CONVERGENCE OF NUMERICAL TIME-AVERAGING AND STATIONARY MEASURES VIA POISSON EQUATIONS

• Published in: SIAM journal on numerical analysis Vol. 48; no. 2; pp. 552 - 577
• Main Authors: MATTINGLY, JONATHAN C., STUART, ANDREW M., TRETYAKOV, M. V.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2010
• Subjects: Approximation; Convergence; Differential equations; Ergodic theory; Error analysis; Errors in statistics; Estimators; Markov processes; Mathematical analysis; Mathematical models; Numerical analysis; Numerical methods; Poisson distribution; Poisson equation; Random variables; Steepest descent method; Studies
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/090770527

Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed.