The paper is devoted to studies of perturbed Markov chains commonly used for description of information networks. In such models, the matrix of transition probabilities for the corresponding Markov chain is usually regularised by adding a special damping matrix multiplied by a small damping (perturbation) parameter ε. In this paper, we present results of the detailed perturbation analysis of Markov chains with damping component and numerical experiments supporting and illustrating the results of this perturbation analysis.
This chapter is devoted to studies of perturbed Markov chains, commonly used for the description of information networks. In such models, the matrix of transition probabilities for the corresponding Markov chain is usually regularized by adding aspecial damping matrix, multiplied by a small damping (perturbation) parameter ε. In this chapter, we present the results of detailed perturbation analysis of Markov chains with damping component and numerical experiments supporting and illustrating the results of this perturbation analysis.
Necessary and sufficient condition for diffusion and stable approximation of ruin probabilities for risk processes are formulated
A classical result in risk theory is the Cramér-Lundberg approximation which says that under some general conditions the exponentially normalized ruin probability converges. In this article, we state an explicit rate of convergence for the Cramér-Lundberg approximation for ruin probabilities in the case where claims are bounded, which is realistic for, e.g., reinsurance models. The method, used to get the corresponding results, is based on renewal and coupling arguments.
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 optimal stopping domains to have one- threshold structure are given. We consider monotone, convex and inhomogeneous-in- time payoff functions. The underlying asset’s price is modelled by an inhomogeneous discrete time Markov process. © 2006 American Mathematical Society.
Conditions, which provide a one-threshold structure for optimal stopping strategies for American type options, are given.
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 optimal stopping domains to have one-threshold structure are given. We consider monotone, convex and inhomogeneous-in-time payoff functions. The underlying asset's price is modelled by an inhomogeneous discrete time Markov process.
New results on a multi-theshold structure of optimal stopping strategies for American type options for Markov type price processes are presented.
Multivariate Markov price processes and American type options for such processes with generalpayoff functions with not more than polynomial rate of growth are considered. Convergence results are obtainedfor optimal reward functionals of American type options for perturbed multivariateMarkov processes and payoff functions. These results are applied to approximation tree type algorithmsfor American type options for exponential diffusion type priceprocesses including mean-reverse stochastic processesused to model stochastic dynamics of energy prices.
We consider the problem of optimal reselling of European options. A bivariate exponential diffusion process is used to describe the reselling model. In this way, the reselling problem is imbedded to the model of finding optimal reward for American type option based on this process. Convergence results are formulated for optimal reward functionals of American type options for perturbed multi-variate Markov processes. An approximation bivariate tree model is constructed and convergence of optimal expected reward for this tree model to the optimal expected reward for the corresponding reselling model is proved.
We consider the problem of optimal reselling of Europeanoptions. A bivariate exponential diffusion process is used todescribe the reselling model. In this way, the reselling problem isimbedded to the model of finding optimal reward for American typeoption based on this process. Convergence results are obtained foroptimal reward functionals of American type options for perturbedmulti-variate Markov processes. An approximation bivariate treemodel is constructed and convergence of optimal expected reward forthis tree model to the optimal expected reward for the correspondingAmerican type option is proved
Numerical algorithms for eveluation of higher order moments for semi-Markov rewards processes are presented. Results of numerical experiments are given and commented.
The model of nonlinearly perturbedcontinuous-time renewal equation is studied in this paper.The perturbation conditions considered involve asymptoticalexpansions with respect to asymptotic scale$\{\varphi_{n,m}(\varepsilon) = \varepsilon^{n +m\omega}\}$,with $n, m$ being non-negative integers and $\omega >1$ beingirrational number. Such asymptotical scale results in non-polynomialtype of asymptotic expansions for solutions for perturbed renewalequations. An example of risk processes with perturbations describedabove and asymptotic expansions in diffusion approximation for ruinprobabilities in this model are given.
A memorial notes about my teacher, Professor Mikhail Yadrenko, our joint work, cooperation, and friendship.
New method of asymptotic analysis for nonlinearly perturbed stochastic processes and systems are presented. Applications to queuing systems, population dynamics systems, and risk processes are described
Necessary and sufficient conditions for weak convergence of first-rare-event times and stochastic flows of first-rare-events for semi-Markov processes are obtained. Also necessary and sufficient conditions for diffusion and stable approximations for ruin probabilities for riosk processes age given.
General condition of convergence are given for optimal rewards of American type options for perturbed Markopv type price processes controlled by market stochastic indices
Conditions of convergence for optimal expected rewards of American type options are given for perturbed Markov type price processes controlled by stochstic indices
A program system for analysis and comparison of reincurance contracts is presented.
