Approximation methods for piecewise deterministic Markov processes and their costs

TL;DR

This paper develops approximation methods for piecewise deterministic Markov processes (PDMPs) to compute quantities like ruin probabilities and expected dividends, using deterministic numerical integration and smoothing techniques, with convergence proofs and practical examples.

Contribution

It introduces a fixed point reformulation and smoothing approach for PDMPs, enabling deterministic integration methods and providing convergence analysis.

Findings

Deterministic cubature rules outperform Monte Carlo in accuracy.

The smoothing technique improves the numerical stability of integrals.

Convergence of the PDMP approximation is rigorously established.

Abstract

In this paper, we analyse piecewise deterministic Markov processes, as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of piecewise deterministic Markov processes. In this context, one is interested in computing certain quantities of interest such as the probability of ruin of an insurance company, or the insurance company's value, defined as the expected discounted future dividend payments until the time of ruin. Instead of explicitly solving the integro-(partial) differential equation related to the quantity of interest considered (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules; this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. On the analytical side, we prove a convergence result for our PDMP approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and provide a comparative study of deterministic and Monte Carlo integration.