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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 quasiMonte 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 phasetype approximations on the process level. We illustrate the smoothing technique for a risktheoretic example, and compare deterministic and Monte Carlo integration.
Original language  English 

Pages (fromto)  308335 
Number of pages  28 
Journal  Scandinavian Actuarial Journal 
Volume  2019 
Issue number  4 
Early online date  Jan 2019 
DOIs  
Publication status  Published  2019 
Keywords
 dividend maximisation
 phasetype approximations
 piecewise deterministic Markov process
 quasiMonte 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
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Projects
 1 Active

Special Research Area (SFB) F55 QuasiMonte Carlo Methods: Theory and Applications
Grabner, P., Tichy, R., Kusner, W. B., Ziefle, J., Brauchart, J., Iaco, M. R. & Aistleitner, C.
1/02/14 → 31/01/22
Project: Research project
Activities
 2 Talk at conference or symposium

Approximation methods for PDMPs and applications in risk theory
Stefan Michael Thonhauser (Speaker)11 Sep 2018Activity: Talk or presentation › Talk at conference or symposium › Science to science

Approximation methods for PDMPs and applications in risk theory
Stefan Michael Thonhauser (Speaker)14 Sep 2018Activity: Talk or presentation › Talk at conference or symposium › Science to science