A Perturbation Theory for Ergodic Markov Chains and Application to Numerical Approximations

• Published in: SIAM journal on numerical analysis Vol. 37; no. 4; pp. 1120 - 1137
• Main Authors: Shardlow, T., Stuart, A. M.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2000
• Subjects: Applied mathematics; Approximation; Banach spaces; Computational mathematics; Differential equations; Ergodic theory; Exact sciences and technology; Integers; Markov analysis; Markov chains; Markov processes; Mathematical analysis; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Noise; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods; Ordinary differential equations; Perturbation theory; Probability and statistics; Probability theory and stochastic processes; Random variables; Sciences and techniques of general use; Spacetime; Stochastic analysis; Markov process; Ergodic theory; Brownian noise; Exponential distribution; Differential equation; Stochastic partial differential equation; Perturbation; Euler scheme; Partial differential equation; Truncation error; Numerical approximation; Markov chain; Brownian motion; Parabolic equation; Sigma algebra; Impulse; Stochastic equation; Stochastic differential equation
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/S0036142998337235

Perturbations to Markov chains and Markov processes are considered. The unperturbed problem is assumed to be geometrically ergodic in the sense usually established through the use of Foster-Lyapunov drift conditions. The perturbations are assumed to be uniform, in a weak sense, on bounded time intervals. The long-time behavior of the perturbed chain is studied. Applications are given to numerical approximations of a randomly impulsed ODE, an Ito stochastic differential equation (SDE), and a parabolic stochastic partial differential equation (SPDE) subject to space-time Brownian noise. Existing perturbation theories for geometrically ergodic Markov chains are not readily applicable to these situations since they require very stringent hypotheses on the perturbations.