experiments show that the MSCD algorithm has more efficient clustering performance and can reveal a clearer community.Central limit theorem for linear processes generated by NSD sequences.https://zbmath.org/1449.600382021-01-08T12:24:00+00:00"Li, Jingyu"https://zbmath.org/authors/?q=ai:li.jingyu"Zhang, Yong"https://zbmath.org/authors/?q=ai:zhang.yong.13|zhang.yong.1|zhang.yong|zhang.yong.9|zhang.yong.2|zhang.yong.10|zhang.yong.7|zhang.yong.14|zhang.yong.8|zhang.yong.11|zhang.yong.5|zhang.yong.4|zhang.yong.12Summary: We consider the linear process \(X_t = \sum\limits_{j = -\infty}^\infty a_j \varepsilon_{t-j}\), \(t \ge 1\), where \(\{{\varepsilon_j}, j \in \mathbb{Z}\}\) is a sequence of strictly stationary negatively superadditive dependent (NSD) random variables with mean zeros and finite variances, \(\{{a_j}, j \in \mathbb{Z}\}\) is a sequence of real numbers with \(\sum\limits_{j = -\infty}^\infty {{a_j} \ne 0}\), \(\sum\limits_{j = -\infty}^\infty {| {a_j}| < \infty}\). Let \(S_n = \sum\limits_{t = 1}^n X_t\), \(n \ge 1\). Then under appropriate hypotheses, by using the moment inequalities for NSD sequences and the convergence of \({S_n}\), we give the central limit theorem for linear processes generated by NSD sequences.Spatial convergence for semi-linear backward stochastic differential equations in Hilbert space: a mild approach.https://zbmath.org/1449.601052021-01-08T12:24:00+00:00"Abidi, Hani"https://zbmath.org/authors/?q=ai:abidi.hani"Pettersson, Roger"https://zbmath.org/authors/?q=ai:pettersson.rogerSummary: In this paper, we present convergence results of a spatial semi-discrete approximation of a Hilbert space-valued backward stochastic differential equations with noise driven by a cylindrical \(Q\)-Wiener process. Both the solution and its space discretization are formulated in mild forms. Under suitable assumptions of the final value and the drift, a convergence rate is established.Steady-state distribution of \({M_t}/H_2^*/\infty\) queue model with non-homogeneous Poisson arrivals.https://zbmath.org/1449.900792021-01-08T12:24:00+00:00"Niu, Xin"https://zbmath.org/authors/?q=ai:niu.xin"Liu, Jianmin"https://zbmath.org/authors/?q=ai:liu.jianmin"Wang, Qingqing"https://zbmath.org/authors/?q=ai:wang.qingqingSummary: The steady-state distribution of an \({M_t}/H_2^*/\infty\) queue model with non-homogeneous Poisson arrivals was considered, and the periodic arrival rate was analyzed. The arrival rate function was expressed by Fourier series, and expressions of mean and variances about busy servers of the system were derived. Considering the special case that the arrival rate function was a sinusoidal function, the steady-state distribution of busy servers and its fluid limit were obtained in the \({M_t}/H_2^*/\infty\) queue model with sinusoidal arrival rate.Characterizations on almost stochastic dominance revisited.https://zbmath.org/1449.600262021-01-08T12:24:00+00:00"Hu, Yunhe"https://zbmath.org/authors/?q=ai:hu.yunhe"Fan, Kun"https://zbmath.org/authors/?q=ai:fan.kunSummary: Almost stochastic dominance has been receiving more attention in the financial and economic literature. In this short note, we characterize the almost first-and second-degree stochastic dominance by requiring one distribution to be ``close to'' a new distribution that dominates or is dominated by another distribution in the traditional sense of the first-and second-order stochastic dominance, respectively. We also investigate the concept of almost stochastic dominance for unbounded random variables.Application of DJ method to Itô stochastic differential equations.https://zbmath.org/1449.600992021-01-08T12:24:00+00:00"Deilami Azodi, H."https://zbmath.org/authors/?q=ai:deilami-azodi.hSummary: This paper develops iterative method described by \textit{V. Daftardar-Gejji} and \textit{H. Jafari} [J. Math. Anal. Appl. 316, No. 2, 753--763 (2006; Zbl 1087.65055)] to solve Itô stochastic differential equations. The convergence of the method for Itô stochastic differential equations is assessed. To verify efficiency of method, some examples are expressed.Remarks on criteria of ergodicity for birth-death processes.https://zbmath.org/1449.370022021-01-08T12:24:00+00:00"Wu, Bingyao"https://zbmath.org/authors/?q=ai:wu.bingyao"Wang, Jian"https://zbmath.org/authors/?q=ai:wang.jian.7|wang.jian.1|wang.jian.4|wang.jian.2|wang.jian.3|wang.jian.5|wang.jian.9Summary: In order to study the relations among various ergodic properties of birth-death processes on \(\textbf{Z}_+\), we establish sufficient conditions under which ergodic property directly yields exponential ergodicity, the existence of discrete spectral and strong ergodicity as well as other stronger properties. Some examples are given which satisfy the ergodicity but the sufficient conditions and conclusions are not valid.The strong consistency of wavelet estimator in the nonparametric regression model under NOD errors.https://zbmath.org/1449.620892021-01-08T12:24:00+00:00"Deng, Xin"https://zbmath.org/authors/?q=ai:deng.xin"Gui, Daiyun"https://zbmath.org/authors/?q=ai:gui.daiyun"Xu, Zhicai"https://zbmath.org/authors/?q=ai:xu.zhicaiSummary: We mainly study the nonparametric regression model with repeated measurements: \(Y^{ (j)} (x_{ni}) = g (x_{ni}) + e^{ (j)} (x_{ni})\), where \(Y^{ (j)} (x_{ni})\) is the \(j\)-th response at the point \(x_{ni}\), \(x_{ni}\)'s are known, and \(g (\cdot)\) is the regression function in \([0, 1]\). By using the Rosenthal type inequality and the Kolmogorov's strong law of large numbers, the strong consistency of the wavelet estimator for \(g (\cdot)\) based on negatively orthant dependent (NOD) random errors is established under some mild conditions, which generalizes the corresponding one for negatively associated (NA) errors.System capacity optimization design and optimal control policy \((N^* ,D^*)\) for \(\mathrm{M}/\mathrm{G}/1\) queue with \(p\)-entering discipline and \(\mathrm{Min}(N,D,V)\)-policy.https://zbmath.org/1449.601322021-01-08T12:24:00+00:00"Luo, Le"https://zbmath.org/authors/?q=ai:luo.le"Tang, Yinghui"https://zbmath.org/authors/?q=ai:tang.yinghuiSummary: This paper considers an \(\mathrm{M}/\mathrm{G}/1\) queueing system with \(p\)-entering discipline and \(\mathrm{Min}(N, D, V)\)-policy, in which the customers who arrive during multiple vacations enter the system with probability \(p\) (\(0 < p \le 1\)). By using the total probability decomposition technique and the Laplace transform, we discuss the transient distribution of queue length at any time \(t\) which started from an arbitrary initial state, and obtain the expressions of the Laplace transform of transient queue-length distribution. Moreover, we obtain the recursion expressions of the steady-state queue length distribution. Meanwhile, we discuss the optimal capacity design by combining the steady-state queue length distribution and numerical example. Finally, the explicit expression of the long-run expected cost rate is derived under a given cost structure. Through numerical calculation, we determine the optimal control policy \((N^*, D^*)\) for minimizing the long-run expected cost per unit time.Complete convergence and complete moment convergence for WOD random variables sequences.https://zbmath.org/1449.600632021-01-08T12:24:00+00:00"Zhang, Qian"https://zbmath.org/authors/?q=ai:zhang.qian"Cai, Guanghui"https://zbmath.org/authors/?q=ai:cai.guanghuiSummary: In this paper, we use a new method to improve corresponding results of previous work by truncating the WOD random variables into five parts.Almost sure exponential stability for some neutral partial integro-differential equations.https://zbmath.org/1449.342672021-01-08T12:24:00+00:00"Ramkumar, K."https://zbmath.org/authors/?q=ai:ramkumar.kasinathan"Mohamed, M. S."https://zbmath.org/authors/?q=ai:mohamed.mohamed-salem"Diop, Mamadou Abdoul"https://zbmath.org/authors/?q=ai:diop.mamadou-abdoulSummary: This paper is concerned with the dynamics of a delay stochastic neutral integro-differential equation in Hilbert spaces by using the theory of resolvent operator. After establishing a result ensuring the existence and uniqueness of a mild solution of this class of equations, we investigate the exponential stability of the moments of a mild solution as well as its sample paths. An example is given to illustrate the results.Valuation of American passport option using a three-time level scheme.https://zbmath.org/1449.911902021-01-08T12:24:00+00:00"Kanaujiya, Ankur"https://zbmath.org/authors/?q=ai:kanaujiya.ankur"Chakrabarty, Siddhartha P."https://zbmath.org/authors/?q=ai:chakrabarty.siddhartha-pSummary: An American passport option whose contingent claim is dependent on the balance of a trading account can be valued by solving a Hamilton-Jacobi-Bellman equation with free boundary. Here, we present the pricing problem for American passport option, as a sequence of linear complementarity problems, using the three-time level finite difference scheme, which typically is suitable for non-smooth payoffs and also applicable in case of large temporal grid size. The option price is obtained through this scheme for the non-symmetric case (when the risk-free rate is different from the cost of carry). It is observed that the numerical approach presented, results in solving the pricing problem using lesser number of grid points as compared to numerical approaches for this problem used previously while maintaining the accuracy of the prices obtained.Probability inequalities and Rosenthal inequalities for the sequence of martingale differences.https://zbmath.org/1449.600292021-01-08T12:24:00+00:00"Xu, Mingzhou"https://zbmath.org/authors/?q=ai:xu.mingzhou"Cheng, Kun"https://zbmath.org/authors/?q=ai:cheng.kun"Ding, Yunzheng"https://zbmath.org/authors/?q=ai:ding.yunzheng"Zhou, Yongzheng"https://zbmath.org/authors/?q=ai:zhou.yongzhengSummary: In this paper, we discuss some inequalities for the sequence of martingale differences. By using properties of conditional expectation and elementary inequalities, we obtain the basic inequalities of Bernstein, Kolomogrov, Hoeffding for the sequence of martingale differences, which extend the results on the case of bounded random vectors. Moreover, we obtain classical Kolmogorov and Rosenthal inequalities for maximum partial sums of martingale differences, which complement the results on the case of independent and negatively dependent random variables under sub-linear expectations.Switch-time distributions and processes.https://zbmath.org/1449.600892021-01-08T12:24:00+00:00"Stoynov, Pavel"https://zbmath.org/authors/?q=ai:stoynov.pavelIn this paper, switch time distributions and processes related to them are investigated. The properties of additive processes are presented by a theorem of \textit{K.-I. Sato} [Lévy processes and infinitely divisible distributions. Cambridge: Cambridge University Press (1999; Zbl 0973.60001)]. Then, additive processes with returns to zero are characterized in Theorem 1. A switch time family of distributions is introduced and simulations of switch time family processes are provided at the end of the paper.
Reviewer: Angela Slavova (Sofia)Short-term travel time prediction model based on secondary correction.https://zbmath.org/1449.900752021-01-08T12:24:00+00:00"Yang, Hang"https://zbmath.org/authors/?q=ai:yang.hang"Wang, Zhongyu"https://zbmath.org/authors/?q=ai:wang.zhongyu"Zou, Yajie"https://zbmath.org/authors/?q=ai:zou.yajie"Wu, Bing"https://zbmath.org/authors/?q=ai:wu.bingSummary: In order to increase both of the accuracy and the robustness for freeway short-term travel time prediction, as well as easing the over-fitting effect, which was brought by the extra training, a hybrid model was proposed on the basis of wavelet neural network and Markov chain. The forecasting performance of different models was examined by three measures, i.e., mean absolute error, mean absolute percentage error, and root mean square error. The results show that the proposed hybrid model enjoys obvious superiority over the other models after the break point of travel time. Furthermore, no prediction-delay was observed in the prediction of break point of travel time. In conclusion, the higher prediction accuracy and the better robustness were found in the hybrid model in peak hours.Time-consistent optimal reinsurance-investment strategy selection for Poisson-Geometric model.https://zbmath.org/1449.911132021-01-08T12:24:00+00:00"Yang, Peng"https://zbmath.org/authors/?q=ai:yang.peng"Yang, Zhijiang"https://zbmath.org/authors/?q=ai:yang.zhijiang"Kong, Xiangxin"https://zbmath.org/authors/?q=ai:kong.xiangxinSummary: In this paper, a time-consistent reinsurance-investment strategy selection for Poisson-geometric model is considered. In risk model, the number of claims is a Poisson-geometric process. When reinsurance is carried out by the insurance company, the premium of reinsurance should be calculated according to the variance principle. When the insurer invest in financial market, the risky asset is assumed to follow a stochastic differential equation with jump. The objective of the insurer is to choose an optimal time-consistent reinsurance-investment strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. We investigate the problem using the stochastic control theory. Explicit solutions for the time-consistent reinsurance-investment strategy and the corresponding value functions are obtained. Finally, the economic significance of the results is analyzed. Numerical calculation is also provided to illustrate the influence of model parameters on optimal strategies.M/G/1 queueing system with \({\mathrm{Min}} (N,D,V)\)-policy control.https://zbmath.org/1449.900772021-01-08T12:24:00+00:00"Luo, Le"https://zbmath.org/authors/?q=ai:luo.le"Tang, Yinghui"https://zbmath.org/authors/?q=ai:tang.yinghuiSummary: In this paper, we consider the \(M/G/1\) queueing system with multiple server vacations and \({\mathrm{Min}} (N, D, V)\)-policy. By using the total probability decomposition technique and the Laplace transformation tool, the transient queue-length distribution and the steady queue-length distribution are discussed. Both the expressions of the Laplace transformation of the transient queue-length distribution and the recursive expressions of the steady queue-length distribution are obtained. Meanwhile, we present the stochastic decomposition result of the steady queue length and the explicit expression of the additional queue length distribution. Furthermore, some special cases are discussed when \(N \to \infty, D \to \infty\), \(p\{V = \infty \} = 1\) or \(p\{V = 0 \} = 1\). Finally, the explicit expression of the long-run expected cost rate is derived under a given cost structure. And through numerical calculation, we determine the optimal control policy \( ({N^*}, {D^*})\) for minimizing the long-run expected cost per unit time as well as compare with the single optimal \({N^*}\)-policy and the single optimal \({D^*}\)-policy.Poisson limit theorems for the generalized allocation scheme.https://zbmath.org/1449.600072021-01-08T12:24:00+00:00"Chuprunov, Alexey"https://zbmath.org/authors/?q=ai:chuprunov.aleksei-n"Fazekas, István"https://zbmath.org/authors/?q=ai:fazekas.istvanIt is a classical observation in probability theory that the binomial distribution in the case of rare events can be approximated by the Poisson distribution. In the paper under review, the authors prove results of this type in the setting of the so-called ``generalized allocation scheme'', as it was introduced by \textit{V. F. Kolchin} [Litov. Mat. Sb. 8, 53--63 (1968; Zbl 0235.60023)]. As a simple case, they for example prove that when allocating \(n\) balls into \(N\) boxes, then the rescaled number of boxes among the first \(K\) boxes containing \(r\) balls converges to a Poisson distribution, as \(K\) and \(n\) tend to \(\infty\). The more general results allow distinguishable/undistinguishable balls, different colors, etc., but also include for example the case of the number of trees having a certain number of non-root vertices in a random forest.
