Recent zbMATH articles in MSC 49Nhttps://zbmath.org/atom/cc/49N2023-12-07T16:00:11.105023ZWerkzeugSplitting methods and short time existence for the master equations in mean field gameshttps://zbmath.org/1522.355112023-12-07T16:00:11.105023Z"Cardaliaguet, Pierre"https://zbmath.org/authors/?q=ai:cardaliaguet.pierre"Cirant, Marco"https://zbmath.org/authors/?q=ai:cirant.marco"Porretta, Alessio"https://zbmath.org/authors/?q=ai:porretta.alessioSummary: We develop a splitting method to prove the well-posedness, in short time, of solutions for two master equations in mean field game (MFG) theory: the second order master equation, describing MFGs with a common noise, and the system of master equations associated with MFGs with a major player. Both problems are infinite-dimensional equations stated in the space of probability measures. Our new approach simplifies and generalizes previous existence results for second order master equations and provides the first existence result for systems associated with MFG problems with a major player.Numerical identification of initial temperatures in heat equation with dynamic boundary conditionshttps://zbmath.org/1522.355752023-12-07T16:00:11.105023Z"Chorfi, S. E."https://zbmath.org/authors/?q=ai:chorfi.salah-eddine"El Guermai, G."https://zbmath.org/authors/?q=ai:el-guermai.ghita"Maniar, L."https://zbmath.org/authors/?q=ai:maniar.lahcen"Zouhair, W."https://zbmath.org/authors/?q=ai:zouhair.walidSummary: We investigate the inverse problem of numerically identifying unknown initial temperatures in a heat equation with dynamic boundary conditions whenever some overdetermination data are provided after a final time. This is a backward parabolic problem which is severely ill-posed. As a first step, the inverse problem is reformulated as a minimization problem for an associated Tikhonov functional. Using the weak solution approach, an explicit formula for the Fréchet gradient of the cost functional is derived from the corresponding sensitivity and adjoint problems. Then, the Lipschitz continuity of the gradient is proved. Next, further spectral properties of the input-output operator are established. Finally, the numerical results for noisy measured data are performed using the regularization framework and the conjugate gradient method. We consider both one- and two-dimensional numerical experiments using finite difference discretization to illustrate the efficiency of the designed algorithm. Aside from dealing with a time derivative on the boundary, the presence of a boundary diffusion makes the analysis more complicated. This issue is handled in the 2-D case by considering the polar coordinate system. The presented method implies fast numerical results.Stochastic elliptic inverse problems. Solvability, convergence rates, discretization, and applicationshttps://zbmath.org/1522.355772023-12-07T16:00:11.105023Z"Dambrine, Marc"https://zbmath.org/authors/?q=ai:dambrine.marc"Khan, Akhtar A."https://zbmath.org/authors/?q=ai:khan.akhtar-ali"Sama, Miguel"https://zbmath.org/authors/?q=ai:sama.miguel"Starkloff, Hans-Jörg"https://zbmath.org/authors/?q=ai:starkloff.hans-jorgSummary: Motivated by the necessity to identify stochastic parameters in a wide range of stochastic partial differential equations, an abstract inversion framework is designed. The stochastic inverse problem is studied in a stochastic optimization framework. The essential properties of the solution map are derived and used to prove the solvability of the stochastic optimization problems. Novel convergence rates for the stochastic inverse problem are presented in the abstract formulation without requiring the so-called smallness condition. Under the assumption of finite-dimensional noise, the stochastic inverse problem is parametrized and solved by using the Stochastic Galerkin discretization scheme. The developed framework is applied to estimate stochastic Lamé parameters in the system of linear elasticity. We present numerical results that are quite encouraging and show the feasibility and efficacy of the developed framework.Spiral-like extremals near a singular surface in a rocket control problemhttps://zbmath.org/1522.370732023-12-07T16:00:11.105023Z"Ronzhina, Mariya I."https://zbmath.org/authors/?q=ai:ronzhina.mariya-igorevna"Manita, Larisa A."https://zbmath.org/authors/?q=ai:manita.larisa-aSummary: In this paper, we