Reachability Estimates of Piecewise Deterministic Markov Processes

Abstract

A piecewise deterministic Markov process (PDMP) is a stochastic process that is governed by random jumps at several time instances and evolves deterministically between those jumps. This paper presents an approximate solution for the reachability problem of such a PDMP. Given a PDMP that is defined on a bounded domain set and over a bounded time period, this paper examines the problem of estimating the probability that the PDMP’s sample paths will remain inside its domain set within the defined time period. The approach proposed in this paper is essentially constructed based on the solution of an initial boundary value problem (IBVP) of the considered PDMP. By imposing certain inequalities on the functions which consists in the solution of such an IBVP, this paper characterizes both under and over approximations of the probability that the PDMP’s sample paths will remain within its bounded domain.