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Perturbed Markov chains are popular models for description of information networks. Insuch models, the transition matrix P0 of an information Markov chain is usually approximated bymatrix Pe = (1-e ...

The paper is devoted to studies of regularly and singularly perturbed Markov chains with damping component. In such models, a matrix of transition probabilities is regularised by adding a special damping matrix multiplied by a small damping (perturbation) parameterε. We perform a detailed perturbation analysis for such Markov chains, particularly, give effective upper bounds for the rate of approximation for stationary distributions of unperturbed Markov chains by stationary distributions of perturbed Markov chains with regularised matrices of transition probabilities, asymptotic expansions for approximating stationary distributions with respect to damping parameter, explicit coupling type upper bounds for the rate of convergence in ergodic theorems forn-step transition probabilities, as well as ergodic theorems in triangular array mode.

New algorithms for construction of asymptotic expansions for exponential and power-exponential moments of hitting times for nonlinearly perturbed semi-Markov processes are presented. The algorithms ...

Perturbed Markov chains are popular models for description of information networks. In such models, the transition matrix \(\mathbf {P}_0\) of an information Markov chain is usually approximated by matrix \(\mathbf {P}_{\varepsilon } = (1 - \varepsilon ) \mathbf {P}_0 + \varepsilon \mathbf {D}\), where \(\mathbf {D}\) is a so-called damping stochastic matrix with identical rows and all positive elements, while \(\varepsilon \in [0, 1\)] is a damping (perturbation) parameter. We perform a detailed perturbation analysis for stationary distributions of such Markov chains, in particular get effective explicit series representations for the corresponding stationary distributions \(\bar{\pi }_\varepsilon \), upper bounds for the deviation \(| \bar{\pi }_{\varepsilon }- \bar{\pi }_0 |\), and asymptotic expansions for \(\bar{\pi }_{\varepsilon }\) with respect to the perturbation parameter \(\varepsilon \).

Necessary and sufficient conditions for convergence in distribution of first-rare-event times and convergence in Skorokhod J-topology of first-rare-event-time processes for perturbed semi-Markov pr ...

New algorithms for computing power moments of hitting times and accumulated rewards of hitting type for semi-Markov processes. The algorithms are based on special techniques of sequential phase space reduction and recurrence relations connecting moments of rewards. Applications are discussed as well as possible generalisations of presented results and results of experimental studies., 24 pages

This paper presents a survey of research results obtained by the author and his collaborators in the areas of limit theorems for Markov-type processes and randomly stopped stochastic processes, ren ...

Asymptotic expansions for stationary and conditional quasi-stationary distributions of nonlinearly perturbed birth-death-type semi-Markov models are presented. Applications to models of population ...

This paper presents a survey of research results obtained by the authorand his collaborators in the areas of limit theorems for Markov-type processes andrandomly stopped stochastic processes, renew ...

Stochastic approximation methods for rewards of American type options are studied. Pay-off functions are non random possibly discontinuous functions or random cadlag functions. General conditions o ...

In Chapter 2, we present operational rules for Laurent asymptotic expansions given in two forms, with remainders given in the standard form o(⋅ ) and with explicit power-type upper bounds for remainders. In the former case, such rules should be possibly considered as generally known, while, in the latter case, asymptotic expansions have some novelty aspects.

In Chapter 4, new algorithms for construction of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes with finite phase spaces are presented. These algorithms are based on the technique of sequential phase space reduction, which can be applied to processes with an arbitrary asymptotic communicative structure of phase spaces. Asymptotic expansions are given in two forms, without and with explicit upper bounds for remainders.

In Chapter 6, we present a background for construction of numerical examples for asymptotic results presented in Chapters 2– 5, some numerical examples as well as a summary survey of applied perturbed stochastic systems.

In Chapter 5, we present asymptotic expansions for stationary and conditional quasi-stationary distributions of nonlinearly perturbed birth-death-type semi-Markov processes, which play an important role in many applications. In this case, the corresponding expansions can be given in a more explicit form.

In Chapter 3, we introduce a model of perturbed semi-Markov processes, formulate basic perturbation conditions, describe a one-step time-space screening procedure of phase space reduction for perturbed semi-Markov processes, introduce hitting times, and prove an invariant property of them with respect to the procedure of phase space reduction. We, also, present algorithms for re-calculation of asymptotic expansions for transition characteristics of nonlinearly perturbed semi-Markov processes with reduced phase spaces and algorithms for sequential reduction of phase space for semi-Markov processes and construction of Laurent asymptotic expansions, without and with explicit upper bounds for remainders, for power moment of hitting times.

New algorithms for computing asymptotic expansions, without and with explicit upper bounds for remainders, for stationary and quasi-stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to models with an arbitrary asymptotic communicative structure of phase spaces.

American type options with general payoff functions possessing polynomial rate of growth are considered for multivariate Markov price processes.Convergence results areobtained for optimal reward fu ...

New algorithms for computing of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces., 83 pages