consider the minimum time problem for a space rocket whose dynamics is given by a control-affine system with drift. The admissible control set is a disc. We study extremals in the neighbourhood of singular points of the second order. Our approach is based on applying the method of a descending system of Poisson brackets and the Zelikin-Borisov method for resolution of singularities to the Hamiltonian system of Pontryagin's maximum principle. We show that in the neighbourhood of any singular point there is a family of spiral-like solutions of the Hamiltonian system that enter the singular point in a finite time, while the control performs an infinite number of rotations around the circle.Well-posedness and regularity for a polyconvex energyhttps://zbmath.org/1522.490262023-12-07T16:00:11.105023Z"Gangbo, Wilfrid"https://zbmath.org/authors/?q=ai:gangbo.wilfrid"Jacobs, Matt"https://zbmath.org/authors/?q=ai:jacobs.matthew"Kim, Inwon"https://zbmath.org/authors/?q=ai:kim.inwon-christinaIn this paper, the authors investigate on well-posedness and regularity for a polyconvex energy. Concretely, the authors prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions.
Reviewer: Savin Treanţă (Bucureşti)Zero-order asymptotics for the solution of one type of singularly perturbed linear-quadratic control problems in the critical casehttps://zbmath.org/1522.490312023-12-07T16:00:11.105023Z"Kurina, G. A."https://zbmath.org/authors/?q=ai:kurina.galina-a"Hoai, Nguyen Thi"https://zbmath.org/authors/?q=ai:hoai.nguyen-thi-xuanSummary: We consider a linear-quadratic control problem in which there is the second power of a small parameter at the derivative of the state variable and the first power of the parameter both in the control term of the state equation and at the quadratic form with respect to the control variable in the performance index; moreover, the state equation represents a critical case of singular perturbation theory. A zero-order asymptotic expansion of the solution is constructed using the so-called direct scheme method in which a postulated asymptotic expansion of the solution is substituted directly into the problem statement, and problems for finding the asymptotic terms are stated.Optimal Quantizer scheduling and controller synthesis for partially observable linear systemshttps://zbmath.org/1522.490322023-12-07T16:00:11.105023Z"Maity, Dipankar"https://zbmath.org/authors/?q=ai:maity.dipankar"Tsiotras, Panagiotis"https://zbmath.org/authors/?q=ai:tsiotras.panagiotisSummary: In networked control systems, the sensory signals are often quantized before being transmitted to the controller. Consequently, performance is affected by the coarseness of this quantization process. Modern communication technologies allow users to obtain resolution-varying quantized measurements based on the prices paid. In this paper, we consider the problem of joint optimal controller synthesis and quantizer scheduling for a partially observed quantized-feedback linear-quadratic-Gaussian system, where the measurements are quantized before being sent to the controller. The system is presented with several choices of quantizers, along with the cost of using each quantizer. The objective is to jointly select the quantizers and synthesize the controller to strike an optimal balance between control performance and quantization cost. When the innovation signal is quantized instead of the measurement, the problem is decoupled into two optimization problems: one for optimal controller synthesis, and the other for optimal quantizer selection. The optimal controller is found by solving a Riccati equation and the optimal quantizer-selection policy is found by solving a linear program -- both of which can be solved offline.Optimal singular LQR problem: a PD feedback solutionhttps://zbmath.org/1522.490332023-12-07T16:00:11.105023Z"Qais, Imrul"https://zbmath.org/authors/?q=ai:qais.imrul"Bhawal, Chayan"https://zbmath.org/authors/?q=ai:bhawal.chayan"Pal, Debasattam"https://zbmath.org/authors/?q=ai:pal.debasattamSummary: Unlike regular linear quadratic regulator (LQR) problems, singular LQR problems, in general, cannot be solved using a static state-feedback controller. This work is primarily focused on the design of feedback controllers which solve the singular LQR problem. We show that such problems can be solved using proportional-derivative (PD) state-feedback