Numerical Methods for Parameter Estimation in Stochastic Systems

Abstract

The objective of this work is to provide numerical simulations in support of a collection of existing results on estimation in two distinct types of stochastic systems. In the first chapter, we consider a linear time-invariant higher-order system of order that is subject to white noise perturbation. We numerically illustrate the result that the quadratic variation estimator of the white noise local variance is asymptotically biased when a forward-difference approach is used for numerically approximating the derivatives of the stochastic process, and that the bias can be eliminated by instead applying a specific alternative numerical differentiation scheme. Moreover, we consider the result that the straightforward discretization of a least squares estimation procedure for unknown parameters in the system leads to an asymptotically biased estimate. In the second chapter, we consider a controlled Markov chain, taking values on a finite state space, whose transition probabilities are assumed to depend on an unknown parameter belonging to a compact set. We first provide numerical illustration of the result that under a particular identifiability condition, the maximum likelihood estimator of this parameter is strongly consistent. Next, we illustrate that under alternative assumptions the sequence of maximum likelihood estimates converges and retains a desirable property relating to the Markov chain's transition probabilities. Additionally, we present a survey of several other related results.