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Approximation methods for piecewise deterministic Markov processes and their costs
Peter Kritzer1, Gunther Leobacher2, Michaela Szölgyenyi3,4
1Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria.
Scandinavian Actuarial Journal
|May 7, 2019
Summary
This study introduces a novel deterministic numerical integration method for analyzing piecewise deterministic Markov processes (PDMPs) in insurance mathematics. The approach enables accurate computation of key metrics like ruin probability, outperforming traditional Monte Carlo methods.
Area of Science:
- Stochastic Processes
- Numerical Analysis
- Insurance Mathematics
Background:
- Piecewise deterministic Markov processes (PDMPs) are crucial for modeling insurance mathematics problems, including ruin probability and company valuation.
- Traditional methods involving integro-(partial) differential equations are often analytically intractable for complex PDMP models.
- Monte Carlo methods, while versatile, can be computationally intensive and less precise for certain insurance calculations.
Purpose of the Study:
- To develop and validate a novel numerical integration approach for PDMPs applicable to insurance mathematics.
- To adapt PDMP problems for deterministic numerical integration, specifically quasi-Monte Carlo rules.
- To provide a more efficient and accurate computational framework for key insurance metrics.
Main Methods:
- Reformulation of cost functionals as fixed points of integral operators for iterative approximation.
- Application of deterministic numerical integration algorithms (quasi-Monte Carlo rules) instead of random ones.
- Introduction of a smoothing technique for integrands to enable the use of error bounds for deterministic cubature rules.
Main Results:
- A novel convergence result for the PDMPs approximation is proven, justifying phase-type approximations on the process level.
- The smoothing technique is demonstrated effectively on a risk-theoretic example.
- Deterministic integration methods show comparable or superior performance to Monte Carlo integration for the analyzed PDMP models.
Conclusions:
- The proposed deterministic integration method offers a viable and efficient alternative for analyzing PDMPs in insurance mathematics.
- The developed techniques facilitate accurate computation of critical insurance quantities, overcoming limitations of traditional analytical and simulation approaches.
- This work provides a theoretical foundation and practical demonstration for applying deterministic cubature to complex stochastic processes in finance and insurance.
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