### 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 |
---|---|

Pages (from-to) | 308-335 |

Number of pages | 28 |

Journal | Scandinavian Actuarial Journal |

Volume | 2019 |

Issue number | 4 |

Early online date | Jan 2019 |

DOIs | |

Publication status | Published - 2019 |

### Fingerprint

### 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

### Cooperations

- NAWI Graz

### Cite this

*Scandinavian Actuarial Journal*,

*2019*(4), 308-335. https://doi.org/10.1080/03461238.2018.1560357

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

Research output: Contribution to journal › Article › Research › peer-review

*Scandinavian Actuarial Journal*, vol. 2019, no. 4, pp. 308-335. https://doi.org/10.1080/03461238.2018.1560357

}

TY - JOUR

T1 - Approximation methods for piecewise deterministic Markov processes and their costs

AU - Thonhauser, Stefan Michael

AU - Leobacher, Gunther

AU - Kritzer, Peter Albin

AU - Szölgyenyi, Michaela

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - dividend maximisation

KW - phase-type approximations

KW - piecewise deterministic Markov process

KW - quasi-Monte Carlo methods

KW - Risk theory

UR - http://www.scopus.com/inward/record.url?scp=85059899894&partnerID=8YFLogxK

U2 - 10.1080/03461238.2018.1560357

DO - 10.1080/03461238.2018.1560357

M3 - Article

VL - 2019

SP - 308

EP - 335

JO - Scandinavian Actuarial Journal

JF - Scandinavian Actuarial Journal

SN - 0346-1238

IS - 4

ER -