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 languageEnglish
Pages (from-to)308-335
Number of pages28
JournalScandinavian Actuarial Journal
Volume2019
Issue number4
Early online dateJan 2019
DOIs
Publication statusPublished - 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

Cooperations

  • 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.
@article{840e6c6ee05a42fc9b72b85023be4ab6,
title = "Approximation methods for piecewise deterministic Markov processes and their costs",
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.",
keywords = "dividend maximisation, phase-type approximations, piecewise deterministic Markov process, quasi-Monte Carlo methods, Risk theory",
author = "Thonhauser, {Stefan Michael} and Gunther Leobacher and Kritzer, {Peter Albin} and Michaela Sz{\"o}lgyenyi",
year = "2019",
doi = "10.1080/03461238.2018.1560357",
language = "English",
volume = "2019",
pages = "308--335",
journal = "Scandinavian Actuarial Journal",
issn = "0346-1238",
publisher = "Taylor and Francis Ltd.",
number = "4",

}

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 -