Almost sure convergence of numerical approximations for Piecewise Deterministic Markov Processes

TL;DR

This paper proves that numerical algorithms for simulating Piecewise Deterministic Markov Processes converge almost surely, with convergence rates matching the deterministic methods used, supported by applications in neuroscience modeling.

Contribution

It provides an almost sure convergence analysis for numerical simulation algorithms of PDMPs, linking stochastic simulation accuracy to deterministic method order.

Findings

Almost sure convergence rate equals the order of the deterministic method used.

Numerical examples demonstrate the theoretical convergence results.

Application to neuroscience models shows practical relevance.

Abstract

Hybrid systems, and Piecewise Deterministic Markov Processes in particular, are widely used to model and numerically study systems exhibiting multiple time scales in biochemical reaction kinetics and related areas. In this paper an almost sure convergence analysis for numerical simulation algorithms for Piecewise Deterministic Markov Processes is presented. The discussed numerical methods arise through discretisina a constructive method defining these processes. The stochastic problem of simulating the random, path-dependent jump times of such processes is reformulated as a hitting time problem for a system of ordinary differential equations with random threshold. Then deterministic continuous methods (methods with dense output) are serially employed to solve these problems numerically. We show that the almost sure asymptotic convergence rate of the stochastic algorithm is identical to the order of the embedded deterministic method. We illustrate our theoretical findings by numerical examples from mathematical neuroscience were Piecewise Deterministic Markov Processes are used as biophysically accurate stochastic models of neuronal membranes.