Numerical method for optimal stopping of piecewise deterministic Markov processes

TL;DR

This paper introduces a numerical method for solving optimal stopping problems in piecewise deterministic Markov processes by combining quantization and path-adapted discretization, providing convergence bounds and an $ ext{epsilon}$-optimal stopping time.

Contribution

It presents a novel numerical approach that effectively approximates the value function and optimal stopping time for PDMPs, with proven convergence properties.

Findings

The method achieves accurate approximations of the value function.

Convergence bounds for the proposed algorithm are established.

Numerical example demonstrates practical effectiveness.

Abstract

We propose a numerical method to approximate the value function for the optimal stopping problem of a piecewise deterministic Markov process (PDMP). Our approach is based on quantization of the post jump location---inter-arrival time Markov chain naturally embedded in the PDMP, and path-adapted time discretization grids. It allows us to derive bounds for the convergence rate of the algorithm and to provide a computable $\epsilon$-optimal stopping time. The paper is illustrated by a numerical example.