Applications of Nonstandard Analysis to Markov Processes and Statistical Decision Theory

Abstract

We use nonstandard analysis to significantly generalize the well-known Markov chain ergodic theorem and establish a fundamentally new complete class theorem, making progress on two core problems in stochastic process theory and statistical decision theory, respectively. In the first part, we study the ergodicity of time-homogenous Markov processes. A time-homogeneous Markov process with stationary distribution $\pi$ is said to be ergodic if its transition probability converges to $\pi$ in total variation distance. In the most general setting of continuous-time Markov processes with general state spaces, there are few results characterizing the ergodicity of the underlying Markov processes. Using the method of nonstandard analysis, for every standard Markov process ${X_t}{t\geq 0}$, we construct a nonstandard Markov process ${X't}{t\in T}$ that inherits most of the key properties of ${X_t}{t\geq 0}$ hence establishing the ergodicity without technical conditions, such as on drift or skeleton chains. In the second part, we study the relationship between frequentist and Bayesian optimality, extending the line of work initiated by Wald in the 1940's. Existing results are subject to technical conditions that rule out semi-parametric decision problems and generally rule out non-parametric ones. Using nonstandard analysis, we show that, among decision procedures with finite risk functions, a decision procedure is extended admissible if and only if its extension has infinitesimal excess Bayes risk. The result holds in complete generality, i.e, without regularity conditions or restrictions on the model or the loss function. This nonstandard characterization of extended admissibility also generates a purely standard theorem: when risk functions are continuous on a compact Hausdorff parameter space, a procedure is extended admissible if and only if it is Bayes.