# Approximation methods for piecewise deterministic Markov processes and their costs

Stefan Michael Thonhauser, Gunther Leobacher, Peter Albin Kritzer, Michaela Szölgyenyi

Research output: Contribution to journalArticleResearchpeer-review

### Abstract

In this paper, we analyse piecewise deterministic Markov processes (PDMPs), as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of PDMPs. There one is interested in computing certain quantities of interest such as the probability of ruin or the value of an insurance company. Instead of explicitly solving the related integro-(partial) differential equation (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. We prove a convergence result for our PDMPs 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 compare deterministic and Monte Carlo integration.

Original language English 308-335 28 Scandinavian Actuarial Journal 2019 4 Jan 2019 https://doi.org/10.1080/03461238.2018.1560357 Published - 2019

### Fingerprint

Piecewise Deterministic Markov Process
Approximation Methods
Smoothing Techniques
Insurance
Costs
Approximation
Integro-partial Differential Equations
Cubature
Probability of Ruin
Monte Carlo Integration
Quasi-Monte Carlo
Integrand
Integral Operator
Convergence Results
Justify
Numerical integration
Error Bounds
Fixed point
Computing
Piecewise deterministic Markov process

### Keywords

• dividend maximisation
• phase-type approximations
• piecewise deterministic Markov process
• quasi-Monte Carlo methods
• Risk theory

### ASJC Scopus subject areas

• Economics and Econometrics
• Statistics and Probability
• Statistics, Probability and Uncertainty

### Fields of Expertise

• Information, Communication & Computing

• NAWI Graz

### Cite this

Approximation methods for piecewise deterministic Markov processes and their costs. / Thonhauser, Stefan Michael; Leobacher, Gunther; Kritzer, Peter Albin; Szölgyenyi, Michaela.

In: Scandinavian Actuarial Journal, Vol. 2019, No. 4, 2019, p. 308-335.

Research output: Contribution to journalArticleResearchpeer-review

Thonhauser, Stefan Michael ; Leobacher, Gunther ; Kritzer, Peter Albin ; Szölgyenyi, Michaela. / Approximation methods for piecewise deterministic Markov processes and their costs. In: Scandinavian Actuarial Journal. 2019 ; Vol. 2019, No. 4. pp. 308-335.
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