Reviewer: Christoph Aistleitner (Graz)Multi-dimensional uncertain differential equation: existence and uniqueness of solution.https://zbmath.org/1449.601172021-01-08T12:24:00+00:00"Ji, Xiaoyu"https://zbmath.org/authors/?q=ai:ji.xiaoyu"Zhou, Jian"https://zbmath.org/authors/?q=ai:zhou.jian.2|zhou.jian|zhou.jian.1|zhou.jian.3Summary: A multi-dimensional uncertain differential equation is a system of uncertain differential equations driven by a multi-dimensional Liu process. This paper first gives the analytic solutions of two special types of multi-dimensional uncertain differential equations. After that, it proves that the multi-dimensional uncertain differential equation has a unique solution provided that its coefficients satisfy the Lipschitz condition and the linear growth condition.Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks.https://zbmath.org/1449.902422021-01-08T12:24:00+00:00"Zhu, Huai-Nian"https://zbmath.org/authors/?q=ai:zhu.huainian"Zhang, Cheng-Ke"https://zbmath.org/authors/?q=ai:zhang.chengke"Jin, Zhuo"https://zbmath.org/authors/?q=ai:jin.zhuoSummary: This paper investigates a continuous-time Markowitz mean-variance asset-liability management (ALM) problem under stochastic interest rates and inflation risks. We assume that the company can invest in \(n+1\) assets: one risk-free bond and \(n\) risky stocks. The risky stock's price is governed by a geometric Brownian motion (GBM), and the uncontrollable liability follows a Brownian motion with drift, respectively. The correlation between the risky assets and the liability is considered. The objective is to minimize the risk (measured by variance) of the terminal wealth subject to a given expected terminal wealth level. By applying the Lagrange multiplier method and stochastic control approach, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation, which can be converted into six partial differential equations (PDEs). The closed-form solutions for these six PDEs are derived by using the homogenization approach and the variable transformation technique. Then the closed-form expressions for the efficient strategy and efficient frontier are obtained. In addition, a numerical example is presented to illustrate the results.Strong limit theorems for arrays of rowwise independent random variables under sublinear expectation.https://zbmath.org/1449.600472021-01-08T12:24:00+00:00"Feng, X."https://zbmath.org/authors/?q=ai:feng.xiaoxue|feng.xinxin|feng.xuenan|feng.xiangchu|feng.xuelin|feng.xiaoxuan|feng.xunli|feng.xiutao|feng.xiaomei|feng.xu|feng.xianggui|feng.xinyang|feng.xiuli|feng.xuerong|feng.xiaobo|feng.xingdong|feng.xingfang|feng.xianchu|feng.xiaoyu|feng.xian|feng.xiaotao|feng.xs|feng.xiaoping|feng.xinan|feng.xuan|feng.xuejian|feng.xiaoning|feng.xizhou|feng.xiaoyun|feng.xinzeng|feng.xingjie|feng.xiaomeng|feng.xiaogao|feng.xijin|feng.xinlei|feng.xianmin|feng.xiating|feng.xiaogang|feng.xiaojian|feng.xing|feng.xiangyu|feng.xiangjun|feng.xiuhua|feng.xiquan|feng.xue|feng.xinyong|feng.xiangnan|feng.xuechao|feng.xiaoyan|feng.xiaoli|feng.xiujuan|feng.xiaojing|feng.xia|feng.xiangdong|feng.xiaowen|feng.xuehua|feng.xingxing|feng.xiaodong|feng.xiufang|feng.xiuqin|feng.xiaoyi|feng.xiuxian|feng.xugang|feng.xiucheng|feng.xiaozhou|feng.xiaoxia|feng.xiaoying|feng.xinwei|feng.xiuhong|feng.xueqin|feng.xiufeng|feng.xiaoqiang|feng.xuxia|feng.xinyu|feng.xiushan|feng.xiaojiang|feng.xiaobing|feng.xunsheng|feng.xiaojun|feng.xiaolin|feng.xiqiao|feng.xueshang|feng.xiang|feng.xinlong|feng.xiangho|feng.xiangbo|feng.xinglai|feng.xiaohua|feng.xuhui|feng.xiaowei|feng.xiaoliang|feng.xuhong|feng.xiangqian|feng.xuning|feng.xiyuan|feng.xiaojiu|feng.xin|feng.xinghua|feng.xiuying|feng.xuemin|feng.xuehao|feng.xiao|feng.xizhong|feng.xianzhi|feng.xuexin|feng.xie|feng.xinxi|feng.xuejun|feng.xisheng|feng.xiangrong|feng.xiaoqin|feng.xiuxia|feng.xixia"Lan, Y."https://zbmath.org/authors/?q=ai:lan.yinghong|lan.yunxu|lan.yingjie|lan.yaoyao|lan.yingchao|lan.yuting|lan.yu|lan.yucheng|lan.yueheng|lan.yongquan|lan.yan|lan.yun|lan.yonghong|lan.yang|lan.yang.1|lan.yizhong|lan.yanxiang|lan.yanfei|lan.yongxin|lan.yanlian|lan.yixin|lan.yongyi|lan.yipeng|lan.yuanqi|lan.ying|lan.youranLet \(\{X_i,i\geq 1 \}\) be a sequence of independent and identically distributed random variables, and let \(1\leq p < 2\).\, Then, the Marcinkiewcz-Zygmund-type strong law of large numbers states that \(\frac{\sum_{i=1}^{n}X_i}{n^{\frac{1}{p}}}\rightarrow 0\) a.s. as \(n\rightarrow \infty\), if and only if \(E[X_1]=0, E[|X_1|^p]<\infty.\) There are numerous generalizations of this result as given in the authors' list itself. In this paper, the authors generalize the above type of strong law of large numbers and the law of the logarithm for arrays of independent random variables under sublinear expectation.
Reviewer: Rasul A. Khan (Forest Glen)Positive spatial autocorrelation impacts on attribute variable frequency distributions.https://zbmath.org/1449.621842021-01-08T12:24:00+00:00"Griffith, Daniel"https://zbmath.org/authors/?q=ai:griffith.daniel-aSummary: Researchers commonly inspect histograms as a first step in data analysis, often finding that these graphs fail to closely align with any of the several hundred ideal frequency distributions. The purpose of this paper is to address how positive spatial autocorrelation -- the most frequently encountered in practice -- can distort histograms when they are constructed with georeferenced data. Normal, Poisson, and binomial random variables -- three widely applicable ones -- are studied after establishing appropriate moment generating functions, and are illustrated with selected simulations. The simulations were designed with an ideal surface partitioning, and with the irregular China county surface partitioning. Results show that even moderate levels of positive spatial autocorrelation, while not affecting means, not only inflate variance, but also modify the probabilities of extreme and/or central values, and can alter skewness and kurtosis. A methodology is outlined for recovering the underlying
unautocorrelated frequency distributions.Statistical analysis of locally stationary processes.https://zbmath.org/1449.621822021-01-08T12:24:00+00:00"Ferreira, Guillermo"https://zbmath.org/authors/?q=ai:ferreira.guillermo-p"Olea, Ricardo"https://zbmath.org/authors/?q=ai:olea.ricardo-a"Palma, Wilfredo"https://zbmath.org/authors/?q=ai:palma.wilfredoSummary: This paper provides an overview of locally stationary processes, a helpful methodology for handling nonstationary time series. These techniques allow for the smooth evolution of the model parameters. This work reviews estimation and predictions techniques, illustrating the application of these methods to real-life data examples. These examples show that the locally stationary methods provide a useful theoretical and practical framework for the statistical analysis of nonstationary time series data.Liggett-Stroock inequalities for time inhomogeneous Markov processes.https://zbmath.org/1449.601272021-01-08T12:24:00+00:00"Song, Juan"https://zbmath.org/authors/?q=ai:song.juan"Zhang, Ming"https://zbmath.org/authors/?q=ai:zhang.mingSummary: We generalize the Liggett-Stroock inequality of the time homogeneous Markov process to the inhomogeneous Markov process, and establish the relationship between the transition semigroup of inhomogeneous Markov process and the Liggett-Stroock inequality.Stochastic representations and a geometric parametrization of the two-dimensional Gaussian law.https://zbmath.org/1449.600112021-01-08T12:24:00+00:00"Dietrich, Thomas"https://zbmath.org/authors/?q=ai:dietrich.thomas"Kalke, Steve"https://zbmath.org/authors/?q=ai:kalke.steve"Richter, Wolf-Dieter"https://zbmath.org/authors/?q=ai:richter.wolf-dieterSummary: Using different types of polar and elliptical polar coordinates, different stochastic representations of the axis-aligned and the regular two-dimensional Gaussian distribution are derived. Advantages and disadvantages of these stochastic representations are discussed. The non-Euclidean geometric measure representation of the axis-aligned two-dimensional Gaussian distribution in [the third author, J. Appl. Anal. 17, No. 2, 165--179 (2011; Zbl 1276.51009)] is taken to derive a new geometric interpretation of the correlation coefficient and to motivate a new geometric parametrization of the regular Gaussian law. Estimators of the new parameters and corresponding distributions are derived. A comparison with different approaches from the literature shows the numerical stability of our results.The beta Weibull Poisson distribution.https://zbmath.org/1449.600192021-01-08T12:24:00+00:00"Percontini, Ana"https://zbmath.org/authors/?q=ai:percontini.ana"Blas, Betsabé"https://zbmath.org/authors/?q=ai:blas.betsabe"Cordeiro, Gauss M."https://zbmath.org/authors/?q=ai:cordeiro.gauss-moutinhoSummary: Providing a wider distribution is always precious for statisticians. A new five-parameter distribution called the beta Weibull Poisson is proposed, which is obtained by compounding the Weibull Poisson and beta distributions. It generalizes several known lifetime models. We obtain some properties of the proposed distribution such as the survival and hazard rate functions, quantile function, ordinary and incomplete moments, order statistics and Rényi entropy. Estimation by maximum likelihood and inference for large samples are addressed. The potentiality of the new model is shown by means of a real data set. In fact, the proposed model can produce better fits than some well-known distributions.A study of exponential-type tails applied to Birnbaum-Saunders models.https://zbmath.org/1449.600922021-01-08T12:24:00+00:00"Ferreira, Marta"https://zbmath.org/authors/?q=ai:ferreira.martaSummary: Birnbaum-Saunders distributions have increasingly been used in environmental sciences applications. A major concern is the adjustment of extreme quantiles. Environmental data have often tails in the Gumbel domain which corresponds to a null tail index and does not allow us to distinguish the different tail weights that might exist between distributions within this domain. Exponential-tail distributions form an important subgroup with the peculiarity of including a parameter that specifies the ``penultimate'' tail behavior. In particular, we analyze the penultimate tail behavior of Birnbaum-Saunders distributions. We find examples with ``heavier'' tails than the classical one that can better accommodate environmental data highly concentrated on the right tail. This is illustrated with an application.A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source.https://zbmath.org/1449.740772021-01-08T12:24:00+00:00"Allam, Allam A."https://zbmath.org/authors/?q=ai:allam.allam-aSummary: A stochastic half-space problem, driven by an additive Gaussian white noise, is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface is traction free and subjected to a time dependent thermal shock. A permeating substance is considered in contact with the bounding surface. Laplace transform technique is used to obtain the solution in the transformed domain by using a direct approach. The mean and variance are derived and analyzed for temperature, displacement, stress, strain, concentration and chemical potential. The asymptotic behavior for the solution is discussed. Numerical results are carried out and represented graphically. The second sound effect is observed in the simulation.Analysis of priority queue with repeated attempts using generalized stochastic Petri nets.https://zbmath.org/1449.680152021-01-08T12:24:00+00:00"Hakmi, Sedda"https://zbmath.org/authors/?q=ai:hakmi.sedda"Lekadir, Ouiza"https://zbmath.org/authors/?q=ai:lekadir.ouiza"Aissani, Djamil"https://zbmath.org/authors/?q=ai:aissani.djamilSummary: The paper deals with the modeling and analysis of a single server queue with repeated attempts and two priority customers involving generalized stochastic Petri nets. Indeed, the consideration of the recalls and priority introduces great analytical difficulties. Therefore, by using the GSPN, we show how this high level formalism allows us to cope with the complexity of this system. The paper extends previous works on this topic and evaluate different performance characteristics of the system. The Markov chain is obtained and some numerical results are presented to illustrate the effect of the system parameters on the developed performance measures.Linear-quadratic optimal control problems for mean-field backward stochastic differential equations with jumps.https://zbmath.org/1449.490322021-01-08T12:24:00+00:00"Tang, Maoning"https://zbmath.org/authors/?q=ai:tang.maoning"Meng, Qingxin"https://zbmath.org/authors/?q=ai:meng.qingxinSummary: This paper is concerned with a linear quadratic optimal control problem for mean-field backward stochastic differential equations (MF-BSDE) driven by a Poisson random martingale measure and a Brownian motion. Firstly, by the classic convex variation principle, the existence and uniqueness of the optimal control are obtained. Secondly, the optimal control is characterized by the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps by the duality method. Thirdly, in terms of a decoupling technique, the stochastic Hamilton system is decoupled by introducing two Riccati equations and an MF-BSDE with jumps. Then an explicit representation for the optimal control is obtained.A class of strong laws of large numbers about countable non-homogeneous Markov chain transferring between sets on a non-homogenous tree.https://zbmath.org/1449.600662021-01-08T12:24:00+00:00"Zhao, Xiumei"https://zbmath.org/authors/?q=ai:zhao.xiumei"Li, Jiaxin"https://zbmath.org/authors/?q=ai:li.jiaxin"Yuan, Hongxing"https://zbmath.org/authors/?q=ai:yuan.hongxing"Liu, Huijie"https://zbmath.org/authors/?q=ai:liu.huijie"Jin, Shaohua"https://zbmath.org/authors/?q=ai:jin.shaohuaSummary: In recent years, the tree model has attracted a great deal of interest among scientists from various research fields such as physics, probability theory, information theory etc. Moreover, stochastic process indexed by a tree has become a hot topic in the field of the probability theory in recent years. The research of the strong laws of large numbers has held an important position in the development process of probability theory, and the strong law of large numbers is one of the central issues of the international probability theory. In this paper, through constructing non-negative martingales and applying Doob's martingale convergence theorem to the research of a.e. convergence, a class of strong laws of large numbers about countable non-homogeneous Markov chain transferring between sets on a non-homogenous tree are given.Martingale transforms and weak Hardy-Orlicz-Karamata spaces of martingales.https://zbmath.org/1449.600842021-01-08T12:24:00+00:00"Zhou, Nian"https://zbmath.org/authors/?q=ai:zhou.nian"Yu, Lin"https://zbmath.org/authors/?q=ai:yu.linSummary: The relations between weak Hardy-Orlicz-Karamata spaces of martingales are characterized by martingale transforms. More precisely, let \({\Phi_1}\underline \prec {\Phi_2}\) be two Young functions and \(b (\cdot)\) be non-decreasing slowly varying function, it is constructively proved that the elements in the weak Hardy-Orlicz-Karamata space \(w{\mathcal{H}_{\Phi_{1,}b}}\) are none other than the martingale transforms of those in weak Hardy-Orlicz-Karamata space \(w{\mathcal{H}_{\Phi_{2,}b}}\). The results obtained here extend the corresponding results in former literature.Recurrence classification of random walk on a strip: near-critical.https://zbmath.org/1449.600882021-01-08T12:24:00+00:00"Zhang, Meijuan"https://zbmath.org/authors/?q=ai:zhang.meijuan"Zhou, Ke"https://zbmath.org/authors/?q=ai:zhou.keSummary: In this paper, we consider the near-critical random walk on a strip. By the explicit criteria for recurrence and transience, with the help of asymptotic theory of the solution of linear difference system with disturbance, and the propositions of matrix norm, we give a recurrence classification in terms of the order of the perturbation matrix.Functional sample path properties of subsequence's \(C\)-\(R\) increments for \({l^p}\)-valued Wiener processes in Hölder norm.https://zbmath.org/1449.600772021-01-08T12:24:00+00:00"Wei, Qicai"https://zbmath.org/authors/?q=ai:wei.qicai"Wang, Wensheng"https://zbmath.org/authors/?q=ai:wang.wensheng.1|wang.wensheng|wang.wensheng.2Summary: This paper obtains the functional sample path properties of subsequence's \(C\)-\(R\) increments for \({l^p}\)-valued, \({1 \le p < \infty}\), Wiener processes. By those properties, functional laws of iterated logarithm for \({l^p}\)-valued Wiener processes are generalized.A note on weak solutions to stochastic differential equations.https://zbmath.org/1449.601032021-01-08T12:24:00+00:00"Ondreját, Martin"https://zbmath.org/authors/?q=ai:ondrejat.martin"Seidler, Jan"https://zbmath.org/authors/?q=ai:seidler.janThe paper presents an elegant and simple proof of a several classical results for weak solutions of stochastic differential equations, namely, their existence under weak assumptions on the coefficients and Yamada-Watanabe result of the form pathwise uniqueness and existence of weak solutions -- existence of strong solutions. It is a continuation of previous works of \textit{M. Hofmanová} and the second author [Stochastic Anal. Appl. 30, No. 1, 100--121 (2012; Zbl 1241.60025); ibid. 31, No. 4, 663--670 (2013; Zbl 1277.60106)], where another simplified proof of the existence result was presented. The present paper takes the method one step further and gives another proof of the identification of the stochastic integral in the equation. To be more precise, the construction of weak solutions relies on a suitable approximation procedure together with a compactness argument: the coefficients are approximated by, e.g., Lipschitz ones, so that existence of unique approximate solutions follows. Then uniform bounds are found to imply tightness of the induced probability laws, and the last step is the identification of the limit. In particular, the passage to the limit in the stochastic integrals is delicate as each approximate stochastic integral is with respect to a different Brownian motion. In the literature, one can find several methods to handle the last point mentioned above: the classical approach by the martingale representation theorem, the method from [2012, loc. cit.; 2013, loc. cit.], a convergence lemma from a paper by \textit{A. Debussche} et al. [Physica D 240, No. 14--15, 1123--1144 (2011; Zbl 1230.60065)] and an auxiliary mollification argument from [\textit{A. Bensoussan}, Acta Appl. Math. 38, No. 3, 267--304 (1995; Zbl 0836.35115)]. The idea in the present paper is similar to that of [Bensoussan, loc. cit.] but the mollification is different. The paper is very carefully written and presents the method in detail and in a nicely readable fashion. The method itself is indeed very elegant.
