Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

TL;DR

This paper establishes conditions under which stochastic integrals with respect to the state occupation measure of a Markov chain converge weakly, linking properties of the chain's generator and Dynkin martingale.

Contribution

It provides new sufficient conditions for weak convergence of these stochastic integrals, connecting state occupation measures, generator scaling, and martingale properties.

Findings

Weak convergence of state occupation measure under generator scaling

Sufficient conditions for weak convergence of stochastic integrals

Connection between occupation measure and Dynkin martingale

Abstract

Our aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale related to the state indicator function, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure.