Numerical methods for the exit time of a piecewise-deterministic Markov process

TL;DR

This paper introduces a numerical method based on quantization to accurately compute the survival function and moments of the exit time for piecewise-deterministic Markov processes, with proven convergence and practical examples.

Contribution

It presents a novel quantization-based numerical approach for analyzing exit times of PDMPs, including convergence proofs and error bounds.

Findings

Method accurately computes survival functions and moments.

Convergence of the algorithm is theoretically established.

Demonstrated effectiveness through academic and reliability models.

Abstract

We present a numerical method to compute the survival function and the moments of the exit time for a piecewise-deterministic Markov process (PDMP). Our approach is based on the quantization of an underlying discrete-time Markov chain related to the PDMP. The approximation we propose is easily computable and is even flexible with respect to the exit time we consider. We prove the convergence of the algorithm and obtain bounds for the rate of convergence in the case of the moments. An academic example and a model from the reliability field illustrate the paper.