Reviewer: Martina Hofmanová (Bielefeld)On weighted \(U\)-statistics for stationary random fields.https://zbmath.org/1449.600362021-01-08T12:24:00+00:00"Klicnarová, Jana"https://zbmath.org/authors/?q=ai:klicnarova.janaSummary: The aim of this paper is to introduce a central limit theorem and an invariance principle for weighted \(U\)-statistics based on stationary random fields. \textit{T. Hsing} and \textit{W. B. Wu} [Ann. Probab. 32, No. 2, 1600--1631 (2004; Zbl 1049.62099)] in their paper introduced some asymptotic results for weighted \(U\)-statistics based on stationary processes. We show that it is possible also to extend their results for weighted \(U\)-statistics based on stationary random fields.Existence of mild solutions for a class of fractional stochastic evolution equations with nonlocal initial conditions.https://zbmath.org/1449.354332021-01-08T12:24:00+00:00"Chen, Pengyu"https://zbmath.org/authors/?q=ai:chen.pengyu"Ma, Weifeng"https://zbmath.org/authors/?q=ai:ma.weifeng"Ahmed, Abdelmonem"https://zbmath.org/authors/?q=ai:ahmed.abdelmonemSummary: This paper obtains the existence results of mild solutions to a class of fractional stochastic evolution equations with nonlocal conditions by applying stochastic analysis theory, Schauder fixed point theorem and approximation method, and assuming that the nonlinear term is Caretheodory continuous and satisfies some weak growth condition, the nonlocal term depends on all the value of independent variable on the whole interval and satisfies some weak growth condition. This work may be viewed as an attempt to develop a general existence theory for fractional stochastic evolution equations with general nonlocal conditions. Finally, as a sample of application, the results are applied to a fractional stochastic partial differential equation with nonlocal integral condition.The joint distribution of ruin related quantities in the classical risk model.https://zbmath.org/1449.910372021-01-08T12:24:00+00:00"Su, Bihao"https://zbmath.org/authors/?q=ai:su.bihao"Li, Jingchao"https://zbmath.org/authors/?q=ai:li.jingchaoSummary: Ruin theory plays a crucial role in risk measurement and risk regulation. Bankruptcy claim is a major focus of ruin theory and the distribution of the aggregate amount of claim can well describe the risk of insurance portfolio. According to the distribution characteristics, we can adopt such means as capital injection and premium re-adjustment to regulate risk. In the classical risk model, the priorities are given to four ruin-related variables: the time of ruin, the aggregate claim amount up to ruin, the total number of claims up to ruin and the deficit at ruin. In this paper, we mainly consider the joint probability density function of the aggregate claim amount up to ruin with other ruin related quantities. The explicit expressions are given for the joint densities when the individual claim follows exponential distribution. In addition, when the individual claim follows a particular decomposition form, the joint density can also be obtained in a decomposition form.Calculation method for frame construction life prediction on the basis of creep and endurance of energy type.https://zbmath.org/1449.742022021-01-08T12:24:00+00:00"Shershneva, Mariya Viktorovna"https://zbmath.org/authors/?q=ai:shershneva.mariya-viktorovnaSummary: The analytical method of estimation of frame construction life prediction in terms of creep by strain failure criterion is suggested. The linearization of stochastic rheological energy equations, is made, the estimations of uptime under the assumptions on the limit values of creep are obtained. The checking of accordance of the method with the experimental data for the creep of samples made of 12Kh18N10T steel under \(T=850^\circ C\) is implemented. There is agreement between the calculated and experimental data.The large deviation of a partial sum of NQD sequences and convergence of a weighted sum of NQD sequences.https://zbmath.org/1449.600422021-01-08T12:24:00+00:00"Du, Xiaoxiao"https://zbmath.org/authors/?q=ai:du.xiaoxiao"Yan, Li"https://zbmath.org/authors/?q=ai:yan.liSummary: By using a moment inequality and a maximum inequality of NQD random variable sequences, this paper mainly discusses the large deviation of a partial sum of NQD sequences and the convergence of a weighted sum of NQD sequences.Local Strassen's law of the iterated logarithm for increments of a \({l^p}\)-valued Wiener processes in Hölder norm.https://zbmath.org/1449.600682021-01-08T12:24:00+00:00"Liu, Yonghong"https://zbmath.org/authors/?q=ai:liu.yonghong"Liu, Haiguo"https://zbmath.org/authors/?q=ai:liu.haiguo"Zhou, Xia"https://zbmath.org/authors/?q=ai:zhou.xiaSummary: Using large deviations of \({l^p}\)-valued wiener processes in Hölder norm, a local Strassen's law of the iterated logarithm for increments of a \({l^p}\)-valued Wiener processes in Hölder norm is investigated.The exponentiated Gompertz generalized family of distributions: properties and applications.https://zbmath.org/1449.620262021-01-08T12:24:00+00:00"Cordeiro, Gauss M."https://zbmath.org/authors/?q=ai:cordeiro.gauss-moutinho"Alizadeh, Morad"https://zbmath.org/authors/?q=ai:alizadeh.morad"Nascimento, Abraão D. C."https://zbmath.org/authors/?q=ai:nascimento.abraao-d-c"Rasekhi, Mahdi"https://zbmath.org/authors/?q=ai:rasekhi.mahdiSummary: The proposal of more flexible distributions is an activity often required in practical contexts. In particular, adding a positive real parameter to a probability distribution by exponentiation of its cumulative distribution function has provided flexible generated distributions having interesting statistical properties. In this paper, we study general mathematical properties of a new generator of continuous distributions with three extra parameters called the exponentiated Gompertz generated (EGG) family. We present some of its special models as well as an essay on its physical motivation. From mathematical point of view, we derive explicit expressions of the EGG family: the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon and Rényi entropies and order statistics, which are valid for any baseline model. We also provide a bivariate EGG extension. The estimation procedure by maximum likelihood of the new class is elaborated and discussed. In order to quantify and to assess the asymptotic behavior of this procedure, we perform a simulation study. Finally, two applications to real data are performed. Results furnish evidence in favor of the use of the EGG beta distribution as a good proposal to these data sets.On the hypothesis testing for the weighted Lindley distribution.https://zbmath.org/1449.620302021-01-08T12:24:00+00:00"Mazucheli, Josmar"https://zbmath.org/authors/?q=ai:mazucheli.josmar"Coelho-Barros, Emílio A."https://zbmath.org/authors/?q=ai:coelho-barros.emilio-a"Louzada, Francisco"https://zbmath.org/authors/?q=ai:louzada.franciscoSummary: Recently, the two-parameter weighted Lindley distribution was proposed as a generalization for the one-parameter Lindley distribution. The proposed new distribution has an additional parameter leading to a more general form for the failure rate function. With appropriate choice of the parameter values, it is possible to model two aging classes of life distributions including bathtub and increasing hazard rates. It thus provides an alternative to many existing life distributions to modeling bathtub hazard rate. In this paper, based on a larger simulation experiment, we study the Type I error rate and power for the likelihood ratio, Wald, modified Wald, Score and Gradient tests used to distinguish the two-parameter weighted Lindley distribution from basic Lindley. With respect to size, under scenarios considered, the simulation study reveals that the likelihood ratio test performs better than the other ones. With respect to power, the Score test is found to perform better than the others.A central limit theorem for random dynamical systems.https://zbmath.org/1449.600402021-01-08T12:24:00+00:00"Lv, Kening"https://zbmath.org/authors/?q=ai:lv.kening"Zheng, Yan"https://zbmath.org/authors/?q=ai:zheng.yanSummary: In this article, we establish a central limit theorem for random dynamical systems, which is a supplement to the ergodic theory of random dynamical system. We can apply the theorem to a certain hyperbolic system to investigate the distribution of stochastic orbits, and further discuss the stochastic stability.Bernoulli difference time series models.https://zbmath.org/1449.621772021-01-08T12:24:00+00:00"Alzaid, Abdulhamid A."https://zbmath.org/authors/?q=ai:alzaid.abdulhamid-a"Omair, Maha A."https://zbmath.org/authors/?q=ai:omair.maha-a"Alhadlaq, Weaam. M."https://zbmath.org/authors/?q=ai:alhadlaq.weaam-mSummary: In this paper, we present a Bernoulli difference Markov model and a Bernoulli difference time series model based on Jacobs-Lewis mixture method. The limiting distribution of Jacobs-Lewis mixture model is obtained. The Bernoulli difference Markov model allows for positive and negative correlation. Maximum likelihood, conditional maximum likelihood, and Yule Walker methods of estimation are considered. Simulations are carried out. The paper concludes with an analysis of a real data set.On exponential decay of the variance of BLUE for the mean of a stationary sequence.https://zbmath.org/1449.600732021-01-08T12:24:00+00:00"Babayan, Nikolay"https://zbmath.org/authors/?q=ai:babayan.nikolay"Ginovyan, Mamikon S."https://zbmath.org/authors/?q=ai:ginovyan.mamikon-sThe problem is concerned with finding the best linear estimator (BLUE) \(\hat m_n\) for the unknown mean \(m\) of a model \(Y(t)=m+X(t)\), where \(X(t)\) is a zero mean, a wide-sense stationary process with a spectral density \(f\) on \([-\pi,\pi]\). That is, it is an estimate of the form \(\hat m_n= \sum_{k=0}^nc_kY(k)\), with the variance \(\mathrm{Var}(\hat m_n)=E|\hat m_n-m|^2\) under the condition \(\sum_{k=0}^nc_k=1\). In this paper, the asymptotic behavior of the variance \(\sigma_n^2(f)=\mathrm{Var}(\hat m_n,f)\) as \(n\rightarrow\infty\) is analyzed, which actually depends on the behavior of \(f\) near origin. To the question wheter it is possible to increase the order of zero of the spectral density \(f\) at the origin to obtain exponential decay of the variance \(\sigma_n^2(f)\) as \(n\rightarrow\infty\), a negative answer is given. Necessary and sufficient conditions for the exponential rate of decrease of the variance \(\sigma_n^2(f)\) are obtained and it is shown that a necessary condition for \(\sigma_n^2(f)\) to decrease to zero exponentially is that the spectral density vanishes on a set of positive Lebesgue measure in any vicinity of zero. The main results of the paper are presented after a short introduction into the topic, and the proofs are given in the last section of the paper after presenting necessary preliminaries and some auxiliary results.
Reviewer: Ilie Valuşescu (Bucureşti)\(\mu \) measure points and two property theorems of probability measures on the product spaces \(X \times X\).https://zbmath.org/1449.600042021-01-08T12:24:00+00:00"Chen, Ping"https://zbmath.org/authors/?q=ai:chen.pingSummary: In this paper, we extend the definitions of Lebesgue points of functions and sets on the Heisenberg group to the case of any separable doubling metric measure space \( (X, d, \mu)\). Based on these two generalized definitions, called \(\mu\) measure points of functions and sets, we also generalize two property theorems of probability measures on \({H^n} \times {H^n}\) and prove that the probability measures on the product spaces \(X \times X\) admit the analogous properties which are essential in solving optimal transportation problems in separable doubling metric measure spaces. These two analogous properties are also fundamental in proving and studying the existence and regularities of optimal transportation maps. The separability of the spaces and the doubling property of measures are mainly used in our proofs.Ruin probability for discrete risk processes.https://zbmath.org/1449.600082021-01-08T12:24:00+00:00"Geček Tuđen, Ivana"https://zbmath.org/authors/?q=ai:gecek-tuden.ivanaA discrete time risk process \(X(n), n=0,1,\dots\), is defined by the formula \(X(n):=n-C(n)\), where \(C(n)\) is a sum of \(m\ge 2\) independent random walks \(C^i(n)\) satisfying \(\mathbb{E}(C^i(1))<\infty\). A generalization to a perturbed discrete time risk process can be obtained by adding a perturbation, presented by an upwards skip-free random walk (or, right-continuous random walk), i.e., a random walk with increments less than or equal to \(1\).
Author's abstract: ``We study the discrete time risk process modelled by the skip-free random walk and derive results connected to ruin probability and crossing a fixed level for this type of process. We use a method relying on the classical ballot theorems to derive the results for crossing a fixed level and compare them to the results known for the continuous time version of the risk process. We generalize this model by adding a perturbation and, still relying on the skip-free structure on that process, we generalize the previous results on crossing the fixed level for the generalized discrete time risk process. We further derive the famous Pollaczek-Khinchine type formula for this generalized process, using the decomposition of the supremum of the dual process at some special instants of time.''
Reviewer: Ljuben Mutafchiev (Sofia)Stability of a deterministic and stochastic SIRS epidemic model with saturated incidence rate.https://zbmath.org/1449.341602021-01-08T12:24:00+00:00"Wang, Laiquan"https://zbmath.org/authors/?q=ai:wang.laiquan"Xamxinur, Abdurahman"https://zbmath.org/authors/?q=ai:xamxinur.abdurahmanSummary: We consider a deterministic and stochastic SIRS epidemic model with saturated incidence rate. We calculate the basic reproduction number \(R_0\) for the stochastic model and obtain the global existence and positivity of the unique solution. Under suitable conditions on the intensity of the noise perturbation, we prove the \(p\)-th exponential stability of the disease free equilibrium by using the differential operator and Itô's formula. We also discuss the asymptotic behavior of the solution of the stochastic model around the equilibrium of the deterministic model. Finally, we analyze the effect of the noise perturbation with respect to the stability of the stochastic model.PDF control of nonlinear stochastic systems based on MGC method.https://zbmath.org/1449.932442021-01-08T12:24:00+00:00"Yang, Hengzhan"https://zbmath.org/authors/?q=ai:yang.hengzhan"Fu, Yueyuan"https://zbmath.org/authors/?q=ai:fu.yueyuan"Gao, Song"https://zbmath.org/authors/?q=ai:gao.song"Qian, Fucai"https://zbmath.org/authors/?q=ai:qian.fucaiSummary: For nonlinear stochastic systems, it is difficult to meet the actual control requirements by taking the low-order statistical characteristics such as mean and variance as the research objects, and the higher order statistical characteristics need to be considered. The probability density function (PDF) contains complete statistical characteristics, therefore, PDF control can achieve effective control of all moments. In this paper, aiming at the nonlinear stochastic system excited by Gaussian white noise, the Fokker-Planck-Kolmogrov (FPK) equation is taken as the research tool, and a PDF control method based on the multi-Gaussian closure (MGC) method is proposed. Firstly, according to the shape of the target PDF, a PDF superimposed by multiple Gaussian PDFs is constructed. Then, an optimization problem is built to make the PDF approximate the target PDF. Furthermore, the state equation of the controlled system is obtained by solving the FPK equation. Finally, the control function is calculated according to the original state equation, and the tracking control of the target PDF is implemented. Simulation results of different target PDFs show the feasibility and the effectiveness of the proposed method.Sobolev-type fractional stochastic differential equations driven by fractional Brownian motion with non-Lipschitz coefficients.https://zbmath.org/1449.601122021-01-08T12:24:00+00:00"Zhan, Wentao"https://zbmath.org/authors/?q=ai:zhan.wentao"Li, Zhi"https://zbmath.org/authors/?q=ai:li.zhi|li.zhi.1Summary: In this paper, we are concerned with the existence and uniqueness of mild solution for a class of nonlinear fractional Sobolev-type stochastic differential equations driven by fractional Brownian motion with Hurst parameter \(H \in (1/2, 1)\) in Hilbert spaces. We obtain the required result by using semigroup theory, stochastic analysis principle, fractional calculus and Picard iteration techniques with some non-Lipschitz conditions.SIS epidemic propagation on hypergraphs.https://zbmath.org/1449.051932021-01-08T12:24:00+00:00"Bodó, Ágnes"https://zbmath.org/authors/?q=ai:bodo.agnesSummary: Mathematical modeling of epidemic propagation on hypergraphs is considered in this paper. The goal is to model the community structure with greater accuracy and to describe the dependence of the infection pressure on the number of infected neighbours with a nonlinear function. The master equation describing the process is derived for an arbitrary hypergraph. The mean-field equations are introduced as an approximation to the master equation and are compared against the stochastic simulations. Simulation results can be used to analyze the effects of the hypergraph structure and the model parameters.Analysis of optimal strategies for customers in the \(\mathrm{M}/\mathrm{M}/1\) queue with a single vacation and setup time.https://zbmath.org/1449.601352021-01-08T12:24:00+00:00"Tian, Ruiling"https://zbmath.org/authors/?q=ai:tian.ruiling"Wang, Yali"https://zbmath.org/authors/?q=ai:wang.yaliSummary: Customers' equilibrium strategies and socially optimal balking strategies are studied in Markovian queues with a single exponential vacation and setup time. Based on the state information of the partially observable system, customers can only observe the server's state when they arrive at the system. According to the reward-cost structure, the customer's profit function and social benefit function are obtained. Then, comparing the socially optimal strategy numerically, the equilibrium strategies are determined.Analysis of waiting time of \(\mathrm{M}/\mathrm{M}/1/m + 1\) queue with customer interjections and balking.https://zbmath.org/1449.601402021-01-08T12:24:00+00:00"Zhang, Yuanyuan"https://zbmath.org/authors/?q=ai:zhang.yuanyuan"Wu, Wenqing"https://zbmath.org/authors/?q=ai:wu.wenqing"Tang, Yinghui"https://zbmath.org/authors/?q=ai:tang.yinghuiSummary: This paper studies the waiting times of an \(\mathrm{M}/\mathrm{M}/1/m + 1\) queue with customer interjections and balking. We first classify the arrival of customers into two categories. The first one joins at the end of the queue called the normal customer, while the second category of customer tries to take place between the queue and occupy a position as close to the front of the queue as possible which is called interjecting customer. Employing the exponential distribution, the Laplace-Stieltjes transform and the formula of the total probability, we determine the waiting time of a customer in position \(n\), the waiting time of a normal customer, and the waiting time of an interjecting customer, and also discuss the impact system parameters on these measures.Stability of a stochastic SIRI model with saturated incidence and relapse.https://zbmath.org/1449.341542021-01-08T12:24:00+00:00"Mu, Yuguang"https://zbmath.org/authors/?q=ai:mu.yuguang"Xu, Rui"https://zbmath.org/authors/?q=ai:xu.rui|xu.rui.2|xu.rui.3|xu.rui.1Summary: In this paper, a stochastic SIRI epidemiological model with saturation incidence and relapse is investigated. Firstly, we show that there exists a unique global positive solution of the stochastic system. Then we discuss the stability of the disease-free equilibrium state and show the extinction of epidemics by using Lyapunov functions. Subsequently, a sufficient condition for persistence is obtained in the mean of the disease. Finally, some numerical simulations are carried out to confirm the analytical results.Long-term behavior of stochastic interest rate models with jumps and memory.https://zbmath.org/1449.911742021-01-08T12:24:00+00:00"Zhou, Wenxin"https://zbmath.org/authors/?q=ai:zhou.wen-xin"Diao, Yifan"https://zbmath.org/authors/?q=ai:diao.yifan"Li, Manman"https://zbmath.org/authors/?q=ai:li.manmanSummary: In order to better reflect the time dependence of the cyclical economic environment or the future monetary forecasts, a stochastic interest rate model is proposed by using a martingale pricing method and related mathematical tools. By assuming that the short-term interest rate model has a random reversion level, we consider three characteristics of the stochastic interest rate model, namely, the delay, jump and time dependence of reversion level. At the same time, based on the overnight data of Shanghai interbank offered rate, the WLS regression analysis method is used to conduct an empirical study on the CIR stochastic interest rate model. It is found that the fitted curve tends to be horizontal. Results show that the long-term benefits almost certainly converge to