For many models it is challenging to study the hitting time directly. We use cookies to distinguish you from other users and to provide you with a better experience on our websites. In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system that is perturbed from one with known eigenvectors and eigenvalues. Two approaches to the construction of perturbation bounds for. Saratov state university abstract weshowthat,forreversiblecontinuoustimemarkovchains,theclosenessofthenonzero eigenvalues of the generator to zero provides complete information about the sensitivity. Perturbation bounds for markov chains with general state space. Perturbation bounds for markov chains with general state. The algorithm is tested for a single equation, coupled two equations, and coupled three equations. The previously developed new perturbationiteration algorithm has been applied to differential equation systems for the first time. The aim of this paper is to investigate the stability of markov chains with general state space. The iteration algorithm for systems is developed first. We establish upper and lower bounds on this condition number in terms of subdominant eigenvalues of the transition map.
By the latter we mean that transition probabilities of a markov chain, with several ergodic classes, are perturbed such that rare transitions among the different ergodic classes of the unperturbed chain are allowed. Moment bounds and ergodicity of switching diffusion systems involving twotimescale markov chains. Other applications of our results to phasetype queues will be. An example in denumerable decision processes fisher, lloyd and ross, sheldon m. Introduction a perturbation in a markov chain can be referred as a slight change in the entries of the corresponding transition stochastic matrix, resulting in structural changes in the underlying process, for example, sets. D and d are derived in terms of a drift condition, where.
Stewart computer science department institute for advanced computer studies university of maryland college park, maryland jiguang sun computing center ofthe chinese academy of sciences beijing, china academic press, inc. We show how to reduce the complex markovswitching problem to solving a system of quadratic polynomial equations. Perturbation theory for markov reward processes with applications to queueing systems volume 20 issue 1 nico m. Rateoptimal perturbation bounds for singular subspaces with. Perturbation bounds for structured robust stability. New perturbation bounds for denumerable markov chains, linear algebra and its applications, 432, 16271649. This is useful for studying how sensitive the original systems eigenvectors and eigenvalues are to changes in the system. Journal of science, engineering and technology, waset world academy of science, engineering and technology ed pp. New perturbation bounds for denumerable markov chains linear algebra and its applications, vol. This paper develops a general perturbation methodology for constructing highorder approximations to the solutions of msdsge models. Puterman skip to main content we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Apr 15, 2008 read strong stability and perturbation bounds for discrete markov chains, linear algebra and its applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Markovswitching dsge msdsge modeling has become a growing body of literature on economic and policy issues related to structural shifts. Introduction a perturbation in a markov chain can be referred as a slight change in the entries of the corresponding transition stochastic matrix, resulting in structural changes in the underlying process, for.
Proceedings ieee conference on decision and control, 2. We perform perturbation analysis in the setting of discretetime markov chains. This article provides series expansions of the stationary distribution of a finite markov chain. Additive perturbation bounds on the eigenvectors of a hermitian matrix. Linear algebra and its applications journal homepage. Under suitable stability conditions, numerical approximations can be derived from the update formulas, and we show that the algorithms converge at a geometric. Perturbation analysis of finite markov chains has received much attention in the literature over recent years see in. We use information technology and tools to increase productivity and facilitate new forms of scholarship.
Twodimensional harmonic oscilator 3 timedependent perturbation theory 4 literature igor luka cevi c. Measurevalued differentiation for stationary markov. Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by. On the existence of quasistationary distributions in denumerable rtransient markov chains authors. New perturbation bounds for denumerable markov chains citeseerx. This thesis is concerned with studying the hitting time of an absorbing state on markov chain models that have a countable state space. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain, where the bounds are given by the. Qbd processes, which constitute a wide class of structured markov chains. Lower bounds, which show that the individual perturbation bounds are rateoptimal, are also given.
Siam journal on numerical analysis society for industrial. Create an aipowered research feed to stay up to date with new papers like this posted to arxiv. We study the parametric perturbation of markov chains with denumerable state spaces. Denumerable markov processes with bounded generators. Our bounds are sharp, we do not impose any structural assumptions on the transition matrix or on the perturbation, and computing the bounds has the same complexity as computing the invariant distribution or computing other bounds in the literature. New perturbation bounds for denumerable markov chains.
