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The Markovian Shot-noise Risk Model: A Numerical Method for Gerber-Shiu Functions.

Simon Pojer1, Stefan Thonhauser1

  • 1Institute of Statistics, Graz University of Technology, Kopernikusgasse 24, Graz, 8010 Austria.

Methodology and Computing in Applied Probability
|February 14, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces a numerical method for approximating Gerber-Shiu functions in Markovian shot-noise environments. The findings demonstrate the convergence of these numerical methods to the exact penalty functions.

Keywords:
Gerber-Shiu functionsMarkov processesRisk theoryShot-NoiseWeak convergence

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Area of Science:

  • Actuarial Science
  • Probability Theory
  • Stochastic Processes

Background:

  • Discounted penalty functions, or Gerber-Shiu functions, are crucial in actuarial mathematics.
  • Markovian shot-noise environments present complex dynamics for analyzing these functions.
  • Existing analytical methods for these functions are often intractable.

Purpose of the Study:

  • To develop a numerical scheme for approximating discounted penalty functions in a Markovian shot-noise environment.
  • To establish the convergence of the numerical approximations to the true Gerber-Shiu functions.
  • To leverage the theory of piecewise-deterministic Markov processes (PDMPs) for this analysis.

Main Methods:

  • Utilizing the structure of piecewise-deterministic Markov processes (PDMPs).
  • Formulating and solving partial integro-differential equations (PIDEs) numerically.
  • Developing and analyzing continuous-time Markov chains with finite state spaces.
  • Employing generator convergence and weak convergence of Markov chains.

Main Results:

  • Demonstrated that Gerber-Shiu functions satisfy specific partial integro-differential equations (PIDEs).
  • Developed a robust numerical scheme to approximate these penalty functions.
  • Showed that the numerical solutions approximate the penalty functions of finite state-space Markov chains.
  • Proved the weak convergence of these Markov chains to the original PDMP.

Conclusions:

  • The proposed numerical scheme provides accurate approximations for discounted penalty functions in Markovian shot-noise environments.
  • The convergence results validate the numerical approach for analyzing complex actuarial models.
  • This work bridges the gap between theoretical PDMP models and practical numerical computation.