a random reversion level.When the sum is a minimal sufficient statistics for the parameter of a discrete distribution with finite values?https://zbmath.org/1449.620052021-01-08T12:24:00+00:00"Zhang, Yingying"https://zbmath.org/authors/?q=ai:zhang.yingying"Rong, Tengzhong"https://zbmath.org/authors/?q=ai:rong.tengzhong"Li, Manman"https://zbmath.org/authors/?q=ai:li.manmanSummary: We prove three theorems for iid discrete random variables taking two values, three values, and \(k\) \((3 \le k < \infty)\) values by using the technique of indicator function. Under some specifications of the probabilities, we prove that the sum is a minimal sufficient statistics for the unknown parameter of interest of the discrete random variable taking two values, three values, and \(k\) \((3 \le k < \infty)\) values. For the dice example, a figure shows that the specifications of the six probabilities are between \(0\) and \(1\) and their sum is \(1\), and a fair dice is possible.Pricing of arithmetic average Asian option under the fractional jump diffusion Heston model.https://zbmath.org/1449.911602021-01-08T12:24:00+00:00"Sun, Yudong"https://zbmath.org/authors/?q=ai:sun.yudong"Tian, Jingren"https://zbmath.org/authors/?q=ai:tian.jingren"Chen, Ying"https://zbmath.org/authors/?q=ai:chen.ying.1|chen.ying.2|chen.yingSummary: In the fractional jump diffusion environment, some results about the Heston financial asset model are studied. By using the Gronwall inequality, the \({L^p}\) boundedness and continuity of Heston financial asset model are given. In addition, the stochastic grid of Heston financial asset model is given, and the price of arithmetic average Asian option is studied by Monte Carlo simulation.Geometric representations of multivariate skewed elliptically contoured distributions.https://zbmath.org/1449.600212021-01-08T12:24:00+00:00"Richter, Wolf-Dieter"https://zbmath.org/authors/?q=ai:richter.wolf-dieter"Venz, John"https://zbmath.org/authors/?q=ai:venz.johnSummary: We derive a wide class of geometric representation formulas for multivariate skewed elliptically contoured distributions and show in a unified geometric way how some of them are related to stochastic representations known in the literature. Furthermore, we make use of the geometric measure representation to explore independence between collections of components of accordingly distributed random vectors, and to investigate contour plots of skewed normal densities from a geometric viewpoint.Hausdorff dimension for range and graph of multi-parameter operator stable Lévy processes.https://zbmath.org/1449.280062021-01-08T12:24:00+00:00"Chen, Xiaoping"https://zbmath.org/authors/?q=ai:chen.xiaoping"Lin, Huonan"https://zbmath.org/authors/?q=ai:lin.huonanSummary: The lower Hausdorff dimension results for the range and the graph of multi-parameter operator stable Lévy processes are established. The consequences are completely determined by the eigenvalues of its exponent matrix.Cramer-type moderate deviations for a stationary Ornstein-Uhlenbeck process under discrete observations.https://zbmath.org/1449.600392021-01-08T12:24:00+00:00"Liu, Hui"https://zbmath.org/authors/?q=ai:liu.hui|liu.hui.3|liu.hui.1|liu.hui.2|liu.hui.4"Jiang, Hui"https://zbmath.org/authors/?q=ai:jiang.huiSummary: This paper considers the asymptotic properties of the drift estimations of a Ornstein-Uhlenbeck process under discrete observations. By using the deviation properties of multiple Wiener-Itô integrals and asymptotic analysis techniques, Cramer-type moderate deviations of the estimators are obtained. For applications, a test statistic which can be used to construct confidence intervals and rejection regions in the hypothesis testing for the drift coefficient is proposed. It is shown that the type II errors tend to zero exponentially.On skewed continuous \(l_{n,p}\)-symmetric distributions.https://zbmath.org/1449.600302021-01-08T12:24:00+00:00"Arellano-Valle, Reinaldo B."https://zbmath.org/authors/?q=ai:arellano-valle.reinaldo-boris"Richter, Wolf-Dieter"https://zbmath.org/authors/?q=ai:richter.wolf-dieterSummary: The general methods from theory of skewed distributions and from the theory of geometric and stochastic representations of \(l_{n,p}\)-symmetric distributions are combined here to introduce skewed continuous \(l_{n,p}\)-symmetric distributions.A class of strong deviation theorems for the random fields associated with bifurcating Markov chains indexed by a binary tree.https://zbmath.org/1449.600562021-01-08T12:24:00+00:00"Min, Fan"https://zbmath.org/authors/?q=ai:min.fan"Yang, Weiguo"https://zbmath.org/authors/?q=ai:yang.weiguoSummary: In this paper, we study a class of strong deviation theorems for random fields which are associated with bifurcating Markov chains indexed by a binary tree. By introducing the asymptotic logarithmic likelihood ratio as a measure of the deviation between the arbitrary random fields and the bifurcating Markov chains on a binary tree, and constructing the martingale, a class of strong deviation theorems for the random fields which are associated with bifurcating Markov chains indexed by a binary tree is established. As corollaries, we obtain the strong law of large numbers and the AEP for the bifurcating Markov chains indexed by a binary tree.A comparison of bounds on three sets of copulas with given degree of non-exchangeability.https://zbmath.org/1449.621132021-01-08T12:24:00+00:00"Xu, Fuxia"https://zbmath.org/authors/?q=ai:xu.fuxia"Wang, Yingjie"https://zbmath.org/authors/?q=ai:wang.yingjieSummary: We establish best-possible supremum bounds of copulas with the degree of non-exchangeability \(t = 3/4\), \(t = 3/5\) and \(t = 3/6 = 1/2\), and study the structures of these sets of copulas. The volumes between the upper and lower bounds are calculated to illustrate that the supremum bounds are specific practical and effective in narrowing the Frechet-Hoeffding bounds.Covariance of the marked process of censored \(\delta\)-shock model.https://zbmath.org/1449.600912021-01-08T12:24:00+00:00"Ye, Jianhua"https://zbmath.org/authors/?q=ai:ye.jianhua"Zheng, Ying"https://zbmath.org/authors/?q=ai:zheng.ying"Liu, Hua"https://zbmath.org/authors/?q=ai:liu.huaSummary: Based on the study of covariance of self-excited filtering Poisson process, the explicit expression of covariance of the marked process of censored \(\delta\) shock model is derived, which extends the theory of \(\delta\)-shock model.Characterizations of the uniform distributions based on upper record values.https://zbmath.org/1449.600242021-01-08T12:24:00+00:00"Lee, Min-Young"https://zbmath.org/authors/?q=ai:lee.min-youngSummary: We obtain two characterizations of uniform distribution based on ratios of upper record values by the properties of independence and identical distribution.The number of failed components in a conditional coherent operating system.https://zbmath.org/1449.622282021-01-08T12:24:00+00:00"Xiong, Wenjie"https://zbmath.org/authors/?q=ai:xiong.wenjie"Li, Danqing"https://zbmath.org/authors/?q=ai:li.danqing"Zhang, Zhengcheng"https://zbmath.org/authors/?q=ai:zhang.zhengchengSummary: Coherent systems are very important in reliability, survival analysis and other life sciences. In this paper, we consider the number of failed components in an \( (n-k+1)\)-out-of-\(n\) system, given that at least \(m\) \((m < k \le n)\) components have failed before time \(t\), and the system is still working at time \(t\). In this case, we compute the probability that there are exactly \(i\) working components. At first the reliability and several stochastic properties are obtained. Furthermore, we extend the results to general coherent systems with absolutely continuous and exchangeable components.An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming.https://zbmath.org/1449.601282021-01-08T12:24:00+00:00"Jin, Haibo"https://zbmath.org/authors/?q=ai:jin.haibo"Hai, Long"https://zbmath.org/authors/?q=ai:hai.long"Tang, Xiaoliang"https://zbmath.org/authors/?q=ai:tang.xiaoliangSummary: An optimal preventive maintenance strategy for multi-state systems based on an integral equation and dynamic programming is described herein. Unlike traditional preventive maintenance strategies, this maintenance strategy is formulated using an integral equation, which can capture the system dynamics and avoid the curse of dimensionality arising from complex semi-Markov processes. The linear integral equation of the system is constructed based on the system kernel. A numerical technique is applied to solve this integral equation and obtain all of the mean elapsed times from each reliable state to each unreliable state. An analytical approach to the optimal preventive maintenance strategy is proposed that maximizes the expected operational time of the system subject to the total maintenance budget based on dynamic programming in which both backward and forward search techniques are used to search for the local optimal solution. Finally, numerical examples concerning two different scales of systems are presented to demonstrate the performance of the strategy in terms of accuracy and efficiency. Moreover a sensitivity analysis is provided to evaluate the robustness of the proposed strategy.Moment and tail inequalities based on uncertain variable sequences.https://zbmath.org/1449.600282021-01-08T12:24:00+00:00"Wang, Zhigang"https://zbmath.org/authors/?q=ai:wang.zhigang.1"Wang, Sizhe"https://zbmath.org/authors/?q=ai:wang.sizhe"Cai, Baiguang"https://zbmath.org/authors/?q=ai:cai.baiguang"He, Wenfeng"https://zbmath.org/authors/?q=ai:he.wenfengSummary: The emphasis in this paper is mainly on some important moments and tails inequalities of uncertain variables within the framework of uncertainty theory. Some uncertain moments and tails inequalities which are consistent with random moments and tails inequalities, for example, \(r\)-order inequality, Kolmogorov's inequality and other important inequalities are presented. The investigations help to lay theoretical foundations for the development of uncertainty theory.On asymptotic distribution of Koenker-Bassett estimator of the parameter of linear regression model with strongly dependent noise.https://zbmath.org/1449.621572021-01-08T12:24:00+00:00"Ivanov, O. V."https://zbmath.org/authors/?q=ai:ivanov.aleksandr-vladimirovich"Kaptur, N. V."https://zbmath.org/authors/?q=ai:kaptur.n-v"Savych, I. M."https://zbmath.org/authors/?q=ai:savych.i-mSummary: Asymptotic properties of Koenker-Bassett estimators of linear regression model parameters with discrete observation time and random noise being nonlinear local transformation of Gaussian stationary time series with singular spectrum are studied. he goal of the work lies in obtaining the requirements to regression function and time series that simulates the random noise, under which the Koenker-Bassett estimators of regression model parameters are asymptotically normal. Linear regression model with discrete observation time and bounded open convex parametric set is the object of the studying. Asymptotic normality of unknown parameters Koenker-Bassett estimators are obtained. For getting these results complicated concepts of time series theory and time series statistics have been used, namely: local transformation of Gaussian stationary time series, stationary time series with singular spectral density, spectral measure of regression function, admissibility of singular spectral density of stationary time series in relation to this measure, expansions by Chebyshev-Hermite polynomials of the transformed Gaussian time series values and its covariances, central limit theorem for weighted sums of the values of such a local transformation.Estimation of distribution of suprema of a strictly \(\varphi\)-sub-Gaussian quasi shot noise process.https://zbmath.org/1449.600752021-01-08T12:24:00+00:00"Vasylyk, O. I."https://zbmath.org/authors/?q=ai:vasylyk.olga-iSummary: In this paper, the author studies properties of a strictly \(\varphi\)-sub-Gaussian quasi shot noise process \(X(t) = \int_{-\infty}^{+\infty} g(t, u)d\xi(u), t\in\mathbb{R}\), generated by the process \(\xi(t)\) and the response function \(g(t,x)\). New estimates for distributions of suprema of such processes are derived. An example of application of the obtained results is given.Splitting algorithm for solving a class of stochastic Hamiltonian systems.https://zbmath.org/1449.601162021-01-08T12:24:00+00:00"Li, Xinyu"https://zbmath.org/authors/?q=ai:li.xinyu"Chen, Xumei"https://zbmath.org/authors/?q=ai:chen.xumei"Yue, Hua"https://zbmath.org/authors/?q=ai:yue.huaSummary: We applied a symmetrical splitting algorithm to decompose the solution of a \(2n\)-dimensional Stratonovich-type stochastic Hamiltonian system into two \(n\)-dimensional subsystems in turn, so as to reduce the dimension and simplify the computation. We obtained the global mean square first order convergence of this method by error analysis. The numerical examples verify the correctness of the theoretical results.Pricing dynamic fund protections under a jump-diffusion model with stochastic protection level.https://zbmath.org/1449.911552021-01-08T12:24:00+00:00"Lv, Wenxin"https://zbmath.org/authors/?q=ai:lv.wenxin"Dong, Yinghui"https://zbmath.org/authors/?q=ai:dong.yinghui"Wu, Sang"https://zbmath.org/authors/?q=ai:wu.sang"Xu, Chao"https://zbmath.org/authors/?q=ai:xu.chaoSummary: Dynamic fund protection provides a guarantee that the account value of the investor never drops below a barrier over the investment period. This article considers the price of the dynamic guaranteed funds with a stochastic protection level under a jump-diffusion model. The explicit formula for the Laplace transform of the dynamic fund protection can be obtained under a hyper-exponential jump-diffusion process. By using the Gaver-Stehfest algorithm, we present some numerical results for the value of the dynamic fund protection.Pricing of perpetual corporate debt with bankruptcy reorganization in a double exponential jump-diffusion model.https://zbmath.org/1449.911882021-01-08T12:24:00+00:00"Lin, Jianwei"https://zbmath.org/authors/?q=ai:lin.jianwei"Li, Huimin"https://zbmath.org/authors/?q=ai:li.huiminSummary: In order to better deal with the risk of the asset jump and the strategy of bankruptcy reorganization faced by the company, based on a structural method and the optimal stopping technique, this paper considers the pricing problem of the perpetual corporate debt with the bankruptcy reorganization scheme of debt-equity swap in a double jump-diffusion model. Pricing analytical solutions of the perpetual corporate debt and the equity are obtained by a differential equation method. Furthermore, this paper also presents a closed-form solution of the optimal bankruptcy boundary and a nonlinear equation satisfied by the optimal coupon. Finally, the numerical results show that the more volatile the corporate asset value is, the more shareholders can gain from the volatile market, but the more unpopular corporate bonds will be. The lower the corporate bond value is, the lower the optimal leverage ratio will be.Local times of the solution to stochastic heat equation with fractional noise.https://zbmath.org/1449.601232021-01-08T12:24:00+00:00"Wang, Zhi"https://zbmath.org/authors/?q=ai:wang.zhi"Yan, Litan"https://zbmath.org/authors/?q=ai:yan.litan"Yu, Xianye"https://zbmath.org/authors/?q=ai:yu.xianyeSummary: In this paper, we study the collision and intersection local times of the solution to stochastic heat equation with additive fractional noise. We mainly prove its existence and smoothness properties through local nondeterminism and chaos expansion.Convergence of controlled models for continuous-time Markov decision processes with constrained average criteria.https://zbmath.org/1449.903572021-01-08T12:24:00+00:00"Zhang, Wenzhao"https://zbmath.org/authors/?q=ai:zhang.wenzhao"Xiong, Xianzhu"https://zbmath.org/authors/?q=ai:xiong.xianzhuSummary: This paper attempts to study the convergence of optimal values and optimal policies of continuous-time Markov decision processes (CTMDP for short) under the constrained average criteria. For a given original model \({\mathcal{M}_\infty}\) of CTMDP with denumerable states and a sequence \(\{{\mathcal{M}_n}\}\) of CTMDP with finite states, we give a new convergence condition to ensure that the optimal values and optimal policies of \(\{{\mathcal{M}_n}\}\) converge to the optimal value and optimal policy of \({\mathcal{M}_\infty}\) as the state space \({S_n}\) of \({\mathcal{M}_n}\) converges to the state space \({S_\infty}\) of \({\mathcal{M}_\infty}\), respectively. The transition rates and cost/reward functions of \({\mathcal{M}_\infty}\) are allowed to be unbounded. Our approach can be viewed as a combination method of linear program and Lagrange multipliers.Optimal dividend strategies for the compound Poisson model with debit interest.https://zbmath.org/1449.911062021-01-08T12:24:00+00:00"Luo, Kui"https://zbmath.org/authors/?q=ai:luo.kui"Liu, Juan"https://zbmath.org/authors/?q=ai:liu.juan"Zhao, Yihui"https://zbmath.org/authors/?q=ai:zhao.yihui"Xiao, Liqun"https://zbmath.org/authors/?q=ai:xiao.liqunSummary: In this paper, we study the optimal dividend strategy for an insurance company with debit interest and dividend payments. By HJB methods, a rule for choosing the strategy that maximizes the expected accumulated discounted dividends until absolute ruin is given. Under the so called threshold strategy, we derive integro-differential equations for the expected accumulated discounted dividends until absolute ruin. Then, explicit expressions for the expected accumulated discounted dividends until absolute ruin with exponential claim amounts are obtained. Finally, based on the explicit expressions, we prove that the optimal strategy is a threshold strategy and the optimal level of threshold is also obtained.White noise driven cubic Ostrovsky equation.https://zbmath.org/1449.601092021-01-08T12:24:00+00:00"Yan, Wei"https://zbmath.org/authors/?q=ai:yan.wei"Zhang, Qiaoqiao"https://zbmath.org/authors/?q=ai:zhang.qiaoqiaoSummary: This paper mainly studies the Cauchy problem for the white noise driven cubic Ostrovsky equation. When \({u_0} (\cdot, \omega) \in {H^s} (R)\) (a.e. \(\omega \in \Omega\)), \(s \ge \frac{1}{4}\) and \(\Phi \in L_2^{0,s}\), the data are \({F_0}\) measurable. By using the Fourier restriction norm method and trilinear estimates as well as the fixed point theorem, we obtain a local well-posedness result.Numerical methods for system parabolic variational inequalities from regime-switching American option pricing.https://zbmath.org/1449.911912021-01-08T12:24:00+00:00"Xing, Jie"https://zbmath.org/authors/?q=ai:xing.jie"Ma, Jingtang"https://zbmath.org/authors/?q=ai:ma.jingtangSummary: The aim of this paper is to study the convergence rates of the trinomial tree methods (TTMs) and perturbed finite difference methods (PFDMs) for system parabolic variational inequalities which govern the value function of regime-switching American option. This paper has three-fold contributions: (i) It establishes the higher-order equivalence between the TTMs and the PFDMs for the regime-switching American options; (ii) It proves the regularities of the solutions to the system of parabolic variational inequalities governing the price of the American options, and studies the comparison principles and the penalty methods. These results are used to prove the convergence rates of the PFDMs; (iii) It proves the convergence rates of the PFDMs for the system of parabolic variational inequalities governing the price of the American options. The convergence rates of the TTMs are obtained by the higher-order equivalence between the TTMs and the PFDMs and the convergence theory for the PFDMs.Initial value randomization of nonlinear evolution equations.https://zbmath.org/1449.370502021-01-08T12:24:00+00:00"Huang, Jianhua"https://zbmath.org/authors/?q=ai:huang.jianhua"Yan, Wei"https://zbmath.org/authors/?q=ai:yan.weiSummary: This paper aims to introduce some nonlinear evolution equations. Firstly, we present Schrödinger equations with random data, KdV equation with random data, and wave equation with random data. Then we give the harmonic analysis tools which are used to solve random data problem. At last, some unsolved problems related to random data are presented.Comparative study of Bhattacharyya and Kshirsagar bounds in Burr XII and Burr III distributions.https://zbmath.org/1449.620322021-01-08T12:24:00+00:00"Nayeban, S."https://zbmath.org/authors/?q=ai:nayeban.samira"Rezaei Roknabadi, A. H."https://zbmath.org/authors/?q=ai:roknabadi.abdol-hamid-rezaei"Mohtashami Borzadaran, G. R."https://zbmath.org/authors/?q=ai:mohtashami-borzadaran.gholam-rezaSummary: A set of families of distributions which might be useful for fitting data was described by \textit{I. W. Burr} [Ann. Math. Stat. 13, 215--232 (1942; Zbl 0060.29602)]. Among them, the families type XII (Burr XII) and type III (Burr III), have gathered special attention in physics, actuarial studies, reliability and applied statistics. Estimating a wide range of functions of their parameters such as reliability, hazard rate and mode, under various conditions, have been done. But, the variances of the estimators are not considered precisely yet.