controllers. It is well known in the literature that the \textit{maximal rank-minimizing} solution of the singular LQR linear matrix inequality (LMI) is pivotal in solving the singular LQR problem. In this paper, we first make use of this maximal rank-minimizing solution to compute the optimal trajectories. Then we provide a PD feedback controller that restricts the trajectories of the closed-loop system to these optimal ones and thus solves the singular LQR problem. While numerous solutions to this problem have been proposed over the course of the extensive research efforts in this field, a controller in the form of a PD state feedback has been long sought after. Our approach is based on the notion of \textit{weakly unobservable (slow)} and \textit{strongly reachable (fast)} subspaces developed in [\textit{M. L. J. Hautus} and \textit{L. M. Silverman}, Linear Algebra Appl. 50, 369--402 (1983; Zbl 0522.93021)]. But, unlike in that reference, we employ these notions to the corresponding Hamiltonian system and not to the plant. This crucial extension of these well-known subspaces to the corresponding Hamiltonian system is key to the optimal PD feedback design that we propose in this paper. It is well known that an optimal state feedback for the singular LQR problem does not exist; the limiting state-feedback controller of the suboptimal ones (high gain controllers) has unbounded coefficients as optimality is approached. We show in this paper that the limiting high gain controller is in fact a PD controller.Non-Markovian impulse control under nonlinear expectationhttps://zbmath.org/1522.490342023-12-07T16:00:11.105023Z"Perninge, Magnus"https://zbmath.org/authors/?q=ai:perninge.magnusSummary: We consider a general type of non-Markovian impulse control problems under adverse non-linear expectation or, more specifically, the zero-sum game problem where the adversary player decides the probability measure. We show that the upper and lower value functions satisfy a dynamic programming principle (DPP). We first prove the dynamic programming principle (DPP) for a truncated version of the upper value function in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. Following this, we use an approximation based on a combination of truncation and discretization to show that the upper and lower value functions coincide, thus establishing that the game has a value and that the DPP holds for the lower value function as well. Finally, we show that the DPP admits a unique solution and give conditions under which a saddle point for the game exists. As an example, we consider a stochastic differential game (SDG) of impulse versus classical control of path-dependent stochastic differential equations (SDEs).Inverse optimal extremum seeking under delayshttps://zbmath.org/1522.490352023-12-07T16:00:11.105023Z"Ferreira, Denis Cesar"https://zbmath.org/authors/?q=ai:ferreira.denis-cesar"Oliveira, Tiago Roux"https://zbmath.org/authors/?q=ai:oliveira.tiago-roux"Krstic, Miroslav"https://zbmath.org/authors/?q=ai:krstic.miroslavIn this paper, the authors investigate on inverse optimal extremum seeking under delays. More concretely, they establish the inverse optimality in the average sense of an earlier Gradient- and Newton-based extremum seeking algorithms for maximizing unknown locally quadratic maps in the presence of constant delays.
Reviewer: Savin Treanţă (Bucureşti)Cooperative optimal control for connected and automated vehicles platoonhttps://zbmath.org/1522.490362023-12-07T16:00:11.105023Z"Chen, Jianzhong"https://zbmath.org/authors/?q=ai:chen.jianzhong"Li, Jing"https://zbmath.org/authors/?q=ai:li.jing.81"Xu, Zhaoxin"https://zbmath.org/authors/?q=ai:xu.zhaoxin"Wu, Xiaobao"https://zbmath.org/authors/?q=ai:wu.xiaobaoSummary: The coordination and the energy consumption are very important for the platooning of connected and automated vehicles (CAVs). A novel cooperative optimal control for CAVs platoon on basic freeway sections is proposed in this paper. A cost function is designed to address the cooperation of followers, the motion synchronization with the leading vehicle and appropriate energy consumptions. A third-order consensus strategy is suggested to design the control input. A more effective and flexible spacing strategy is introduced. The asymptotically stability and string stability conditions of the system are established. By constructing and solving the LMI optimization problem, the optimal control gains are derived and the global cost function is minimized to a specific upper bound. Numerical simulations are performed on several specific traffic scenarios. The results demonstrate the effectiveness of the presented modeling method.Error estimates of a theta-scheme for second-order mean field gameshttps://zbmath.org/1522.651372023-12-07T16:00:11.105023Z"Bonnans, J. Frédéric"https://zbmath.org/authors/?q=ai:bonnans.joseph-frederic"Liu, Kang"https://zbmath.org/authors/?q=ai:liu.kang"Pfeiffer, Laurent"https://zbmath.org/authors/?q=ai:pfeiffer.laurentSummary: We introduce and analyze a new finite-difference scheme, relying on the theta-method, for solving monotone second-order mean field games. These games consist of a coupled system of the Fokker-Planck and the Hamilton-Jacobi-Bellman equation. The theta-method is used for discretizing the diffusion terms: we approximate them with a convex combination of an implicit and an explicit term. On contrast, we use an explicit centered scheme for the first-order terms. Assuming that the running cost is strongly convex and regular, we first prove the monotonicity and the stability of our thetascheme, under a CFL condition. Taking advantage of the regularity of the solution of the continuous problem, we estimate the consistency error of the theta-scheme. Our main result is a convergence rate of order \(\mathcal{O}(h^r)\) for the theta-scheme, where \(h\) is the step length of the space variable and \(r \in (0, 1)\) is related to the Hölder continuity of the solution of the continuous problem and some of its derivatives.Entanglement islands, fire walls and state paradox from quantum teleportation and entanglement swappinghttps://zbmath.org/1522.810482023-12-07T16:00:11.105023Z"Wang, Xuanhua"https://zbmath.org/authors/?q=ai:wang.xuanhua"Zhang, Kun"https://zbmath.org/authors/?q=ai:zhang.kun"Wang, Jin"https://zbmath.org/authors/?q=ai:wang.jin.3Summary: Recent discovery of the fine-grained entropy formula in gravity succeeded in reconstructing the Page curves that are compatible with unitary evolution. The formula of generalized entropy derived from the gravitational path integration, nevertheless, does not provide a concrete insight on how information comes out from a black hole. In this paper, we start from a qubit model and provide a quantum informational interpretation of entanglement islands. We propose an identification of entanglement islands with quantum measurements and remark on the parallel between the black hole information problem and the old problem of quantum measurements. We show that the Page curve can still be realized even if information is lost so that the information paradox can be explained as one manifestation of measurement problem. We show that such interpretation is necessary for a quantum informational model if smooth horizons and bulk reconstruction are assumed, and demonstrate explicitly that Page curves of solvable 2D gravity can be obtained through teleportation and entanglement swapping. We argue that the similarities between the black hole information problem and the measurement problem suggest links in the origins of the two problems.Regularization for Wasserstein distributionally robust optimizationhttps://zbmath.org/1522.900592023-12-07T16:00:11.105023Z"Azizian, Waïss"https://zbmath.org/authors/?q=ai:azizian.waiss"Iutzeler, Franck"https://zbmath.org/authors/?q=ai:iutzeler.franck"Malick, Jérôme"https://zbmath.org/authors/?q=ai:malick.jeromeSummary: Optimal transport has recently proved to be a useful tool in various machine learning applications needing comparisons of probability measures. Among these, applications of distributionally robust optimization naturally involve Wasserstein distances in their models of uncertainty, capturing data shifts or worst-case scenarios. Inspired by the success of the regularization of Wasserstein distances in optimal transport, we study in this paper the regularization of Wasserstein distributionally robust optimization. First, we derive a general strong duality result of regularized Wasserstein distributionally robust problems. Second, we refine this duality result in the case of entropic regularization and provide an approximation result when the regularization parameters vanish.A fictitious-play finite-difference