Jul 17, 2006 2011 perturbation analysis of continuoustime absorbing markov chains. Meyer 1992 has developed inequalities in terms of the nonunit eigenvalues h, j 2. Regular perturbation of vgeometrically ergodic markov chains. Comparison of perturbation bounds for the stationary distribution of a markov chain. Australia received september 1992 revised november 1992 abstract. Perturbation theory for markov reward processes with. New perturbation bounds for denumerable markov chains core. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain, where. Numerical examples are given to illustrate the performance of the algorithm. Strong stability and perturbation bounds for discrete markov chains strong stability and perturbation bounds for discrete markov chains rabta, boualem. Perturbation analysis for denumerable markov chains with application to queueing models. We present update formulas that allow us to express the stationary distribution of a continuoustime markov process with denumerable state space having generator matrix q through a continuoustime markov process with generator matrix q.
We provide a unified approach to pamc for finite and denumerable markov. Perturbation bounds for quantum markov processes and their. New perturbation bounds for denumerable markov chains new perturbation bounds for denumerable markov chains mouhoubi, zahir. Timedependent perturbation theory literature 1 timeindependent nondegenerate perturbation theory general formulation firstorder theory secondorder theory 2 timeindependent degenerate perturbation theory general formulation example. Moment bounds and ergodicity of switching diffusion systems. We consider both regular and singular perturbations. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Strong stability and perturbation bounds for discrete markov. Strong stability and perturbation bounds for discrete markov chains. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of inhomogeneous markov chains with continuous time and a finite or countable state space. New perturbation bounds for denumerable markov chains, linear. Abstractthis paper is devoted to perturbation analysis of denumerable markov chains.
We consider three basic matrix norms to capture the perturbation distance, and focus on the computational aspect. Solutions are compared with those of variational iteration method and numerical solutions, and a good. In the present paper we propose an approach to the construction of general estimates for the perturbation bounds of markov chains in terms of special weighted. Summary in this paper, our interest is in the perturbation analysis of level.
Strong stability and perturbation bounds for discrete. In the first part, we introduce a condition number that measures the sensitivity of fixed points of a quantum channel to perturbations. Perturbation bounds perturbation analysis of markov chains residual matrix norm ergodicity coef. Series expansions for continuoustime markov processes. Perturbation analysis for continuoustime markov chains. Harcourt bruce jovanovich, publishers boston san diego new york london sydney tokyo toronto. Sensitivity of finite markov chains under perturbation e. Gaussian elimination, perturbation theory and markov. Moreover, we obtain perturbation bounds on the stationary distributions, which extends the results by liu 2012 for. Error bounds for augmented truncation approximations of markov. Singular perturbation analysis for countable markov chains. For finite irreducible markov chains many perturbation bounds for the.
Sensitivity of finite markov chains under perturbation. Moment bounds and ergodicity of switching diffusion. A critical feature of the technique is a middle step that breaks the problem into solvable and perturbation parts. Denumerable markov chains can be used to represent many real systems. Nunezqueijaperturbation analysis for denumerable markov chains with. I present a perturbative approach that allows one to uniformly bound the difference between the hitting time moment generating functions of two markov chains in a neighbourhood of the origin. To address the effects of uncertainty in probability estimates, in previous work we have developed a variety of techniques for perturbation analysis of discrete and continuoustime markov chains dtmcs and ctmcs.
This paper is devoted to perturbation analysis of denumerable markov chains. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain, where the bounds are given by the weighted. Reliability modelling and data analysis of vacuum circuit breaker subject to random shocks. A critical account of perturbation analysis of markov chains.
Markov chains, deviation matrix, linear pogramming, perturbation matrix analysis 1. We study general statespace markov chains that depend on a parameter, say, sufficient conditions are established for the stationary performance of such a markov chain to be differentiable with respect to. On perturbation bounds for continuoustime markov chains. Perturbation theory and finite markov chains researchgate. This section may be regarded as a complement of daleys work 3. Perturbation results for nearly uncoupled markov chains with applications to iterative methods. June 29, 2018 abstract in this paper, new conditions for the stability of vgeometrically ergodic markov chains are introduced. In the present paper we propose an approach to the construction of general estimates for the perturbation bounds of markov chains in terms of special weighted norms related to total variation. Perturbation analysis for denumerable markov chains with. Qbd processes, which constitute a wide class of structured markov. Introduction the purpose of this paper is to describe the special problems that emerge when gaussian elimination is used to determinine the stead. Finite continuous time markov chains theory of probability. Introduction to stochastic processes, prenticehall, new jersey. Singularly perturbed discretetime markov chains, siam journal on applied mathematics, 6, 834854.
Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain, where the. Rateoptimal perturbation bounds for singular subspaces. We investigate the stability of quantum markov processes with respect to perturbations of their transition maps. Mar 19, 20 we investigate the stability of quantum markov processes with respect to perturbations of their transition maps. Taylor series expansions for stationary markov chains. Finally, in section 4, we explicitly obtain the quasistationary distributions of a leftcontinuous random walk to demonstrate the usefulness of our results. However, there are only few references available on perturbation analysis of markov chains with an in.
Apr 30, 2015 we investigate perturbation for continuoustime markov chains ctmcs on a countable state space. Perturbation methods for markovswitching models andrew foerstery juan rubioramirezz dan waggonerx tao zhaaugust 2, 2012 abstract markov switching models are a way to consider discrete changes in the economic environment, such as policy changes, and allow agents in the economy to form expectations over these changes. For finite irreducible markov chains many perturbation bounds for the stationary vector are available. This leads to an efficient numerical algorithm for computing the stationary distribution of a finite markov chain. Perturbation methods for markovswitching dsge models.
Siam journal on numerical analysis volume 25, issue 3. Robust stability of a linear multivariable system, in the sense of robustness under multiplicative transfer function perturbation, is necessarily preserved under sufficiently small perturbations in the model parameters i. We present new conditions for the strong stability of markov chains after a small perturbation of their transition kernels. The problems of stability and the corresponding estimates were considered for new classes of processes in zeifman. We study the parametric perturbation of markov chains with denumerable. Seneta school of mathematics and statistics, university of sydney, nsu. This paper compares and analyzes bounds found in the literature for finite and denumerable markov chains and introduces new bounds based. Mar 15, 2010 new perturbation bounds for denumerable markov chains new perturbation bounds for denumerable markov chains mouhoubi, zahir. Sharp entrywise perturbation bounds for markov chains. Dec 19, 2017 the aim of this paper is to investigate the stability of markov chains with general state space.
We study general statespace markov chains that depend on a parameter, say, sufficient conditions are established for the stationary performance of such a markov chain to be differentiable with respect to specifically, we study the case of unbounded performance functions and thereby extend the result on weak differentiability of stationary distributions of markov chains to unbounded. Regular perturbation of vgeometrically ergodic markov chains deborah ferre, loic herve, james ledoux. For many markov chains of practical interest, the invariant distribution is extremely sensitive to perturbations of some entries of the. Let p be the transition matrix of a positive recurrent markov chain on the integers.
On the existence of quasistationary distributions in. Semigroups of conditioned shifts and approximation of markov processes kurtz, thomas g. Discretetime markov chains with twotime scales and a countable. We investigate perturbation for continuoustime markov chains ctmcs on a countable state space. Also, we obtain perturbation bounds with respect to different quantities. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal. The results are based on an extension of the standard perturbation theory formulated by keller and liverani. Two approaches to the construction of perturbation bounds. Perturbations of countable markov chains and processes. In this paper, our interest is in the perturbation analysis of level. Measurevalued differentiation for stationary markov chains.
If p is a transition matrix of a markov chain, andequation is derived by perturbing. Haverford college 2005 dissertation submitted in partial satisfaction of. Gaussian elimination, perturbation theory, and markov chains g. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain, where the bounds are given by the weighted supremum norm. Xiaoyue li a,1, rui wang a,b, george yin c,2 a school of mathematics and statistics, northeast normal university, changchun, jilin, 024, china department of economics, university of kansas, lawrence, ks 66045, usa c department of. Pdf series expansions for continuoustime markov chains. Series expansions for finitestate markov chains semantic. Perturbationiteration method for firstorder differential. Bounds on convergence of entropy rate approximations in.