In this paper, we consider two well-known lower bounds for the variance of any unbiased estimator, which are Bhattacharyya and Kshirsagar bounds for the Burr XII and Burr III distributions. In these distributions, the general forms of the Bhattacharyya and Kshirsagar matrices are obtained. In addition, we evaluate different Bhattacharyya and Kshirsagar bounds for the variance of any unbiased estimator of the reliability, hazard rate, mode and median due to Burr XII and Burr III distributions and conclude that in each case, which bound has higher convergence and is better to use. Also via some figures, we compare the two bounds with bootstrap method in approximating the variance of the unbiased estimator of the reliability, median and mean of the Burr XII distributions.Strong convergence of the semi-implicit Euler method for a kind of stochastic Volterra integro-differential equations.https://zbmath.org/1449.601142021-01-08T12:24:00+00:00"Gao, Jianfang"https://zbmath.org/authors/?q=ai:gao.jianfang"Ma, Shufang"https://zbmath.org/authors/?q=ai:ma.shufang"Liang, Hui"https://zbmath.org/authors/?q=ai:liang.huiSummary: This paper is mainly concerned with the strong convergence analysis of the semi-implicit Euler method for a kind of stochastic Volterra integro-differential equations (SVIDEs). The solvability and the mean-square boundedness of numerical solutions are presented. In view of the properties of the Itô integral, different from the known stochastic problems, it is proved that the strong convergence order of the semi-implicit Euler method is 1, although the approximation order of the Itô integral is 0.5. The theoretical results are illustrated by extensive numerical examples.Percolation of the prolate ellipsoids of rotation in the continuum.https://zbmath.org/1449.601412021-01-08T12:24:00+00:00"Buzmakova, Mariya Mikhaĭlovna"https://zbmath.org/authors/?q=ai:buzmakova.mariya-mikhailovnaSummary: Continuum percolation of hard prolate ellipsoids of rotation with permeable shell are investigated. It is the model of phase transition sol-gel. Ellipsoids are located in the cube randomly. For each set of parameters 100 tests are spent. For each test the finding of the percolation cluster is the main task. The fraction of the packing for which the probability of the percolation cluster appearance is equal 0.5, is called a percolation threshold. The value of the percolation threshold corresponds to the gel point. The dependence of value of the percolation threshold on thickness of permeable shell and aspect ratio is obtained. In addition to the percolation threshold, other characteristics of the model are obtained, such as: the size distribution of clusters, the average cluster size, the strength and the fractal dimension of the percolation cluster, the average value and the distribution of neighbors of an element, and critical exponents.Using of special Hermite functions for investigation of power properties of Grubbs' criterion.https://zbmath.org/1449.621102021-01-08T12:24:00+00:00"Shiryaeva, Lyudmila Konstantinovna"https://zbmath.org/authors/?q=ai:shiryaeva.lyudmila-konstantinovnaSummary: We consider a normal sample with a single upper outlier. A distribution of studentized form of outlier's deviation from the sample mean is obtained. This distribution uses Hermite special functions with negative integer-valued index. The integral relationships for David's power measures of Grubbs criteria are obtained. We discuss the case, when Grubbs statistic is the likelihood-ratio statistic. We find the maximal deviation of power function for Grubbs criteria from the probability that the contaminant is the outlier and it is identified as discordant. We receive the region of critical values of Grubbs statistic, where the second power measure of David equals to the third and forth power measures of David. We make calculations of power function for Grubbs criteria in the case of normal samples with a single upper outlier with the right shift. The results of calculations are similar to the theoretically expected facts.Approximation of smooth stable invariant manifolds for stochastic partial differential equations.https://zbmath.org/1449.370342021-01-08T12:24:00+00:00"Guo, Zhongkai"https://zbmath.org/authors/?q=ai:guo.zhongkai"Yan, Xingjie"https://zbmath.org/authors/?q=ai:yan.xingjie"Yang, Xinguang"https://zbmath.org/authors/?q=ai:yang.xinguangSummary: Invariant manifolds are complicated random sets used for describing and understanding the qualitative behavior of nonlinear dynamical systems. The purpose of the present paper is to try to approximate smooth stable invariant manifolds for a type of stochastic partial differential equations with multiplicative white noise near the fixed point. Two examples are given to illustrate our results.Stability of \(\theta\)-Heun methods for nonlinear stochastic delay differential equations.https://zbmath.org/1449.601152021-01-08T12:24:00+00:00"Jiang, Qian"https://zbmath.org/authors/?q=ai:jiang.qian"Zhang, Yindi"https://zbmath.org/authors/?q=ai:zhang.yindi"Wang, Caixia"https://zbmath.org/authors/?q=ai:wang.caixiaSummary: The \(\theta\)-Heun method for solving stochastic delay differential equations is presented. Based on this method, sufficient conditions for general nonlinear stochastic delay differentials equations are obtained for the MS-stability, GMS-stability and exponential stability of mean square. Compared with the Heun method, the \(\theta\)-Heun method has less restriction on the step size. The numerical experiments at the end of the paper verify the relevant conclusions.Generalized stochastic model of creep and creep rupture beams in pure bending and its application to the estimation of reliability.https://zbmath.org/1449.742012021-01-08T12:24:00+00:00"Radchenko, Vladimir Pavlovich"https://zbmath.org/authors/?q=ai:radchenko.vladimir-pavlovich"Shershneva, Mariya Viktorovich"https://zbmath.org/authors/?q=ai:shershneva.mariya-viktorovich"Tsvetkov, Vitaliĭ Vladimirovich"https://zbmath.org/authors/?q=ai:tsvetkov.vitalii-vladimirovichSummary: Generalized stochastic model of creep and creep rupture beams under pure bending in terms ``generalized load'', ``generalized displacement'', ``time'' is offered. Beam is considered as a single entity (the specific model). The complete analogy between the curves of uniaxial creep sample under constant stress and generalized creep curves beams in the curvature of the beam coordinates ``curvature beams -- time'' under the constant bending moment is determined. On the basis of this analogy the stochastic equation of state beam is formed. Method of reliability estimating of the beams bending under creep on parametric criteria of failure in a significant scatter of the data is developed. Calculation results and recommendations for lifelength assigning are presented.On a question of limiting distribution of series in random binary sequence.https://zbmath.org/1449.601192021-01-08T12:24:00+00:00"Barvinok, Vitaliĭ Alekseevich"https://zbmath.org/authors/?q=ai:barvinok.vitalii-alekseevich"Bogdanovich, Valeriĭ Iosifovich"https://zbmath.org/authors/?q=ai:bogdanovich.valerii-iosifovich"Plotnikov, Andreĭ Nikolaevich"https://zbmath.org/authors/?q=ai:plotnikov.andrei-nikolaevichSummary: Limiting forms of distribution of length of the maximum series of successes in random binary sequences, formed in Bernulli-Markov's chain and in Polya's scheme which is an equivalent to local trends of a time series of strictly stationary process are investigated. More simple and added proofs of theorems of the law of the big numbers for series of both types are offered. For series of the second type, the effect is established of the cyclic bimorphism of the limiting law with degeneration on one of the phases and the convergence according to the probability on set no more, than four consequent values of the natural series.Hypothesis testing for ergodicity of inhomogeneous diffusions.https://zbmath.org/1449.621042021-01-08T12:24:00+00:00"Shao, Jin"https://zbmath.org/authors/?q=ai:shao.jin"Jiang, Hui"https://zbmath.org/authors/?q=ai:jiang.huiSummary: For a class of time inhomogeneous diffusions, to test their ergodicity, we construct a suitable statistic. Then the unbiasedness and consistency of our test could be proved. Moreover, we also apply our main results to \(\alpha\)-Wiener bridge.Optimum management of the network of city bus routes based on a stochastic dynamic model.https://zbmath.org/1449.900612021-01-08T12:24:00+00:00"Wang, Shi'an"https://zbmath.org/authors/?q=ai:wang.shian"Ahmed, N. U."https://zbmath.org/authors/?q=ai:ahmed.nasir-uddinSummary: In this paper, we develop a stochastic dynamic model for the network of city bus routes subject to resource and other practical constraints. We define an objective function on the basis of four terms: fuel cost, operating cost, customers waiting time, and revenue of the bus company. Hereafter, an optimization problem is formulated and solved by use of nonlinear integer programming. If the technique presented here is implemented, it is expected to boost the bus company's revenue, reduce waiting time and therefore promote customer satisfaction. A series of numerical experiments is carried out and the corresponding optimization problems are addressed giving the optimal number of buses allocated to each of the bus routes in the network. Since the dynamic model proposed here can be applied to any network of bus routes, it is believed that the procedure developed in this paper is of great potential for both the city bus company and the customers.Cubic transmuted uniform distribution: an alternative to beta and Kumaraswamy distributions.https://zbmath.org/1449.600202021-01-08T12:24:00+00:00"Rahman, Md. Mahabubur"https://zbmath.org/authors/?q=ai:rahman.md-mahabubur"Al-Zahrani, Bander"https://zbmath.org/authors/?q=ai:al-zahrani.bander-m"Shahbaz, Saman Hanif"https://zbmath.org/authors/?q=ai:shahbaz.saman"Shahbaz, Muhammad Qaiser"https://zbmath.org/authors/?q=ai:shahbaz.muhammad-qaiserSummary: In this article, a new cubic transmuted (CT) family of distributions has been proposed by adding one more parameter. We have introduced cubic transmuted uniform (CTU) distribution by using the proposed class. We have also provided a detail description of the statistical properties of the proposed CTU distribution along with its estimation and real-life application.Equilibrium analysis in the M/M/1 queue with two types of breakdowns.https://zbmath.org/1449.900832021-01-08T12:24:00+00:00"Zhang, Songtai"https://zbmath.org/authors/?q=ai:zhang.songtai"Xu, Xiuli"https://zbmath.org/authors/?q=ai:xu.xiuliSummary: This paper considers the equilibrium behavior of customers in a Markovian queue with two types of breakdowns, where the normal server can get a breakdown at any time. The system does not admit a new arrival once a breakdown happens, and there may exist two independent types of breakdowns: (1) partial breakdown: the server continues to serve the customers on spot at a low rate and is repaired when the system is empty; (2) full breakdown: the server stagnates service and is repaired immediately. When the repair is over, new arrivals will be accepted. Assuming that all the customers have the option of joining or balking in order to maximize their own benefits and based on a linear reward-cost structure, we analyze the equilibrium joining strategies of the customers and the average social benefits of the system in the fully observable case and the almost unobservable case, respectively. On this basis, the effect of several parameters on customers' strategic behavior is presented by some numerical examples.The conditions of existence with probability one of generalized solutions of Cauchy problem for the heat equation with a random right part.https://zbmath.org/1449.352462021-01-08T12:24:00+00:00"Tylyshchak, A. I."https://zbmath.org/authors/?q=ai:tylyshchak.a-iSummary: The subject of this work is at the intersection of two branches of mathematics: mathematical physics and stochastic processes. The influence of random factors should often be taken into account in solving problems of mathematical physics. The heat equation with random conditions is a classical problem of mathematical physics. In this paper we consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on a line with a random right part. We consider the right part as a random function of the space \(\text{Sub}_{\varphi}(\Omega)\). The conditions of existence with probability one generalized solution of the problem are investigated. Using this results one can construct modeless, which approximate solutions of such equations with given accuracy and reliability in the uniform metric.On the convergence rate for the estimation of impulse response function in the space \(L_p(T)\).https://zbmath.org/1449.600742021-01-08T12:24:00+00:00"Rozora, I. V."https://zbmath.org/authors/?q=ai:rozora.iryna-vSummary: The problem of estimation of a stochastic linear system has been a matter of active research for the last years. One of the simplest models considers a `black box' with some input and a certain output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The sphere of applications of these models is very extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models). In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function. We assume that impulse function is square-integrable. Input signal is supposed to be Gaussian stationary stochastic process with known spectral density. A sample input-output cross-correlogram is taken as an estimator of the response function. An upper bound for the tail of the distribution of the estimation error is found that gives a convergence rate of estimator to impulse response function in the space \(L_p(T)\).Equivalence between tails, grand Lebesgue spaces and Orlicz norms for random variables without Cramer's condition.https://zbmath.org/1449.600762021-01-08T12:24:00+00:00"Kozachenko, Yu."https://zbmath.org/authors/?q=ai:kozachenko.yuriy-v"Ostrovsky, E."https://zbmath.org/authors/?q=ai:ostrovskii.e-i"Sirota, L."https://zbmath.org/authors/?q=ai:sirota.leaSummary: We propose non-asymptotical pairwise bilateral exact up to multiplicative constant interrelations between the tail behaviour, moments (grand Lebesgue spaces) norm and Orlicz's norm for random variables, which does not satisfy in general the Cramer condition.Analysis of a \(k/n (g)\) system with expert repairman's multiple vacations and replaceable repair facility.https://zbmath.org/1449.900902021-01-08T12:24:00+00:00"Zhang, Yuanyuan"https://zbmath.org/authors/?q=ai:zhang.yuanyuan"Wu, Wenqing"https://zbmath.org/authors/?q=ai:wu.wenqingSummary: This paper studies a repairable \(k/n (G)\) system with expert repairman's multiple vacations and replaceable repair facility. The expert repairman leaves for a vacation when there is no broken component. Once an operating component breaks down during his vacation period, it is repaired immediately by an ordinary repairman. The ordinary repairman becomes inactivated when there is no broken component or the expert returns from his vacation. By using the Markov process theory and the matrix solution method, we obtain the transient and the stationary of the system availability and the rate of occurrence of failures, the system reliability, the mean time to system failure, and the probability that the repair facility is being replaced. Further, we discuss the time-dependent behavior of these reliability measures under different initial states. Finally, special cases of the system are presented to show the correctness of our results.Transient and equilibrium solutions of queue length distribution for \(\mathrm{M}/\mathrm{G}/1\) queueing system with \({\mathrm{Min}}(N, D, V)\)-policy and single server vacation.https://zbmath.org/1449.601362021-01-08T12:24:00+00:00"Wang, Min"https://zbmath.org/authors/?q=ai:wang.min.1|wang.min.2|wang.min"Tang, Yinghui"https://zbmath.org/authors/?q=ai:tang.yinghuiSummary: This paper considers the \(\mathrm{M}/\mathrm{G}/1\) queueing system with single server vacation which can be interrupted immediately according to the \(\mathrm{Min}(N, D, V)\)-policy. By applying the total probability decomposition technique and the Laplace transformation, the transient and steady-state properties of the queue length from any initial state are discussed, and the Laplace transformation expression of the transient solution of queue length distribution is obtained. Moreover, we derive recursive expressions of the equilibrium solution of queue length distribution for convenient calculation. Furthermore, we propose stochastic decomposition structures of the steady-state queue length, explicit expressions for the probability distribution of the additional queue length and the corresponding results for some special cases. Finally, by numerical examples, we discuss the sensitivity of the steady state queue length distribution towards system parameters and analyze the influence of different parameters on the performance of the system.Second and third order moment inequalities for probability distributions.https://zbmath.org/1449.600272021-01-08T12:24:00+00:00"Simić, S."https://zbmath.org/authors/?q=ai:simic.sasa|simic.suzana|simic.slobodan-k|simic.srboljub-s|simic.slobodan-m|simic.slobodan-n|simic.svetlana|simic.slavkoSummary: We give refinements of some convex and log-convex moment inequalities. We derive second and third order inequalities using a special kind of algebraic positive semi-definite forms. An open problem concerning an eight parameter refinement of the third order is also stated. We suggest some applications of our results in information theory concerning relative divergence of type \(s\) and in theory of means.Stochastic permanence of solution to stochastic non-autonomous logistic equation with jumps.https://zbmath.org/1449.600972021-01-08T12:24:00+00:00"Borysenko, O. D."https://zbmath.org/authors/?q=ai:borysenko.oleksandr-d"Borysenko, D. O."https://zbmath.org/authors/?q=ai:borysenko.d-oSummary: It is investigated a non-autonomous logistic differential equation with disturbance of coefficients by white noise, centered and non-centered Poisson noises. The coefficients of equation are locally Lipschitz continuous but do not satisfy the linear growth condition. This equation describes the dynamics of population in the Verhulst model which takes into account the logistic effect: an increase of the population size produces a decrease in fertility and an increase in mortality; since resources are limited, if the population size exceeds some threshold level, the habitat cannot support the growth. The property of stochastic permanence is desirable since it means the long time survival in a population dynamics. Sufficient conditions for the stochastic permanence of population in the considered model are obtained.Optimal behavioral portfolio selection for an individual under inflation risk.https://zbmath.org/1449.911232021-01-08T12:24:00+00:00"Guo, Wenjing"https://zbmath.org/authors/?q=ai:guo.wenjing"Jiang, Haiwen"https://zbmath.org/authors/?q=ai:jiang.haiwenSummary: It is well known that inflation risk is an important factor that affects investors' making decisions. Also, the influence of investors' behavioral characteristics on portfolio selections can not be ignored. This paper discusses the problem of optimal behavioral portfolio selection for an individual under inflation risk. At first, in the financial market, we introduce an inflation-linked index bond, which can be used to hedge the inflation risk. Meanwhile, investors are assumed to be loss averse. Thus, we