Approximating a Diffusion by a Hidden Markov Model

TL;DR

This paper establishes the equivalence between certain spectral and approximation properties of continuous-time Markov processes, showing they can be approximated by finite-state hidden Markov models and have discrete spectra under specific conditions.

Contribution

It proves the equivalence of Donsker-Varadhan conditions, finite-state HMM approximation, and resolvent kernel separability for a broad class of diffusions, linking spectral properties with approximation capabilities.

Findings

Markov processes satisfy equivalent spectral and approximation conditions.

Transition semigroup can be approximated by finite-state hidden Markov models.

Processes have purely discrete spectra under these conditions.

Abstract

For a wide class of continuous-time Markov processes, including all irreducible hypoelliptic diffusions evolving on an open, connected subset of $\RL^d$, the following are shown to be equivalent: (i) The process satisfies (a slightly weaker version of) the classical Donsker-Varadhan conditions; (ii) The transition semigroup of the process can be approximated by a finite-state hidden Markov model, in a strong sense in terms of an associated operator norm; (iii) The resolvent kernel of the process is `$v$-separable', that is, it can be approximated arbitrarily well in operator norm by finite-rank kernels. Under any (hence all) of the above conditions, the Markov process is shown to have a purely discrete spectrum on a naturally associated weighted $L_\infty$ space.