×

Approximation of passage times of \(\gamma\)-reflected processes with FBM input. (English) Zbl 1303.60027

Let \((X_t)_{t\geq 0}\) be a centered Gaussian process with almost surely continuous sample paths and covariance function \(\text{cov}(X_t,X_s)=(t^{2H}+s^{2H}-|t-s|^{2H})/2\), \(t,s\geq 0\), i.e., \((X_t)_{t\geq 0}\) is a standard fractional Brownian motion with Hurst index \(H\in(0,1)\).
The \(\gamma\)-reflected process with input process \(Y_t:=X_t-ct\) is defined by \(W_t:=Y_t-\gamma \inf_{s\in[0,t]}Y_s\), \(t\geq 0\), where \(\gamma\in[0,1]\) and \(c>0\) are two fixed constants.
The first and last passage times of the process \((W_t)_{t\geq 0}\) to a constant threshold \(u>0\) are \(\tau_1(u):=\inf\{t\geq 0:\,W_t>u\}\), \(\tau_2(u):=\sup\{t\geq 0:\,W_t>u\}\). Put \((\tau_1^*(u),\tau_2^*(u)):=(\tau_1(u),\tau_2(u))\) under the condition that \(\tau_1(u)<\infty\). The main result of this article is the convergence in distribution \(((\tau_1^*(u)-\tilde t_0 u)/A(u),(\tau_2^*(u)-\tilde t_0 u)/A(u))\to_D (\xi,\xi)\) as \(u\to\infty\), where \(\xi\) is a standard normal distributed random variable, \(\tilde t_0=H/(c(1-H))\) and \(A(\cdot)\) is a suitable norming function.
This joint convergence implies in particular \((\tau_2^*(u)-\tau_1^*(u))/A(u)\to 0\) in probability as \(u\to\infty\).
The paper also links ruin problems with extremes of nonhomogeneous Gaussian random fields defined by \((Y_t)_{t\geq 0}\).

MSC:

60G15 Gaussian processes
60G70 Extreme value theory; extremal stochastic processes
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry . Springer, New York. · Zbl 1149.60003
[2] Albin, J. M. P. (1990). On extremal theory for stationary processes. Ann. Prob. 18 , 92-128. · Zbl 0704.60029 · doi:10.1214/aop/1176990940
[3] Asmussen, S. (1987). Applied Probability and Queues . John Wiley, Chichester. · Zbl 0624.60098
[4] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities , 2nd edn. World Scientific, Hackensack, NJ. · Zbl 1247.91080
[5] Awad, H. and Glynn, P. (2009). Conditional limit theorems for regulated fractional Brownian motion. Ann. Appl. Prob. 19 , 2102-2136. · Zbl 1204.60084 · doi:10.1214/09-AAP605
[6] Dȩbicki, K. (2002). Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98 , 151-174. · Zbl 1059.60047 · doi:10.1016/S0304-4149(01)00143-0
[7] Dȩbicki, K. and Mandjes, M. (2003). Exact overflow asymptotics for queues with many Gaussian inputs. J. Appl. Prob. 40 , 704-720. · Zbl 1041.60036 · doi:10.1239/jap/1059060897
[8] Dȩbicki, K. and Mandjes, M. (2011). Open problems in Gaussian fluid queueing theory. Queueing Systems 68 , 267-273. · Zbl 1275.60039 · doi:10.1007/s11134-011-9237-y
[9] Dȩbicki, K. and Tabiś, K. (2011). Extremes of the time-average stationary Gaussian processes. Stoch. Process. Appl. 121 , 2049-2063. · Zbl 1227.60045 · doi:10.1016/j.spa.2011.05.005
[10] Dȩbicki, K., Hashorva, E. and Ji, L. (2014). Gaussian risk models with financial constraints. Scand. Actuarial J. DOI: 10.1080/03461238.2013.850442. · Zbl 1401.91130
[11] Dȩbicki, K., Michna, Z. and Rolski, T. (2003). Simulation of the asymptotic constant in some fluid models. Stoch. Models 19 , 407-423. · Zbl 1039.60040 · doi:10.1081/STM-120023567
[12] Dieker, A. B. and Yakir, B. (2014). On asymptotic constants in the theory of extremes for Gaussian processes. Bernoulli 20 , 1600-1619. · Zbl 1298.60043 · doi:10.3150/13-BEJ534
[13] Duncan, T. E. and Jin, Y. (2008). Maximum queue length of a fluid model with an aggregated fractional Brownian input. In Markov Processes and Related Topics: a Festschrift for Thomas G. Kurtz (Inst. Math. Statist. Collect. 4 ), Institute of Mathematical Statistics, Beachwood, OH, pp. 235-251. · Zbl 1167.60358 · doi:10.1214/074921708000000408
[14] Embrechts, P., Klüpelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. For Insurance and Finance . Springer, Berlin. · Zbl 0873.62116
[15] Griffin, P. S. (2013). Convolution equivalent Lévy processes and first passage times. Ann. Appl. Prob. 23 , 1506-1543. · Zbl 1347.60054 · doi:10.1214/12-AAP879
[16] Griffin, P. S. and Maller, R. A. (2012). Path decomposition of ruinous behavior for a general Lévy insurance risk process. Ann. Appl. Prob. 22 , 1411-1449. · Zbl 1259.60051 · doi:10.1214/11-AAP797
[17] Griffin, P. S., Maller, R. A. and Roberts, D. (2013). Finite time ruin probabilities for tempered stable insurance risk processes. Insurance Math. Econom. 53, 478-489. · Zbl 1304.91106
[18] Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems . John Wiley, New York. · Zbl 0659.60112
[19] Hashorva, E., Ji, L. and Piterbarg, V. I. (2013). On the supremum of \(\gamma\)-reflected processes with fractional Brownian motion as input. Stoch. Process. Appl. 123 , 4111-4127. · Zbl 1316.60054 · doi:10.1016/j.spa.2013.06.007
[20] Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83 , 257-271. · Zbl 0997.60057 · doi:10.1016/S0304-4149(99)00041-1
[21] Hüsler, J. and Piterbarg, V. (2008). A limit theorem for the time of ruin in a Gaussian ruin problem. Stoch. Process. Appl. 118 , 2014-2021. · Zbl 1151.60313 · doi:10.1016/j.spa.2007.11.006
[22] Hüsler, J. and Zhang, Y. (2008). On first and last ruin times of Gaussian processes. Statist. Prob. Lett. 78 , 1230-1235. · Zbl 1145.60021 · doi:10.1016/j.spl.2007.11.028
[23] Kozachenko, Y., Melnikov, A. and Mishura, Y. (2014). On drift parameter estimation in models with fractional Brownian motion. Statistics DOI: 10.1080/02331888.2014.907294. · Zbl 1396.62190
[24] Mandjes, M. (2007). Large Deviations for Gaussian Queues . John Wiley, Chichester. · Zbl 1125.60103
[25] Pickands, J., III (1969). Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145 , 75-86. · Zbl 0206.18901 · doi:10.2307/1995059
[26] Piterbarg, V. I. (1972). On the paper by J. Pickands ‘Upcrosssing probabilities for stationary Gaussian processes’. Vestnik Moscov. Univ. Ser. I Mat. Meh. 27 , 25-30 (in Russian). English translation: Moscow Univ. Math. Bull. 27, 19-23. · Zbl 0262.60020
[27] Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Transl. Math. Monogr. 148 ) American Mathematical Society, Providence, RI. · Zbl 0841.60024
[28] Piterbarg, V. I. (2001). Large deviations of a storage process with fractional Browanian motion as input. Extremes 4 , 147-164. · Zbl 1003.60053 · doi:10.1023/A:1013973109998
[29] Whitt, W. (2002). Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and Their Application to Queues . Springer, New York. · Zbl 0993.60001
[30] Zeevi, A. J. and Glynn, P. W. (2000). On the maximum workload of a queue fed by fractional Brownian motion. Ann. Appl. Prob. 10 , 1084-1099. \endharvreferences · Zbl 1073.60089 · doi:10.1214/aoap/1019487607
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.