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Approximation methods for piecewise deterministic Markov processes and their costs
| Content Provider | Scilit |
|---|---|
| Author | Kritzer, Peter Leobacher, Gunther Szölgyenyi, Michaela Thonhauser, Stefan |
| Copyright Year | 2019 |
| Description | 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. |
| Related Links | https://www.tandfonline.com/doi/pdf/10.1080/03461238.2018.1560357?needAccess=true https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6474733/pdf |
| Ending Page | 335 |
| Page Count | 28 |
| Starting Page | 308 |
| ISSN | 03461238 |
| e-ISSN | 16512030 |
| DOI | 10.1080/03461238.2018.1560357 |
| Journal | Scandinavian Actuarial Journal |
| Issue Number | 4 |
| Volume Number | 2019 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2019-04-21 |
| Access Restriction | Open |
| Subject Keyword | Applied Mathematics Risk Theory Piecewise Deterministic Markov Process Quasi-monte Carlo Methods Phase-type Approximations Dividend Maximisation |
| Content Type | Text |
| Resource Type | Article |
| Subject | Statistics and Probability Economics and Econometrics Statistics, Probability and Uncertainty |