get the optimal individual behavioral portfolio selection model under inflation risk. Then, maximizing the expected utility of the part investor's terminal wealth exceeds the reference level, the explicit solutions for the optimal strategies and terminal wealth are derived by martingale approach, and the properties of optimal strategies are discussed by property analysis and numerical simulation. Finally, the numerical results show that the inflation risk and loss aversion indeed have a significant effect on the optimal strategies.An analogue of Sylvester's four-point problem on the sphere.https://zbmath.org/1449.600132021-01-08T12:24:00+00:00"Maehara, H."https://zbmath.org/authors/?q=ai:maehara.hiroshi"Martini, H."https://zbmath.org/authors/?q=ai:martini.horstSummary: A finite subset \(X\) of the unit sphere \(\mathbb{S}^{d-1}\) in \(\mathbb{R}^d\) is called extremal if, for every \(x\in X\), there is a hemisphere that contains \(X\setminus \{x\}\) in its interior and has \(x\) on its boundary. Let \(P\) denote the probability that a random sample of \(d+1\) points, chosen uniformly from \(\mathbb{S}^{d-1}\), is extremal. We show that \(P=1-(d+2)/2^d\).Mathematical aspect of the combinatorial game ``Mahjong''.https://zbmath.org/1449.050152021-01-08T12:24:00+00:00"Cheng, Yuan"https://zbmath.org/authors/?q=ai:cheng.yuan"Li, Chikwong"https://zbmath.org/authors/?q=ai:li.chi-kwong"Li, Sharon H."https://zbmath.org/authors/?q=ai:li.sharon-hSummary: We illustrate how one can use basic combinatorial theory and computer programming technique (Python) to analyze the combinatorial game: Mahjong. The results confirm some folklore concerning the game, and expose some unexpected results. Related results and possible future research in connection to artificial intelligence are mentioned. Readers interested in the subject may further develop the techniques to deepen the study of the game, or study other combinatorial games.Convergence rates of the semi-discrete method for stochastic differential equations.https://zbmath.org/1449.601042021-01-08T12:24:00+00:00"Stamatiou, I. S."https://zbmath.org/authors/?q=ai:stamatiou.ioannis-s"Halidias, N."https://zbmath.org/authors/?q=ai:halidias.nikolaosSummary: We study the convergence rates of the semi-discrete (SD) method originally proposed by the second author [Int. J. Comput. Math. 89, No. 6, 780--794 (2012; Zbl 1255.65020)]. The SD numerical method was originally designed mainly to reproduce qualitative properties of nonlinear stochastic differential equations (SDEs). The strong convergence property of the SD method has been proved, but except for certain classes of SDEs, the order of the method was not studied. We study the order of \(L_2\) -convergence and show that it can be arbitrarily close to 1/2. The theoretical findings are supported by numerical experiments.Pricing measure thoughts and option pricing.https://zbmath.org/1449.911612021-01-08T12:24:00+00:00"Tian, Yingxu"https://zbmath.org/authors/?q=ai:tian.yingxuSummary: A two-factor spot underlying model is studied which contributes to the derivatives pricing in energy market. This model contains a stochastic mean reverting which is described by an Ornstein-Uhlenbeck process. In practice, as opposed to the classical assumptions of the derivatives pricing in a complete market, the scholars cannot find the risk-neural measure in energy market due to the non-storable property of some traded energies like electricity. The problem is addressed on how to find a suitable pricing measure and derive the new measure dynamics of the spot price model. Then, some probabilistic properties of the new measure dynamics are explored. Meanwhile, the European call option price is computed and the future price is written on the new underlying dynamics under pricing measure. All the results are closed form.Limit theorem for perturbed random walks.https://zbmath.org/1449.600712021-01-08T12:24:00+00:00"Ngo, Hoang-Long"https://zbmath.org/authors/?q=ai:ngo.hoang-long"Peigné, Marc"https://zbmath.org/authors/?q=ai:peigne.marcSummary: We consider random walks perturbed at zero which behave like (possibly different) random walk with independent and identically distributed increments on each half lines and restarts at 0 whenever they cross that point. We show that the perturbed random walk, after being rescaled in a proper way, converges to a skew Brownian motion whose parameter is defined by renewal functions of the simple random walk and the transition probabilities from 0.A novel bat algorithm based on cross boundary learning and uniform explosion strategy.https://zbmath.org/1449.350622021-01-08T12:24:00+00:00"Yong, Jia-Shi"https://zbmath.org/authors/?q=ai:yong.jia-shi"He, Fa-Zhi"https://zbmath.org/authors/?q=ai:he.fazhi"Li, Hao-Ran"https://zbmath.org/authors/?q=ai:li.haoran"Zhou, Wei-Qing"https://zbmath.org/authors/?q=ai:zhou.wei-qingSummary: Population-based algorithms have been used in many real-world problems. Bat algorithm (BA) is one of the states of the art of these approaches. Because of the super bat, on the one hand, BA can converge quickly; on the other hand, it is easy to fall into local optimum. Therefore, for typical BA algorithms, the ability of exploration and exploitation is not strong enough and it is hard to find a precise result. In this paper, we propose a novel bat algorithm based on cross boundary learning (CBL) and uniform explosion strategy (UES), namely BABLUE in short, to avoid the above contradiction and achieve both fast convergence and high quality. Different from previous opposition-based learning, the proposed CBL can expand the search area of population and then maintain the ability of global exploration in the process of fast convergence. In order to enhance the ability of local exploitation of the proposed algorithm, we propose UES, which can achieve almost the same search precise as that of firework explosion algorithm but consume less computation resource. BABLUE is tested with numerous experiments on unimodal, multimodal, one-dimensional, high-dimensional and discrete problems, and then compared with other typical intelligent optimization algorithms. The results show that the proposed algorithm outperforms other algorithms.Local time and Tanaka formula of \(G\)-martingales.https://zbmath.org/1449.601022021-01-08T12:24:00+00:00"Liu, Guo-Min"https://zbmath.org/authors/?q=ai:liu.guominSummary: The objective of this paper is to study the local time and Tanaka formula of symmetric \(G\)-martingales. We introduce the local time of \(G\)-martingales and show that it belongs to the \(G\)-expectation space \(L_G^2(\ohm_T)\). By a localization argument, we obtain the bicontinuous modification of local time. Furthermore, we give the Tanaka formula for convex functions of \(G\)-martingales.An optimal dividend strategy in the discrete Sparre Andersen model when payments are subject to transaction costs.https://zbmath.org/1449.911112021-01-08T12:24:00+00:00"Wang, Shaofeng"https://zbmath.org/authors/?q=ai:wang.shaofeng"Yin, Chuancun"https://zbmath.org/authors/?q=ai:yin.chuancun"Shen, Ying"https://zbmath.org/authors/?q=ai:shen.ying.1Summary: In this paper, we study the optimal dividend problem of dividend transaction costs under the discrete Sparre Andersen model, in which payments are subject to transaction costs. The optimal value function is obtained by updating the initial time. In addition, we prove that the optimal value function is the only bounded solution of Hamilton-Jacobi-Bellman equation and obtain the optimal dividend by the optimal value transformation according to Bellman recursive algorithm.On exponential decay of a distance between solutions of an SDE with non-regular drift.https://zbmath.org/1449.600962021-01-08T12:24:00+00:00"Aryasova, O."https://zbmath.org/authors/?q=ai:aryasova.olga-v"Pilipenko, A."https://zbmath.org/authors/?q=ai:pilipenko.andrey-yuSummary: We consider a multidimensional stochastic differential equation with a Gaussian noise and a drift vector having a jump discontinuity along a hyperplane. The large time behavior of the distance between two solutions starting from different points is studied. We find a sufficient condition for the exponential decay of the distance if the drift does not satisfy a dissipative condition on a given hyperplane.Stochastic models of just-in-time systems and windows of vulnerability in terms of the processes of birth and death.https://zbmath.org/1449.600462021-01-08T12:24:00+00:00"Butov, Aleksandr Aleksandrovich"https://zbmath.org/authors/?q=ai:butov.aleksandr-aleksandrovich"Kovalenko, Anatoliĭ Aleksandrovich"https://zbmath.org/authors/?q=ai:kovalenko.anatolii-aleksandrovichSummary: The paper proposes a method for constructing models based on the analysis of birth and death processes with linear growth in semimartingale terms. Based on this method, stochastic models of simple just-in-time systems (analyzed in the theory of productive systems) and windows of vulnerability (widely discussed in risk theory) are considered. The main results obtained in the work are presented in terms of the average values of the time during which the processes reach zero values. At the same time, they are considered and used in the study of assessment models for local times of the processes. Here, simple Markov processes with a linear growth of intensities (perhaps, depending on time) are analyzed. At the same time, the obtained and used estimates are of theoretical interest. Thus, for example, the average value of the stopping time, at which the process reaches zero, depends on functions such as the harmonic number and the remainder term for the logarithmic function in the Taylor theorem.
As the main result, the method of mathematical modeling of just-in-time systems and windows of vulnerability is proposed. The semimartingale description method used here should be considered as the first step of such a modeling, since, being a trajectory method, it allows diffusion (including non-Markov processes) generalizations when constructing stochastic models of windows of \textit{vulnerability} and \textit{just-in-time}. In the theoretical part of the article, we formulate statements for the average values of the local time and the stopping times when the birth and death processes reach a given value. This allows us to uniformly present estimates for the models of the \textit{just-in-time} system and for \textit{windows of vulnerability}, the result for which is given in the form of a limit theorem. The main results are formulated as theorems and lemmas. The proofs use semimartingale methods.Study on viral dynamics model with uncertainty.https://zbmath.org/1449.920492021-01-08T12:24:00+00:00"Xu, Liandi"https://zbmath.org/authors/?q=ai:xu.liandi"Guo, Manjing"https://zbmath.org/authors/?q=ai:guo.manjing"Nie, Linfei"https://zbmath.org/authors/?q=ai:nie.linfeiSummary: Considering the influence of uncertainties on the spread and reproduction of viruses in organisms, a virus infection model with uncertainty is proposed. The existence and uniqueness of the global positive solution of the model are proved. By constructing a suitable Lyapunov function, the criteria for determining the extinction and persistence of viruses are established. The theoretical results show that the uncertain factors can inhibit the virus to some extent, and the elimination of the virus can be accelerated when the intensity of uncertainty is high.Some selected topics for the bootstrap of the empirical and quantile processes.https://zbmath.org/1449.600672021-01-08T12:24:00+00:00"Alvarez-Andrade, Sergio"https://zbmath.org/authors/?q=ai:alvarez-andrade.sergio"Bouzebda, Salim"https://zbmath.org/authors/?q=ai:bouzebda.salimSummary: In the present work we consider the asymptotic distribution of \(L_p\) functionals of bootstrapped weighted uniform quantile and empirical processes. The asymptotic laws obtained are represented in terms of Gaussian integrals. We investigate the strong approximations for the bootstrapped Vervaat process and the weighted bootstrap for Bahadur-Kiefer process. We obtain new results on the precise asymptotics in the law of the logarithm related to complete convergence and a.s. convergence, under some mild conditions, for the weighted bootstrap of empirical and the quantile processes. In addition we consider the strong approximation of the hybrids of empirical and partial sums processes when the sample size is random.Berry-Esseen bounds for drift parameter estimation of discretely observed fractional Vasicek-type process.https://zbmath.org/1449.600782021-01-08T12:24:00+00:00"Alazemi, Fares"https://zbmath.org/authors/?q=ai:alazemi.fares"Douissi, Soukaina"https://zbmath.org/authors/?q=ai:douissi.soukaina"Es-Sebaiy, Khalifa"https://zbmath.org/authors/?q=ai:es-sebaiy.khalifaSummary: In this paper, we study statistical estimation problems of drift parameters of Vasicek-type processes driven by fractional Brownian motion. Based on fixed-time-step observations and using Malliavin calculus combined with the recent Nourdin-Peccati analysis, we provide estimators of the drift parameters and analyze their asymptotic behaviors. More precisely, we study the strong consistency and the asymptotic distribution of the estimators and we give the rate of their convergence in law.A probabilistic approach toward finite commutative rings.https://zbmath.org/1449.130172021-01-08T12:24:00+00:00"Rehman, Shafiqur"https://zbmath.org/authors/?q=ai:rehman.shafiqur"Baig, Abdul Qudair"https://zbmath.org/authors/?q=ai:baig.abdul-qudair"Haider, Kamran"https://zbmath.org/authors/?q=ai:haider.kamranSummary: Let \(m\), \(n\) be positive integers and \(m \le n\). Suppose that we select two elements at random (with replacement) from the ring \({\mathbb{Z}_n}\). Then a question arises that ``what is the probability that the product of these two elements is \({\overline m}\)?'' We derive explicit formulas to compute the probability that the product of two elements chosen at random (with replacement) from \({\mathbb{Z}_n}\) is \({\overline m}\). Also we obtain some bounds for this probability.Local functional Chung's law of the iterated logarithm for increments of a Brownian motion in Hölder norm.https://zbmath.org/1449.600692021-01-08T12:24:00+00:00"Liu, Yonghong"https://zbmath.org/authors/?q=ai:liu.yonghong"Wang, Weina"https://zbmath.org/authors/?q=ai:wang.weinaSummary: Using large deviations and small deviations of Brownian motion in Hölder norm, local functional Chung's law of the iterated logarithm for increments of a Brownian motion in Hölder norm is obtained.General central limit theorems under sublinear expectations.https://zbmath.org/1449.600372021-01-08T12:24:00+00:00"Lan, Yuting"https://zbmath.org/authors/?q=ai:lan.yuting"Zhang, Ning"https://zbmath.org/authors/?q=ai:zhang.ning.1|zhang.ning|zhang.ning.2Summary: In this paper, we investigate the generalized central limit theorem under sublinear expectations based on three weaker conditions with the notion of \(G\)-normal distribution. Initially, the condition \(\mathbb{E}\left[ {{X_n}} \right] = \mathcal{E}\left[ {{X_n}} \right] = 0\) is replaced by \(\left| {\mathbb{E}\left[ {{X_n}} \right]} \right| + \left| {\mathcal{E}\left[ {{X_n}} \right]} \right| = O\left({\frac{1}{n}} \right)\). Furthermore, the original 2-nd and \(\left({2 + \delta} \right)\)-th moments conditions are weakened through the truncation of random variables. Finally, we develop the theorem for convolutionary random variables.On a characterization theorem on non-discrete totally disconnected locally compact fields.https://zbmath.org/1449.600052021-01-08T12:24:00+00:00"Feldman, Gennadiy M."https://zbmath.org/authors/?q=ai:feldman.gennadiy-m"Myronyuk, Margaryta V."https://zbmath.org/authors/?q=ai:myronyuk.margarytaSummary: We prove the following theorem. Let \(X\) be a non-discrete totally disconnected locally compact field, \(R\) be its ring of integers, \(P\) be the nonzero prime ideal of \(R\). Assume that the residue field \(R/P\) is a field of characteristic \(p > 2\). Let \(\xi\) and \(\eta\) be independent identically distributed random variables with values in \(X\) and distribution \(\mu\), such that \(\mu\) has a continuous density with respect to a Haar measure on \(X\). This implies that the random variables \(S = \xi + \eta\) and \(D = (\xi-\eta)^2\) are independent if and only if \(\mu\) is a shift of the Haar distribution of a compact subgroup of \(X\).Inversion of initial-value problem by means of quasi-reversibility regularization method combined with discrete random noise.https://zbmath.org/1449.652282021-01-08T12:24:00+00:00"Yang, Fan"https://zbmath.org/authors/?q=ai:yang.fan.1"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.4|zhang.yan.3|zhang.yan.2"Li, Xiaoxiao"https://zbmath.org/authors/?q=ai:li.xiaoxiaoSummary: The inversion of initial value problem of fractional diffusion equation is explored with discrete random noise. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the measured data. The quasi-reversibility regularization method is used to obtain a regularized approximate solution and the convergence estimate is given under a priori parameter choice rule. Numerical results show that this method will be effective and stable.A strong limit theorem of \(m\)-ordered non-homogeneous Markov chains on a non-homogeneous tree.https://zbmath.org/1449.600522021-01-08T12:24:00+00:00"Jin, Shaohua"https://zbmath.org/authors/?q=ai:jin.shaohua"Peng, Zhibing"https://zbmath.org/authors/?q=ai:peng.zhibing"Wang, Dong"https://zbmath.org/authors/?q=ai:wang.dong.1"Xu, Zeling"https://zbmath.org/authors/?q=ai:xu.zelingSummary: In this paper, we obtain a strong limit theorem of \(m\)-ordered non-homogeneous Markov chains on a non-homogeneous tree.Moments of additive statistics with respect to the Ewens sampling formula.https://zbmath.org/1449.110912021-01-08T12:24:00+00:00"Manstavičius, Eugenijus"https://zbmath.org/authors/?q=ai:manstavicius.eugenijus"Stepas, Vytautas"https://zbmath.org/authors/?q=ai:stepas.vytautasSummary: The additive semigroup of vectors with non-negative integer coordinates endowed with the Ewens probability measure plays an important role as a probabilistic space for many statistical models. In the present paper, we obtain upper estimates of the power moments of additive statistics defined on the semigroup. The statistics are sums of dependent random variables; however, our results have the form of the Rosenthal and von Bahr-Esseen inequalities. The arguments perfected in probabilistic number theory are adopted in the proofs.Pricing of American catastrophe disaster insurance futures options with martingale method.https://zbmath.org/1449.911702021-01-08T12:24:00+00:00"Zhao, Yuexu"https://zbmath.org/authors/?q=ai:zhao.yuexu"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie.7|liu.jie.5|liu.jie.4|liu.jie.2|liu.jie|liu.jie.3|liu.jie.1Summary: Based on the logarithmic normal diffusion model with jumps, the catastrophe disaster insurance futures and options pricing are investigated by the martingale and the modified actuarial methods. Then the European call insurance futures and options at anytime are obtained. Finally the empirical analysis based on R software gives the difference and the relationship between the above two methods, which shows that the actuarial pricing method is more accurate.Mathematical programming models for estimating the initial states of Markov chains by maximum likelihood.https://zbmath.org/1449.903532021-01-08T12:24:00+00:00"Lou, Zhenkai"https://zbmath.org/authors/?q=ai:lou.zhenkai"Hou, Fujun"https://zbmath.org/authors/?q=ai:hou.fujun"Lou, Xuming"https://zbmath.org/authors/?q=ai:lou.xumingSummary: Parameters estimation is a common issue in Markov models. Owing to the importance of the initial state, in this paper we estimate the initial state for Markov chain models in which the initial state is unknown. According to whether the states are visible, we divide the models into general Markov models and hidden Markov models. We build linear programming models and non-linear programming models by maximum likelihood with considering the amount of states or observation symbols, and prove that the probabilities of state at each stage meet the normalization of probability. For linear programming models, we point out that they can be solved by the simplex method, and show the expression of solution. For non-linear programming models, we account for the existence of the optimal solution, and turn models into equations by K-T condition. In the examples, we apply lingo to obtain the optimal solution while the equations are hard to solve.Clustering: Markov algorithm.https://zbmath.org/1449.600062021-01-08T12:24:00+00:00"Malyk, I. V."https://zbmath.org/authors/?q=ai:malyk.igor-v"Knignits'ka, T. V."https://zbmath.org/authors/?q=ai:knignitska.t-v"Gorbatenko, M. Yu."https://zbmath.org/authors/?q=ai:gorbatenko.mykola-yuSummary: This paper deals with the problem of clustering on graphs based on the eigenvalues of the stochastic matrix of the graph. It is proved, that the eigenvalues of the stochastic matrix for large graphs \( (N > 100)\) are divided into three groups, one of which is decisive for the number of clusters in the graph. Using the theory of random matrices, it is possible to show, that the asymptotic distribution of the subgroup of real parts of eigenvalues of the stochastic matrix of the graph is described by a semicircular Winger distribution, and the parameter of this distribution is \(O(1/\sqrt{N})\). The use of stochastic matrices allows to accurately localize the eigenvalues responsible for the number of clusters, which was not possible for adjacency matrices. The basic assumptions of the model are related to the properties of Markov discrete chains, which allows to extend the scope of the obtained results to a wider class of objects. Theoretical results are verified by clustering time series describing the value of \(N = 470\) shares of S\&P 500 data for the 6 year period (2013--2018). The clustering of these time series show the results of the presence of 5 clearly defined groups, consistent with the data used. A covariance matrix with zeros on the diagonal elements is used to construct a stochastic matrix for time series. This approach allows to localize the main clusters more precisely. The main result of the work is devoted to asymmetric matrices with mathematical expectations not equal 0, which allow to generalize some results of big data theory. Also, the dependence of clustering results on the cluster size, the number of clusters and the asymmetry of cluster size is noted in the paper. Using the Monte-Carlo method, it is shown that the proposed Markov algorithm is more stable to noise in the graph than some of classical algorithms.Geometry of permutation limits.https://zbmath.org/1449.600102021-01-08T12:24:00+00:00"Rahman, Mustazee"https://zbmath.org/authors/?q=ai:rahman.mustazee"Virág, Bálint"https://zbmath.org/authors/?q=ai:virag.balint"Vizer, Máté"https://zbmath.org/authors/?q=ai:vizer.mateThis article concerns permutons (a measure on a square with uniform marginal distributions) and permuton processes (a process \(X\) with continuous sample paths such that each \(X(t)\) is uniformly distributed on some fixed interval). In the graph of the symmetric group \(\mathfrak{S}_n\), a \textit{sorting network} is a geodesic from the permutation \((1 2 3 \dots n)\) to \((n (n - 1) (n - 2) \dots 1)\). This article is a hopeful step towards resolving the Archimedean path conjecture of \textit{O. Angel} et al. [Adv. Math. 215, No. 2, 839--868 (2007; Zbl 1132.60008)], which states that as \(n \to \infty\), the random sorting network process on \(\mathfrak{S}_n\) converges in probability to the ``Archimedean process'' \[ \mathcal{A}(t) = \cos (\pi t) \mathbb{A}_x + \sin (\pi t) \mathbb{A}_y, t \in [0,1]\] for an appropriate random variable \((\mathbb{A}_x, \mathbb{A}_y)\) on the plane, defined in terms of an ``Archimedean measure''.
The primary results of this article are:
A process is a permuton process if and only if it is the limit of a sequence of permutation-valued processes.
Among permuton processes \(X(t)\), \(0 \leq t \leq 1\), on the square \([-1, 1]\), such that \(X(1) = - X(0)\), there is a unique process with minimal Dirichlet energy, and it is Archimedean.
Given a permuton-valued path from the identity permuton (of support \(\{ (x, x) : x \in [-1, 1] \}\)) to the reverse permuton (of support \(\{ (x, -x) : x \in [-1, 1] \}\)), the Dirichlet energy (via the Wasserstein metric) of the path is bounded below by the energy of the Archimedean path, that minimum achieved only by the Archimedean path.
However, there is a function of permutons which, if minimized by a nontrivial permuton, is minimized by at least four distinct permutons.
There is a qualitative discussion in which it is asserted that second statement above and the results of \textit{M. Kotowski} [Limits of random permuton processes and large deviations of the interchange process. Toronto: University of Toronto (PhD Thesis) (2016)] together establish the Archimedean path conjecture for ``relaxed'' random sorting networks.
Reviewer: Gregory Loren McColm (Tampa)Option pricing under mixed exponential jump diffusion model based on the FST method.https://zbmath.org/1449.911692021-01-08T12:24:00+00:00"Zhang, Sumei"https://zbmath.org/authors/?q=ai:zhang.sumei"Zhao, Jieqiong"https://zbmath.org/authors/?q=ai:zhao.jieqiongSummary: The mixed exponential jump-diffusion model that can approximate any distribution is widely used to describe the actual trend of stock price. Based on the Fourier space time-stepping (FST) method, this paper considers European option pricing under the mixed exponential jump-diffusion model. By the Fourier transform and the characteristic exponent, the partial integro-differential equation for pricing European options is transformed into an ordinary differential equations and solved to obtain European option prices. Numerical results indicate that the FST method is accurate and fast. Moreover, by collecting real market data and the nonlinear least squares method, we apply the obtained option price to model calibration to obtain the model parameters which match the real market. By examining the impact of jump parameters on the implied volatility, we find that the mixed exponential jump-diffusion model can well reflect the volatility ``smile'' of asset returns.Mean-square stability of stochastic age-dependent delay population systems with Poisson jumps.https://zbmath.org/1449.350522021-01-08T12:24:00+00:00"Li, Qiang"https://zbmath.org/authors/?q=ai:li.qiang.3"Kang, Ting"https://zbmath.org/authors/?q=ai:kang.ting"Chen, Feifei"https://zbmath.org/authors/?q=ai:chen.feifei"Zhang, Qimin"https://zbmath.org/authors/?q=ai:zhang.qiminSummary: This paper deals with the mean-square stability problem of stochastic age-dependent delay population systems with Poisson jumps. Under certain conditions, the definition of mean-square stability of the numerical solution is given. By utilizing the compensated stochastic \(\theta\) methods, the mean-square stability of the numerical solution is investigated and a sufficient condition for mean-square stability of the numerical solution is presented. It is shown that the compensated stochastic \(\theta\) methods are mean-square stable for any stepsize \(\Delta \tau /m\) when \(1/2 \le \theta \le 1\), and they are exponentially mean-square stable if the stepsize \(\Delta t \in (0, \Delta {t_0})\) when \(0 \le \theta < 1\). Finally, the theoretical results are also confirmed by a numerical experiment.Fluid models driven by a working vacation-queue with PH-service time distribution.https://zbmath.org/1449.900802021-01-08T12:24:00+00:00"Wang, Huining"https://zbmath.org/authors/?q=ai:wang.huining"Xu, Xiuli"https://zbmath.org/authors/?q=ai:xu.xiuliSummary: This paper is concerned with the fluid model which is driven by a PH-service time and single-server queue with server working vacation. To analyze the fluid model. we first establish the stationary distribution of the queue length process. Based on the steady-state distribution, the matrix-type ordinary differential equation is obtained for the joint distribution characterizing the fluid model dynamics. With the help of the Laplace transform and the Laplace-Stieltjes transform, as usual, the probability for the system empty and the average fluid level are given. An application of these obtained results to the mobile Ad Hoc networks is provided. The sensitivities about the system primitive parameters to the performance measures such as the average fluid level are discussed by some numerical experiments.On the occupation times in a dual delayed Sparre Andersen risk model.https://zbmath.org/1449.600832021-01-08T12:24:00+00:00"Zhang, Wanlu"https://zbmath.org/authors/?q=ai:zhang.wanlu"Yin, Xiaolong"https://zbmath.org/authors/?q=ai:yin.xiaolong"Zhao, Xianghua"https://zbmath.org/authors/?q=ai:zhao.xianghuaSummary: In this paper, we study the joint Laplace transform of the occupation times until ruin in a dual delayed Sparre Andersen risk model with exponential jumps. Using the transformation method and the fluctuation theory, an explicit expression of the joint Laplace transform is derived.A limit property of random transition probability for a Markov chain indexed by a tree in Markovian environment.https://zbmath.org/1449.600582021-01-08T12:24:00+00:00"Shi, Zhiyan"https://zbmath.org/authors/?q=ai:shi.zhiyan"Bao, Dan"https://zbmath.org/authors/?q=ai:bao.dan"Wu, Baihui"https://zbmath.org/authors/?q=ai:wu.baihuiSummary: In this paper, we study a tree-indexed Markov chain in a random environment under discrete state and prove the realization of this stochastic process in the probability space. Meanwhile, the equivalence between tree-indexed Markov chains in Markov environment and tree-indexed Markov double chains is provided in this paper. Finally, the strong limit property of the harmonic mean of the random transition probability of tree-indexed Markov chains in the Markovian environment is obtained under the finite state.Qualitative analysis of an SIRI epidemic model with stochastic effects.https://zbmath.org/1449.341332021-01-08T12:24:00+00:00"Gao, Jianzhong"https://zbmath.org/authors/?q=ai:gao.jianzhong"Zhang, Tailei"https://zbmath.org/authors/?q=ai:zhang.taileiSummary: An SIRI bilinear epidemic model with stochastic effects is studied. The global existence, uniqueness and boundedness of its positive solution are proved by using stopping time theory and Lyapunov analysis method. It is also shown that the solution of the stochastic model oscillates around the corresponding deterministic disease-free equilibrium and endemic equilibrium points, and sufficient conditions for persistence in mean of the solution of the stochastic model and disease extinction are obtained. Finally, numerical simulations are carried out to illustrate of theoretical results.Valuation on outer performance option in a non-affine stochastic volatility jump-diffusion model.https://zbmath.org/1449.911522021-01-08T12:24:00+00:00"He, Jiawen"https://zbmath.org/authors/?q=ai:he.jiawen"Wei, Zhu'e"https://zbmath.org/authors/?q=ai:wei.zhueSummary: In this paper, the pricing of the carry option is investigated under the non-affine jump diffusion model in which the two underlying asset prices satisfy a class of random interest rates, random volatility and jumps exist in asset prices and volatility. Firstly, by using the Taylor formula to solve the nonlinear differential equation of linear problems, we obtain an approximate solution for characteristic function for the underlying log-asset price. Then, a semi-analytical pricing formula for the price of outer performance option is attained by means of Fourier inversion transform. We then apply the model to price performance option. Numerical examples show that the non-affine stochastic volatility jump-diffusion option pricing model is more accurate than the affine stochastic model. Moreover, the volatilities in both diffusion and jumps have significant effects on the option price.Existience and uniqueness of solutions for a class of stochastic fuzzy forest diffusion system with Poisson jump.https://zbmath.org/1449.601082021-01-08T12:24:00+00:00"Lv, Shuting"https://zbmath.org/authors/?q=ai:lv.shuting"Ma, Zeling"https://zbmath.org/authors/?q=ai:ma.zelingSummary: In this paper, we introduce a class of stochastic fuzzy forest diffusion system with Poisson jump, which is subjected to two kinds of uncertainties: stochastic and fuzziness, simultaneously. Under a boundedness condition, which is weaker than linear growth condition and the Lipschits condition, we obtain the existence and uniqueness of solution to the stochastic system by the Picard iteration method. The estimation formula of the approximate solution's error of the Picard iteration is given.Linear combination and reliability of generalized logistic random variables.https://zbmath.org/1449.600152021-01-08T12:24:00+00:00"de Sena Monteiro Ozelim, Luan Carlos"https://zbmath.org/authors/?q=ai:ozelim.luan-carlos-de-s-m"Rathie, Pushpa Narayan"https://zbmath.org/authors/?q=ai:rathie.pushpa-narayanSummary: Experimental random data, in general, present a skewed behaviour. Thus, asymmetrical generalized distributions are of interest. The generalized logistic distributions (GLDs) are good candidates to model skewed data because their probability density functions (p.d.f.) and characteristic functions are mathematically simple. In this paper, exact expressions in terms of the H-function are, for the first time, derived for the p.d.f. and for the cummulative distribution function of the linear combination of GLDs of type IV with different location, scale and shape parameters. Also, exact and approximate expressions are derived for \(R=P(X<Y)\). Numerical examples illustrate the correctness of the expressions derived.On the existence and uniqueness of solution of stochastic differential system driven by G-Brownian motion.https://zbmath.org/1449.600942021-01-08T12:24:00+00:00"Chalabi, El-Hacene"https://zbmath.org/authors/?q=ai:chalabi.el-hacene"Boutabia, Hacene"https://zbmath.org/authors/?q=ai:boutabia.haceneSummary: In this paper we prove the existence and the uniqueness of the solution of system of stochastic differential equations driven by G-Brownian motion by using the Caratheodory approximation scheme.The asymptotic behavior of a stochastic chemostat model with Michaelis-Menten food chain.https://zbmath.org/1449.341812021-01-08T12:24:00+00:00"Zhao, Yihan"https://zbmath.org/authors/?q=ai:zhao.yihan"Yang, Zhichun"https://zbmath.org/authors/?q=ai:yang.zhichunSummary: This paper investigates the asymptotic behavior of a stochastic chemostat model with Michaelis-Menten food chain in which the dilution rate is disturbed by white noise. First, the global existence and uniqueness of the positive solution of the model is proved. Then, by constructing Lyapunov function and using Itô's formula, sufficient condition for the stochastic global asymptotic stability of the washout equilibrium of the model is obtained. Finally, the long-time asymptotic behavior of the solution of the model are studied, which mainly reveals the oscillatory behavior of the solution around the predator-free equilibrium and positive equilibrium of the corresponding deterministic model under different conditions. The results improve and extend the relevant work of the existing literature.Pricing a chained dynamic fund protection with exponential protection level.https://zbmath.org/1449.911642021-01-08T12:24:00+00:00"Xu, Chao"https://zbmath.org/authors/?q=ai:xu.chao"Dong, Yinghui"https://zbmath.org/authors/?q=ai:dong.yinghuiSummary: Dynamic fund protection (DFP) provides a guarantee that the account value of the investor never drops below a barrier in the investment period. In order to reduce the downside risk taken by vendors, some previous researchers proposed a chained dynamic fund protection (CDFP), whose protection is activated only if the value of basic fund reaches predefined upper constant protection line. Motivated by them, we consider the pricing of a CDFP with exponential upper and down protection lines. By using the reflection properties of Brownian motion, we present the explicit pricing formula for a CDFP with exponential protection lines.Time-stepping error bound for a stochastic parabolic Volterra equation disturbed by fractional Brownian motions.https://zbmath.org/1449.652562021-01-08T12:24:00+00:00"Qi, Ruisheng"https://zbmath.org/authors/?q=ai:qi.ruisheng"Lin, Qiu"https://zbmath.org/authors/?q=ai:lin.qiuSummary: In this paper, we consider a stochastic parabolic Volterra equation driven by the infinite dimensional fractional Brownian motion with Hurst parameter \(H \in \left[ {\frac{1}{2}, 1} \right)\). We apply the piecewise constant, discontinuous Galerkin method to discretize this equation in the temporal direction. Based on the explicit form of the scalar resolvent function and the refined estimates for the Mittag-Leffler function, we derive sharp mean-square regularity results for the mild solution. The sharp regularity results enable us to obtain the optimal error bound of the time discretization. These theoretical findings are finally accompanied by several numerical examples.On existence and asymptotic behavior of the time-dependent solution of the \(\mathrm{M}/\mathrm{G}/1\) queueing model with optional deterministic server vacations.https://zbmath.org/1449.601302021-01-08T12:24:00+00:00"Kasim, Ehmet"https://zbmath.org/authors/?q=ai:kasim.ehmet"Gupur, Geni"https://zbmath.org/authors/?q=ai:gupur.geniSummary: In this paper, we consider the \(\mathrm{M}/\mathrm{G}/1\) queueing model with optional deterministic server vacations. Firstly, we convert the system into an abstract Cauchy problem, then we prove well-posedenss of the system by using the operator semigroup methods. Next, we investigate asymptotic behavior of its time-dependent solution by studying spectral properties of the corresponding operator. Therefore, we conclude that the time-dependent solution of the model strongly converges to its steady-state solution.On the complete convergence of sequences of random elements in Banach spaces.https://zbmath.org/1449.600512021-01-08T12:24:00+00:00"Huan, N. V."https://zbmath.org/authors/?q=ai:huan.n-vA Banach space \(X\) is said to have Rademacher type \(p > 0\) if there exist a constant \(T < \infty\) such that for every \({x_1},\dots,{x_n} \in X\), \(E{\Vert {\sum\limits_{i = 1}^n {{\varepsilon_i}{x_i}}} \Vert ^p} \le {T^p}\sum\limits_{i = 1}^n {{{\Vert {{x_i}} \Vert}^p}} \) where \(E\) denotes the expectation with respect to uniformly chosen \(\varepsilon = ({\varepsilon_1},\dots,{\varepsilon_n}) \in {[ - 1,1]^n}\). The paper considers a sequence \(\{ {X_n},n \ge 1\} \) of independent random variables with values in \(X\) with the partial \(k\)-th sum \({S_k}\) and gives criteria for the convergence of \(\sum\nolimits_{n = 1}^\infty {\frac{1}{n}P({{\max}_{1 \le k \le n}}\Vert {{S_k}} \Vert > \varepsilon {n^\alpha}})\) and \(\sum\nolimits_{n = 1}^\infty {\frac{{\log n}}{n}P({{\max}_{1 \le k \le n}}\Vert {{S_k}} \Vert > \varepsilon {n^\alpha}})\) for every \(\varepsilon > 0\).
Reviewer: Oleg K. Zakusilo (Kyïv)Improvements to exact Boltzmann sampling using probabilistic divide-and conquer and the recursive method.https://zbmath.org/1449.600022021-01-08T12:24:00+00:00"DeSalvo, Stephen"https://zbmath.org/authors/?q=ai:desalvo.stephen-aSummary: We demonstrate an approach for exact sampling of certain discrete combinatorial distributions, which is a hybrid of exact Boltzmann sampling and the recursive method, using probabilistic divide-and-conquer (PDC). The approach specializes to exact Boltzmann sampling in the trivial setting, and specializes to PDC deterministic second half in the first non-trivial application. A large class of examples is given for which this method broadly applies, and several examples are worked out explicitly.Selection of conditional independence graph models when the distribution is extended skew normal.https://zbmath.org/1449.621242021-01-08T12:24:00+00:00"Pacillo, Simona"https://zbmath.org/authors/?q=ai:pacillo.simonaSummary: The extended skew-normal family of distributions is a slight extension of the skewnormal one, which achieves closure under conditioning. In this paper, we discuss its application in conditional independence graphs selection. First, we derive a test for a single edge exclusion/inclusion based on a Wald-type statistic. Then, we show how the asymptotic null distribution of the Wald test changes when some regularity conditions of the parameter space fail to hold. Finally, we propose an alternative test and carry out numerical experiments to assess the performances, in finite samples, of the two methods.Baire categorical aspects of first passage percolation. II.https://zbmath.org/1449.540422021-01-08T12:24:00+00:00"Maga, B."https://zbmath.org/authors/?q=ai:maga.balazsSummary: In this paper we continue our earlier work [\textit{B. Maga}, Acta Math. Hung. 156, No. 1, 145--171 (2018; Zbl 1424.54065)] about topological first passage percolation and answer certain questions asked in our previous paper. Notably, we prove that apart from trivialities, in the generic configuration there exists exactly one geodesic ray, in the sense that we consider two geodesic rays distinct if they only share a finite number of edges. Moreover, we show that in the generic configuration any not too small and not too large convex set arises as the limit of a sequence \(B(t_n)/t_n\) for some \(t_n\to\infty\). Finally, we define topological Hilbert first passage percolation, and amongst others we prove that certain geometric properties of the percolation in the generic configuration guarantee that we consider a setting linearly isomorphic to the ordinary topological first passage percolation.Software system for simulation of study of queuing system with heterogeneous servers and independent queues.https://zbmath.org/1449.600012021-01-08T12:24:00+00:00"Kotenko, Andreĭ Petrovich"https://zbmath.org/authors/?q=ai:kotenko.andrey-petrovich"Bukarenko, Maksim Borisovich"https://zbmath.org/authors/?q=ai:bukarenko.maksim-borisovichSummary: A software package is described for the simulation of study and automated visualization of the state of a queuing system with heterogeneous servers and independent limited queues. The base of the algorithm is the method of description of such queues with the use of Mealy finite-state machines.On the convergence of sequences of probability measures.https://zbmath.org/1449.280052021-01-08T12:24:00+00:00"Zaharopol, Radu"https://zbmath.org/authors/?q=ai:zaharopol.raduIn this paper \((X, d)\) is a Polish space, \(\mathcal{M}(X)\) all real valued Borel measures on \(X\) and \(C_b(X)\) all bounded continuous functions on \(X\). A function \(f\in C_b(X)\) is said to have bounded support if the set \(\overline{\{x\in X:f(x)\neq 0\}}\) is contained in an open ball. \(C^{(b)}_{bs}(X)\) are those functions in \(C_b(X)\) which have bounded support and \(C^{(ucb)}_{bs}(X)\) are those in \(C^{(b)}_{bs}(X)\) which are uniformly continuous. For a non-negative \(\nu\in\mathcal{M}(X)\), a Borel set \(A\subset X\) is said to be \(\nu\)-continuous if \(\nu(\bar A) =\nu(A^\circ)\). In this paper the author gives an extension of the Portmanteau Theorem. The main result is: Let \(\{\mu_n\}\) be a sequence of probability measures in \(\mathcal{M}(X)\) and \(\mu\in \mathcal{M}(X), \mu\ge 0\). Then following statements are equivalent: (a) \(\mu_n\to\mu\) pointwise on \(C^{(b)}_{bs}(X)\); (b) \(\mu_n\to\mu\) pointwise on \(C^{(ucb)}_{bs}(X)\); (c) for every closed bounded set \(F\subset X\) we have lim sup \(\mu_n(F) \le \mu(F)\), and for every open bounded set \(G\subset X\) we have lim inf \(\mu_n(G) \ge \mu(G)\); (d) \(\mu_n(A)\to\mu(A)\) for every \(\mu\)-continuous Borel set \(A\subset X\). Many sophisticated lemmas are proved to reach this result.
Reviewer: Surjit Singh Khurana (Iowa City)New proof of the Novikov criterion using backward stochastic differential equations.https://zbmath.org/1449.600982021-01-08T12:24:00+00:00"Chikvinidze, B."https://zbmath.org/authors/?q=ai:chikvinidze.besikSummary: Using backward stochastic differential equations we give a new proof of the well known Novikov's criterion.The work of Fernando de Helguero on non-normality arising from selection.https://zbmath.org/1449.620012021-01-08T12:24:00+00:00"Azzalini, Adelchi"https://zbmath.org/authors/?q=ai:azzalini.adelchi"Regoli, Giuliana"https://zbmath.org/authors/?q=ai:regoli.giulianaSummary: The current literature on so-called `skew-symmetric distributions' is closely linked to the idea of a selection mechanism operated by some latent variable. We illustrate the pioneering work of Fernando de Helguero who in 1908 [Rom. 4. Math. Kongr. 8. 288--299 (1909; JFM 40.0294.03)] put forward a formulation for the genesis of non-normal distributions via a selection mechanism, which perturbs a normal distribution, hence employing a closely connected argument with the one now widely used in this context. Arguably, de Helguero can then be considered the precursor of the current idea of skew-symmetric distributions. Unfortunately, a tragic quirk of fate did not allow him to pursue his project beyond the initial formulation and his work went unnoticed for the rest of the 20th century.Asymptotic stability of impulsive neutral stochastic functional differential equation driven by fractional Brownian motion.https://zbmath.org/1449.342792021-01-08T12:24:00+00:00"Cui, Jing"https://zbmath.org/authors/?q=ai:cui.jing"Liang, Qiuju"https://zbmath.org/authors/?q=ai:liang.qiuju"Bi, Nana"https://zbmath.org/authors/?q=ai:bi.nanaSummary: In this paper, we consider the asymptotic stability in the \(p\)-th moment of mild solutions of impulsive neutral stochastic functional differential equations driven by fractional Brownian motion in a real separable Hilbert space. A fixed point approach is used to achieve the required result. A practical example is provided to illustrate the viability of the abstract result of this work.Finite difference schemes for the tempered fractional Laplacian.https://zbmath.org/1449.652132021-01-08T12:24:00+00:00"Zhang, Zhijiang"https://zbmath.org/authors/?q=ai:zhang.zhijiang"Deng, Weihua"https://zbmath.org/authors/?q=ai:deng.weihua"Fan, Hongtao"https://zbmath.org/authors/?q=ai:fan.hongtaoSummary: The second and all higher order moments of the \(\beta \)-stable Lévy process diverge, the feature of which is sometimes referred to as shortcoming of the model when applied to physical processes. So, a parameter \(\lambda \) is introduced to exponentially temper the Lévy process. The generator of the new process is the tempered fractional Laplacian \({\left ({\Delta + \lambda} \right)^{\beta /2}}\) In this paper, we first design the finite difference schemes for the tempered fractional Laplacian equation with the generalized Dirichlet type boundary condition, their accuracy depends on the regularity of the exact solution on \({\bar \Omega}\). Then the techniques of effectively solving the resulting algebraic equation are presented, and the performances of the schemes are demonstrated by several numerical examples.The limit behaviour of random walks with arrests.https://zbmath.org/1449.600722021-01-08T12:24:00+00:00"Prykhodko, O. O."https://zbmath.org/authors/?q=ai:prykhodko.o-oSummary: Let \(\tilde S\) be a random walk which behaves like a standard centred and square-integrable random walk except when hitting 0. Upon the \(i\)-th hit of 0 the random walk is arrested there for a random amount of time \(\eta_i\geq 0\); and then continues its way as usual. The random variables \(\eta_1,\eta_2,\dots\) are assumed i.i.d. We study the limit behaviour of this process scaled as in the Donsker theorem. In case of \(E\eta_i <\infty\), weak convergence towards a Wiener process is proved. We also consider the sequence of processes whose arrest times are geometrically distributed and grow with \(n\). We prove that the weak limit for the last model is either a Wiener process, a Wiener process stopped at 0 or a Wiener process with a sticky point.Discounted penalty function of generalized compound Poisson model of two type insurance when funds fall to initial surplus.https://zbmath.org/1449.911042021-01-08T12:24:00+00:00"Li, Jingbin"https://zbmath.org/authors/?q=ai:li.jingbin"Wang, Xiulian"https://zbmath.org/authors/?q=ai:wang.xiulian"Zou, Hua"https://zbmath.org/authors/?q=ai:zou.huaSummary: The generalized compound Poisson model of two type insurance is considered. The discounted penalty function about the stopping time when funds fall to initial surplus is studied. The integral differential equation and renewal equation for the discounted penalty function are deduced by using probability theory and the Laplace transform, and then the concrete expression of the discounted penalty function and the moment of stopping time are obtained. The explicit expression of the discounted penalty function is given when the claim amount distribution is exponential.Long time behaviors of the solutions to stochastic wave equations with damping.https://zbmath.org/1449.350602021-01-08T12:24:00+00:00"Wang, Suxin"https://zbmath.org/authors/?q=ai:wang.suxinSummary: The long time behavior of the solution to a stochastic wave equation with damping is considered. Under some appropriate conditions, the exponential stability of the solutions holds almost surely. Finally two examples to illustrate the results are given.Particle filtration algorithm with variable-frequency mutation.https://zbmath.org/1449.600812021-01-08T12:24:00+00:00"Yu, Ping"https://zbmath.org/authors/?q=ai:yu.ping"Cao, Jie"https://zbmath.org/authors/?q=ai:cao.jie"Huang, Kaijie"https://zbmath.org/authors/?q=ai:huang.kaijieSummary: Aimed at the problem of particle diversity reduction on particle filtration algorithms induced by particle degeneracy and resampling, a strategy of adaptive frequency conversion is employed in mutative operation of immune theory, and cooperated with particle filtration, a novel particle filtration algorithm with variable-frequency mutation is designed. An adaptive variable-frequency operator is introduced into the algorithm to regulate the present mutation frequency in real time and control the quantity of the mutational particles. Further, various strategies are adopted to carry out the mutational operation for the particles and improve the adaptability of the particles to the change of the system state. Finally, the weight value calculation of the regenerated particles is conducted and a new particle set is composed from the particles with greater weight value to improve filtration accuracy. It is shown that this approach could fulfill the high-accurate estimation task with less quantity of particles and would have higher filtration precision, particle variety, and comprehensive ratio of credit to cost of calculation speed. Meantime, the particle distribution would be more rational and a certain quantity of particles would still exist outside of the high likelihood region, so that providing a condition for keeping a better estimation accuracy in the case of system mutation.General result of precise asymptotics for length of longest increasing subsequences.https://zbmath.org/1449.600652021-01-08T12:24:00+00:00"Zhao, Li"https://zbmath.org/authors/?q=ai:zhao.liSummary: By using the central limit theorem and the inequalities of the tail probability for the length of the longest increasing subsequence and properties of the Tracy-Wisdom distribution, the author gives a general result of precise asymptotics in complete moment convergence for the length of the longest increasing subsequence for more general boundary functions and quasi-weight functions.First order convergence of weak Wong-Zakai approximations of Levy-driven Marcus SDEs.https://zbmath.org/1449.650062021-01-08T12:24:00+00:00"Kosenkova, Tetyana"https://zbmath.org/authors/?q=ai:kosenkova.tetyana"Kulik, Alexei"https://zbmath.org/authors/?q=ai:kulik.aleksei|kulik.alexey-m"Pavlyukevich, Ilya"https://zbmath.org/authors/?q=ai:pavlyukevich.ilyaSummary: For solutions \(X=(X_t)_{t\in[0,T]}\) of a Lévy-driven Marcus (canonical) stochastic differential equation we study the Wong-Zakai type time discrete approximations \(\overline{X}=(\overline{X}_{kh})_{0\leq k\leq T/h}\) and establish the first order convergence \( |E_xf(X_T) -E_xf(X^h_T)|\leq Ch\) for \(f\in C^4_b\).Research on systematic financial risk measurement of joint-stock commercial banks based on high-dimensional dynamic R-Vine Copula.https://zbmath.org/1449.911822021-01-08T12:24:00+00:00"Han, Chao"https://zbmath.org/authors/?q=ai:han.chao"Zhou, Bing"https://zbmath.org/authors/?q=ai:zhou.bing"Xiong, Ya"https://zbmath.org/authors/?q=ai:xiong.yaSummary: The main objective of this paper is to accurately measure systematic financial risk. The R-vine copula method is used to solve the problem of precise measurement of systematic risk in joint-stock commercial banks. In the research process, GJR-GARCH and GPD models are fitted and filtered to get PIT series, which are used to realize dynamic R-vine copula modeling and simulation. VaR is simulated by the principle of historical recurrence, and compared with the static model. Finally, the conclusion that dynamic R-Vine Copula model is better is drawn. The significance of this paper lies in providing new model support for the measurement of systematic financial risk, providing more precise quantitative methods for the early warning of systematic financial risk, and expecting to play an important role in the supervision, prevention and control of financial risk and academic research in the new era.Random semilinear system of differential equations with state-dependent delay.https://zbmath.org/1449.342782021-01-08T12:24:00+00:00"Blouhi, Tayeb"https://zbmath.org/authors/?q=ai:blouhi.tayeb"Ferhat, Mohamed"https://zbmath.org/authors/?q=ai:ferhat.mohamedSummary: In this paper, we prove the existence of mild solutions for a first-order semilinear differential equation with state-dependent delay. The existence results are established by means of a new version of Perov's fixed point principles.