Nonlinearly perturbed

Dmitrii Silvestrov

Teoriâ slučajnyh processov, Tome 14 (2008) no. 4, pp. 129-164

Résumé

This paper is a survey of results presented in the recent book [25]. This book is devoted to studies of quasi-stationary phenomena in nonlinearly perturbed stochastic systems. New methods of asymptotic analysis for nonlinearly perturbed stochastic processes based on new types of asymptotic expansions for perturbed renewal equation and recurrence algorithms for construction of asymptotic expansions for Markov type processes with absorption are presented. Asymptotic expansions are given in mixed ergodic (for processes) and large deviation theorems (for absorption times) for nonlinearly perturbed regenerative processes, semi-Markov processes, and Markov chains. Applications to analysis of quasi-stationary phenomena in nonlinearly perturbed queueing systems, population dynamics and epidemic models, and risk processes are presented. The book also contains an extended bibliography of works in the area.

Détail

Keywords: Nonlinear perturbation, quasi-stationary phenomenon, pseudo-stationary phenomenon, stochastic system, renewal equation, asymptotic expansion, ergodic theorem, limit theorem, large deviation, regenerative process, regenerative stopping time, semi-Markov process, queueing system, population dynamics, epidemic model, lifetime, risk process, ruin probability, Cramér-Lundberg approximation, diffusion approximation.Nonlinear perturbation, quasi-stationary phenomenon, pseudo-stationary phenomenon, stochastic system, renewal equation, asymptotic expansion, ergodic theorem, limit theorem, large deviation, regenerative process, regenerative stopping time, semi-Markov process, queueing system, population dynamics, epidemic model, lifetime, risk process, ruin probability, Cramér-Lundberg approximation, diffusion approximation.Nonlinearly perturbed stochastic processes.
Mots-clés : Markov chain, absorption time, Markov chain, absorption time