method for linearly solvable mean field gameshttps://zbmath.org/1522.910102023-12-07T16:00:11.105023Z"Inoue, Daisuke"https://zbmath.org/authors/?q=ai:inoue.daisuke"Ito, Yuji"https://zbmath.org/authors/?q=ai:ito.yuji"Kashiwabara, Takahito"https://zbmath.org/authors/?q=ai:kashiwabara.takahito"Saito, Norikazu"https://zbmath.org/authors/?q=ai:saito.norikazu"Yoshida, Hiroaki"https://zbmath.org/authors/?q=ai:yoshida.hiroakiSummary: An iterative finite difference scheme for mean field games (MFGs) is proposed. The target MFGs are derived from control problems for multidimensional systems with advection terms. For such MFGs, linearization using the Cole-Hopf transformation and iterative computation using fictitious play are introduced. This leads to an implementation-friendly algorithm that iteratively solves explicit schemes. The convergence properties of the proposed scheme are mathematically proved by tracking the error of the variable through iterations. Numerical calculations show that the proposed method works stably for both one- and two-dimensional control problems.A zero-sum deterministic impulse controls game in infinite horizon with a new HJBI-QVIhttps://zbmath.org/1522.910222023-12-07T16:00:11.105023Z"El Asri, Brahim"https://zbmath.org/authors/?q=ai:el-asri.brahim"Lalioui, Hafid"https://zbmath.org/authors/?q=ai:lalioui.hafid"Mazid, Sehail"https://zbmath.org/authors/?q=ai:mazid.sehailSummary: In the present paper, we study a two-player, zero-sum, deterministic differential game with both players adopting impulse controls in infinite-time horizon, under rather weak assumptions on the cost functions. We prove by means of the dynamic programming principle that the lower and upper value functions are continuous and viscosity solutions to the corresponding Hamilton-Jacobi-Bellman-Isaacs (HJBI) quasi-variational inequality (QVI). We define a new HJBI-QVI for which, under a proportional property assumption on the maximizing player cost, the value functions are the unique viscosity solution. We then prove that the lower and upper value functions coincide.Generic properties of first-order mean field gameshttps://zbmath.org/1522.910392023-12-07T16:00:11.105023Z"Bressan, Alberto"https://zbmath.org/authors/?q=ai:bressan.alberto"Nguyen, Khai T."https://zbmath.org/authors/?q=ai:nguyen.khai-tSummary: We consider a class of deterministic mean field games, where the state associated with each player evolves according to an ODE which is linear w.r.t. the control. Existence, uniqueness, and stability of solutions are studied from the point of view of generic theory. Within a suitable topological space of dynamics and cost functionals, we prove that, for ``nearly all'' mean field games (in the Baire category sense) the best reply map is single-valued for a.e. player. As a consequence, the mean field game admits a strong (not randomized) solution. Examples are given of open sets of games admitting a single solution, and other open sets admitting multiple solutions. Further examples show the existence of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the best reply map, and another open set of MFG having a unique solution which is unstable. We conclude with an example of a MFG with terminal constraints which does not have any solution, not even in the mild sense with randomized strategies.Random lift of set valued maps and applications to multiagent dynamicshttps://zbmath.org/1522.910402023-12-07T16:00:11.105023Z"Capuani, Rossana"https://zbmath.org/authors/?q=ai:capuani.rossana"Marigonda, Antonio"https://zbmath.org/authors/?q=ai:marigonda.antonio"Ricciardi, Michele"https://zbmath.org/authors/?q=ai:ricciardi.micheleSummary: We introduce an abstract framework for the study of general mean field games and mean field control problems. Given a multiagent system, its macroscopic description is provided by a time-depending probability measure, where at every instant of time the measure of a set represents the fraction of (microscopic) agents contained in it. The trajectories available to each of the microscopic agents are affected also by the overall state of the system. By using a suitable concept of random lift of set valued maps, together with fixed point arguments, we are able to derive properties of the macroscopic description of the system from properties of the set valued map expressing the admissible trajectories for the