Optimal stopping for partially observed piecewise-deterministic Markov processes

TL;DR

This paper develops a recursive filtering approach and a numerical method for solving optimal stopping problems in partially observed piecewise-deterministic Markov processes, providing convergence guarantees and practical algorithms.

Contribution

It introduces a new recursive formulation of the optimal filter and a quantization-based numerical method for approximating the value function in this setting.

Findings

Proposed a convergent numerical algorithm for the problem.

Bounded the convergence rate of the approximation.

Derived the dynamic programming equation for the partially observed process.

Abstract

This paper deals with the optimal stopping problem under partial observation for piecewise-deterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed optimal stopping problem. Then, we propose a numerical method, based on the quantization of the discrete-time filter process and the inter-jump times, to approximate the value function and to compute an actual $\epsilon$-optimal stopping time. We prove the convergence of the algorithms and bound the rates of convergence.