Department of Applied Mathematics

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Variance estimation for Markov processes
(Stellenbosch : Stellenbosch University, 2021-03) Blomerus, Wessel; Touchette, Hugo; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Applied Mathematics.
ENGLISH ABSTRACT: We study the asymptotic variance of additive functionals of Markov processes, used in statistics and stochastic modelling as estimators of model parameters. The observations generated by these processes are correlated, which complicates the estimation of the asymptotic variance. In practice, methods for estimating the asymptotic variance are based on either estimating the correlation function or the segmentation of the additive observable (batch mean method). In this thesis, we propose and study three new estimators, based on a link between the asymptotic variance, large deviation theory, and an equation of probability theory called the Poisson equation. The first two estimators rely on the fact that the solution of the Poisson equation can be represented as a conditional expectation. The third estimator is based on a stochastic approximation of the solution of the Poisson equation, suggested by recent works in large deviation theory, which describe the solution as an eigenfunction that can be iteratively estimated in an ‘online’ way as a simulation unfolds. We illustrate these three estimators for simple Markov processes, including Markov chains and diffusion processes, for which the asymptotic variance is exactly known.