Approximations of Piecewise Deterministic Markov Processes and their convergence properties

TL;DR

This paper introduces discretisation schemes for simulating piecewise deterministic Markov processes (PDMPs), analyzing their convergence properties and applying them to models in statistics and biology.

Contribution

The paper develops first and higher order approximation schemes for PDMPs, enabling their simulation when exact characteristics are unavailable, and studies their convergence behavior.

Findings

Pathwise convergence to the continuous PDMP as step size decreases

Convergence in law to the invariant measure of the PDMP

Application to models in computational statistics and biology

Abstract

Piecewise deterministic Markov processes (PDMPs) are a class of stochastic processes with applications in several fields of applied mathematics spanning from mathematical modeling of physical phenomena to computational methods. A PDMP is specified by three characteristic quantities: the deterministic motion, the law of the random event times, and the jump kernels. The applicability of PDMPs to real world scenarios is currently limited by the fact that these processes can be simulated only when these three characteristics of the process can be simulated exactly. In order to overcome this problem, we introduce discretisation schemes for PDMPs which make their approximate simulation possible. In particular, we design both first order and higher order schemes that rely on approximations of one or more of the three characteristics. For the proposed approximation schemes we study both pathwise convergence to the continuous PDMP as the step size converges to zero and convergence in law to the invariant measure of the PDMP in the long time limit. Moreover, we apply our theoretical results to several PDMPs that arise from the computational statistics and mathematical biology literature.