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, renewal theory and ergodic theorems forperturbed stochastic processes, quasi-stationary distributions for perturbed stochas-tic systems, methods of stochastic approximation for price processes, asymptoticexpansions for nonlinearly perturbed semi-Markov processes and applications ofthe above results to queuing systems, reliability models, stochastic networks, bio-stochastic systems, perturbed risk processes, and American-type options.
Exponential asymptotic expansion for distributions of the surplus prior and at the time of ruin are given for perturbed risk processes
Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes are given
Let X and Y be two complete, separable, metric spaces, xi(epsilon)(x), x is an element of X and nu(epsilon) be, for every epsilon is an element of[0, 1], respectively, a random field taking values in space Y and a random variable taking values in space X. We present general conditions for convergence in distribution for random variables xi(epsilon)(nu(epsilon)) that is the conditions insuring holding of relation, xi(epsilon)(nu(epsilon)) d ->xi(0)(nu(0)) as epsilon -> 0.
A survey on functional limit theorems for compositions of stochastic processes ispresented. Applications to stochastic processes with random scaling of time, randomsums, extremes with random sample size, generalised exceeding processes, sum- andmax-processes with renewal stopping, and shock processes are discussed.
This chapter presents a survey of results on improved asymptotics for ruin probabilities in the Cramér–Lundberg, diffusion, and stable approximations of ruin probabilities for perturbed risk processes, obtained by the author and his collaborators. These results are: exponential asymptotic expansions for ruin probabilities in the Cramér–Lundberg and diffusion approximations of ruin probabilities; necessary and sufficient conditions for convergence of ruin probabilities in the model of diffusion and stable approximations; and explicit exponential rates of convergence in the Cramér–Lundberg approximation for ruin probabilities for reinsurance risk processes.
The paper presents results of complete analysis and classification of individual ergodic theorems for perturbed alternating regenerative processes with semi-Markov modulation. New short, long and super-long time ergodic theorems for regularly and singular type perturbed alternating regenerative processes are presented.
Necessary and sufficient conditions for convergence in distribution offirst-rare-event times and convergence in Skorokhod J-topology of first-rare-event-time processes for perturbed semi-Markov processes with finite phase space are obtained.
This paper is a survey of results presented in the recent book [25]1) .This book is devoted to studies of quasi-stationary phenomena innonlinearly perturbed stochastic systems. New methods of asymptoticanalysis for nonlinearly perturbed stochastic processes basedon new types of asymptotic expansions for perturbed renewal equationand recurrence algorithms for construction of asymptotic expansionsfor Markov type processes with absorption are presented.Asymptotic expansions are given in mixed ergodic (for processes) andlarge deviation theorems (for absorption times) for nonlinearly perturbedregenerative processes, semi-Markov processes, and Markovchains. Applications to analysis of quasi-stationary phenomena innonlinearly perturbed queueing systems, population dynamics andepidemic models, and risk processes are presented. The book alsocontains an extended bibliography of works in the area.
The paper is a survey of the latest results on quasi-stationary phenomena in nonlinearly perturbed stochastic systems. New methods of asymptotic analysis for nonlinearly perturbed stochastic processes based on new type of asymptotic expansions for perturbed renewal equation and recurrence algorithms for the constructing of 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 phenomenon in nonlinearly perturbed queuing systems, population dynamics and epidemic models, and for risk processes are presented.
This book is the first volume of a two-volume monograph devoted to the study of limit and ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and alternating regenerative processes with semi-Markov modulation. The first volume presents necessary and sufficient conditions for weak convergence for first-rare-event times and convergence in the topology J for first-rare-event processes defined on regularly perturbed finite Markov chains and semi-Markov processes; new asymptotic recurrent algorithms of phase space reduction and effective conditions of weak convergence for distributions of hitting times and convergence of expectations of hitting times for regularly and singularly perturbed finite Markov chains and semi-Markov processes.
This book is the second volume of two volumes monograph devoted to the study of limit and ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and alternating regenerative processes with semi-Markov modulation. The second volume presents new super-long, long and short time ergodic theorems for perturbed alternating regenerative processes and multi-alternating regenerative processes modulating by regularly and singularly perturbed finite semi-Markov processes.
New methods of asymptotic analysis for nonlinearly perturbed stochstic processes and systems are presented.
necessary and sufficient condition of weak convergence for first-rara-event times for semi-Markov processes are formulated. Applications to asymptotic analysis of ruin probabilities for risk processes are discussed.
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 new type of asymptotic expansions for perturbed renewal equation and recurrence algorithms for the constructing of 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 phenomenon in nonlinearly perturbed queuing systems, population dynamics and epidemic models, and for risk processes are presented. The book also contains an extended bibliography of works in the area.