Dmitrii Silvestrov

Quasi-stationary distributions of a stochastic metapopulation model

A stochastic metapopulation model which explicitly considers first order interactions between local populations is constructed. The model takes the spatial arrangement of patches into account and keeps track of which patches are occupied and which are empty. The time-evolution of the meta-population is governed by a Markov chain with finite state space. We give a detailed description of the long term behaviour of the Markov chain. Many interesting biological issues can be addressed using the model. As an especially important example we discuss the so-called core and satellite species hypothesis in the light of the model.

Limit theorems for randomly stopped stochastic processes

The book is devoted to studies of weak limit theorems for randomly stopped stochastic processes and functional limit theorems for compositions of stochastic processes. A survey of basic results rel ...

Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems

The 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 ...

Limit theorems for mixed max-sum processes with renewal stopping

This article is devoted to the investigation of limit theorems for mixed max-sum processes with renewal type stopping indexes. Limit theoremsof weak convergence type are obtained as well as functional limit theorems.

Nonlinearly perturbed regenerative processes and pseudo-stationary phenomena for stochastic systems

New types of mixed large deviation and ergodic theorems are obtained for nonlinearly perturbed regenerative processes, semi-Markov processes, and continuous-time Markov chains with absorption. Applications to the analysis of pseudo-stationary phenomena for stochastic systems are discussed. Examples related to models of population dynamics and highly reliable queuing systems are considered.

Quasi-Stationary Phenomena for Semi-Markov Processes

New types of nonlinear asymptotical expansions are obtained for distribution of first hitting times and quasi-stationary distributions for nonlinearly perturbed semi-Markov processes.

Cramér-Lundberg approximation for nonlinearly perturbed risk processes

Abstract An extension of the classical Cramer–Lundberg approximation for ruin probabilities to a model of nonlinearly perturbed risk processes is presented. We introduce correction terms for the Cramer–Lundberg and diffusion type approximations, which provide the right asymptotic behaviour of relative errors in a perturbed model. The dependence of these correction terms on relations between the rate of perturbation and the speed of growth of an initial capital is investigated. Various types of perturbations of risk processes are discussed. The results are based on a new type of exponential asymptotics for perturbed renewal equations.

Threshold structure of optimal stopping strategies for American type option. I

The paper presents results of theoretical studies of optimal stopping domains of American type options in discrete time. Sufficient conditions on the payoff functions and the price process for the ...

Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

New algorithms for computing asymptotic expansionsfor power moments of hitting times and stationary andquasi-stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are basedon special techniques of sequential phase space reduction, which can be applied to models with an arbitrary asymptoticcommunicative structure of phase spaces.

A Pricing Process with Stochastic Volatility Controlled by a Semi-Markov Process

Abstract This paper is devoted to the investigation of the geometrical Brownian motion as a price process where the drift and volatility are controlled by a semi-Markov process. Conditions of risk-neutral measure are given as well as a formula for the risk-neutral price for European options. The discrete version, the binomial model controlled by a semi-Markov chain, is examined and limit theorems describing the transition from discrete time binomial to continuous time model are given. A system of partial differential equations for distribution functions of average volatility is given. Related Monte Carlo algorithms are described.