microscopical agents. The techniques used can be applied to consider a broad class of dependence between the trajectories of the single agent and the state of the system. We apply the results in the case in which the admissible trajectories of the agents are the minimizers of a suitable integral functional depending also from the macroscopic evolution of the system.On a constrained infinite-time horizon linear quadratic gamehttps://zbmath.org/1522.910502023-12-07T16:00:11.105023Z"Krastanov, Mikhail I."https://zbmath.org/authors/?q=ai:krastanov.mikhail-ivanov"Rozenov, Rossen"https://zbmath.org/authors/?q=ai:rozenov.rossen"Stefanov, Boyan K."https://zbmath.org/authors/?q=ai:stefanov.boyan-kSummary: A linear quadratic differential game on an infinite-time horizon is studied in the case when the controls of the minimizing player are subject to constraints. A sufficient condition for a saddle point equilibrium is provided based on the conversion of the infinite-time horizon game to a game on a finite-time horizon. The method is applied to a simple monetary policy model as an illustrative example.Evasion from several pursuers in the game with coordinate-wise integral constraintshttps://zbmath.org/1522.910522023-12-07T16:00:11.105023Z"Ibragimov, Gafurjan"https://zbmath.org/authors/?q=ai:ibragimov.gafurjan-i"Salleh, Yusra"https://zbmath.org/authors/?q=ai:salleh.yusra"Alias, Idham Arif"https://zbmath.org/authors/?q=ai:alias.idham-arif"Pansera, Bruno Antonio"https://zbmath.org/authors/?q=ai:pansera.bruno-antonio"Ferrara, Massimiliano"https://zbmath.org/authors/?q=ai:ferrara.massimilianoSummary: We consider a simple motion evasion differential game of one evader from many pursuers in \({\mathbb{R}}^n\). The control functions of players are subjected to coordinate-wise integral constraints. If the position of the evader never coincides with the position of any pursuer, then we say that evasion is possible. In the present paper we obtain sufficient conditions of evasion. For any positive number \(\varepsilon\), a strategy for the evader is constructed, which guarantees the evasion in \(\varepsilon\)-neighborhood of a coordinate axis.Linear group pursuit problem with fractional derivatives, simple matrices, and different possibilities of playershttps://zbmath.org/1522.910532023-12-07T16:00:11.105023Z"Petrov, N. N."https://zbmath.org/authors/?q=ai:petrov.nikolai-nikandrovich"Machtakova, A. I."https://zbmath.org/authors/?q=ai:machtakova.alena-igorevnaSummary: In a finite-dimensional Euclidean space, we consider the problem of pursuit by a group of pursuers of one evader, which is described by a system of equations with a Caputo derivative of order \(\alpha \), where the sets of feasible controls are convex compact sets. We obtain sufficient conditions for the solvability of pursuit and evasion problems, in the study of which the method of resolving functions is used.Stackelberg and Nash equilibria in games with linear-quadratic payoff functions as models of public goodshttps://zbmath.org/1522.910642023-12-07T16:00:11.105023Z"Gorelik, Victor"https://zbmath.org/authors/?q=ai:gorelik.viktor-aleksandrovich"Zolotova, Tatiana"https://zbmath.org/authors/?q=ai:zolotova.tatianaSummary: The paper proposes a game model with an additive convolution of two criteria, describing public and personal interests. The first (general) criterion depends on strategies of all players and represents losses from the intensity of their activity. The second (particular) criterion for each player is a function of his strategy and reflects the income from his activities. The negative definite quadratic form is taken as a general criterion. The particular criterion of each player is linear, which is quite natural for the formalization of the income function. It turns out that the resulting game with linear-quadratic payoff functions has good properties, in particular, the independence of the leader's strategy in the Stackelberg equilibrium from the parameters of the follower's linear functions (in contrast to the Nash equilibrium). This property means that the leader does not need accurate information about the follower's objective function, and his strategy has the property of robustness.
