Stochastic Approximation of the Paths of Killed Markov Processes Conditioned on Survival

TL;DR

This paper develops a stochastic pathwise approximation method for killed Markov processes conditioned on survival, extending existing representations and providing new insights into their quasi-limiting distributions.

Contribution

It introduces a novel pathwise construction for killed Markov processes and derives a stochastic approximation for their quasi-limiting distributions.

Findings

Pathwise description of killed Markov processes achieved

Stochastic approximation method for quasi-limiting distributions developed

Applicable to reducible killed Markov processes

Abstract

Reinforced processes are known to provide a stochastic representation for the quasi-stationary distribution of a given killed Markov process - describing the killed Markov process at fixed time instants. In this paper we shall adapt the construction to provide a pathwise description. We also obtain a stochastic approximation for the quasi-limiting distributions of reducible killed Markov processes as a corollary.