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

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.