For the entire collection see [Zbl 1508.90001].Imitative innovation or independent innovation strategic choice of emerging economies in non-cooperative innovation competitionhttps://zbmath.org/1522.911582023-12-07T16:00:11.105023Z"Liu, Yang"https://zbmath.org/authors/?q=ai:liu.yang.238"Liu, Zhiying"https://zbmath.org/authors/?q=ai:liu.zhiying"Xu, Kaifei"https://zbmath.org/authors/?q=ai:xu.kaifeiSummary: The importance of knowledge and technology is self-evident, especially the core technology of key nodes in the industrial chain, which will change the country's status in the supply chain, and even the national economic security. This scenario has led to a global non-cooperative innovation competition. In order to ensure the safety of local industrial chain and shorten the technological distance with developed countries, emerging economies can adopt imitative innovation by observing the core technologies from developed countries, or choose independent innovation strategy. How should emerging economies make the choice? We analyze this problem by establishing a dynamic non-cooperative technology development model. The research results show that when the innovation capacity gap between emerging economies and developed regions is large, the choice of imitation strategy is highly necessary. And when the gap is small, the independent innovation strategy can be selected. In addition, due to the existence of both domestic and foreign markets, developed countries can adopt strict policies to restrict the sale of products containing core technologies to overseas markets to limit the spillover of important technologies. We also consider the impact of policies that limit technology spillovers and show the impact of local market capacity in emerging economies.Stable dividends under linear-quadratic optimisationhttps://zbmath.org/1522.913032023-12-07T16:00:11.105023Z"Avanzi, B."https://zbmath.org/authors/?q=ai:avanzi.benjamin"Falden, D. K."https://zbmath.org/authors/?q=ai:falden.debbie-kusch"Steffensen, M."https://zbmath.org/authors/?q=ai:steffensen.mogensSummary: The optimisation criterion for dividends from a risky business is most often formalised in terms of the expected present value of future dividends. That criterion disregards a potential, explicit demand for the stability of dividends. In particular, within actuarial risk theory, the maximisation of future dividends has been studied as the so-called de Finetti problem [\textit{B. De Finetti}, ``On an alternative approach to the collective theory of risk'' (Italian), in: Transactions of the XV international congress of actuaries. New York, NY: The Society. 433--443 (1957)]. However, there the optimal strategies typically become so-called barrier strategies. These are far from stable, and suboptimal affine dividend strategies have recently received attention. In contrast, in the class of linear-quadratic problems, the demand for stability is explicitly stressed. These have often been studied in diffusion models different from the actuarial risk models. We bridge the gap between these thinking patterns by deriving optimal affine dividend strategies under a linear-quadratic criterion for an additive process. We characterise the value function by the Hamilton-Jacobi-Bellman equation, solve it, and compare the objective and the optimal controls to the classical objective of maximising the expected present value of future dividends. Thereby we provide a framework within which stability of dividends from a risky business, e.g. in classical risk theory, is explicitly demanded and obtained.Deep empirical risk minimization in finance: looking into the futurehttps://zbmath.org/1522.913122023-12-07T16:00:11.105023Z"Reppen, Anders Max"https://zbmath.org/authors/?q=ai:reppen.anders-max"Soner, Halil Mete"https://zbmath.org/authors/?q=ai:soner.halil-meteSummary: Many modern computational approaches to classical problems in quantitative finance are formulated as empirical loss minimization (ERM), allowing direct applications of classical results from statistical machine learning. These methods, designed to directly construct the optimal feedback representation of hedging or investment decisions, are analyzed in this framework demonstrating their effectiveness as well as their susceptibility to generalization error. Use of classical techniques shows that over-training renders trained investment decisions to become anticipative, and proves overlearning for large hypothesis spaces. On the other hand, nonasymptotic estimates based on Rademacher complexity show the convergence for sufficiently large training sets. These results emphasize the importance of synthetic data generation and the appropriate calibration of complex models to market data. A numerically studied stylized example illustrates these possibilities, including the importance of problem dimension in the degree of overlearning, and the effectiveness of this approach.
{{\copyright} 2022 Wiley Periodicals LLC.}Simulation of COVID-19 propagation scenarios in the Republic of Kazakhstan based on regularization of agent modelhttps://zbmath.org/1522.920652023-12-07T16:00:11.105023Z"Krivorotko, O. I."https://zbmath.org/authors/?q=ai:krivorotko.olga-i"Kabanikhin, S. I."https://zbmath.org/authors/?q=ai:kabanikhin.sergei-i"Bektemesov, M. A."https://zbmath.org/authors/?q=ai:bektemesov.maktagali-a"Sosnovskaya, M. I."https://zbmath.org/authors/?q=ai:sosnovskaya.m-i"Neverov, A. V."https://zbmath.org/authors/?q=ai:neverov.a-vSummary: An algorithm for modeling scenarios for new diagnosed cases of COVID-19 in the Republic of Kazakhstan is proposed. The algorithm is based on the treatment of incomplete epidemiological data and the inverse problem solving for the agent-based model (ABM) using a set of available epidemiological data. The main tool for building the ABM is the open library Covasim. In the event of a sudden change in the situation (appearance of a new strain, removal or introduction of restrictive measures, etc.), the model parameters are updated with additional information for the previous month (data assimilation). The inverse problem was solved by tree Parzen estimates optimization. As an example, two scenarios of COVID-19 propagation are given, calculated on December 12, 2021 for the period up to January 20, 2022. The scenario, which took into account the New Year holidays (published on December 12, 2021 on \url{covid19-modeling.ru}), almost coincided with what happened in reality (the error was 0,2\%).Infinite horizon stochastic impulse control with delay and random coefficientshttps://zbmath.org/1522.931882023-12-07T16:00:11.105023Z"Djehiche, Boualem"https://zbmath.org/authors/?q=ai:djehiche.boualem"Hamadène, Said"https://zbmath.org/authors/?q=ai:hamadene.said"Hdhiri, Ibtissem"https://zbmath.org/authors/?q=ai:hdhiri.ibtissem"Zaatra, Helmi"https://zbmath.org/authors/?q=ai:zaatra.helmiSummary: We study a class of infinite horizon impulse control problems with execution delay when the dynamics of the system is described by a general stochastic process adapted to the Brownian filtration. The problem is solved by means of probabilistic tools relying on the notion of Snell envelope and infinite horizon reflected backward stochastic differential equations. This allows us to establish the existence of an optimal strategy over all admissible strategies.Stochastic linear-quadratic control with a jump and regime switching on a random horizonhttps://zbmath.org/1522.931892023-12-07T16:00:11.105023Z"Hu, Ying"https://zbmath.org/authors/?q=ai:hu.ying"Shi, Xiaomin"https://zbmath.org/authors/?q=ai:shi.xiaomin"Xu, Zuo Quan"https://zbmath.org/authors/?q=ai:xu.zuoquanSummary: In this paper, we study a stochastic linear-quadratic control problem with random coefficients and regime switching on a random horizon \([0,T \wedge \tau]\), where \(\tau\) is a given random jump time for the underlying state process and \(T\) a constant. We obtain the explicit optimal feedback control and explicit optimal value of the problem by solving a system of stochastic Riccati equations (SREs) with jumps on the random horizon \([0,T \wedge \tau]\). By the decomposition approach stemming from filtration enlargement theory, we express the solution to the system of SREs with jumps in terms of another system of SREs involving only Brownian filtration on the deterministic horizon \([0,T]\). Solving the latter system is the key theoretical contribution of this paper and we accomplish this under three different conditions, one of which seems to be new in the literature. The above results are then applied to study a mean-variance hedging problem with random parameters that depend on both Brownian motion and Markov chain. The optimal portfolio and optimal value are presented in closed forms with the aid of a system of linear backward stochastic differential equations with jumps and unbounded coefficients in addition to the SREs with jumps.