Abstract
We discuss aspects of numerical methods for the computation of Gerber-Shiu or discounted penalty-functions in renewal risk models. We take an analytical point of view and link this function to a partial-integro-differential equation and propose a numerical method for its solution. We show weak convergence of an approximating sequence of piecewise-deterministic Markov processes (PDMPs) for deriving the convergence of the procedures. We will use estimated PDMP characteristics in a subsequent step from simulated sample data and study its effect on the numerically computed Gerber-Shiu functions. It can be seen that the main source of instability stems from the hazard rate estimator. Interestingly, results obtained using MC methods are hardly affected by estimation.
Highlights
In this article we study the computation of Gerber-Shiu functions
We show that our approximation converges to the exact functional by showing that the related generators of the piecewise-deterministic Markov processes (PDMPs) converge in an appropriate sense and applying techniques from the theory of Markov processes in addition to results from Kritzer et al (2019)
From the above results we obtain that the Gerber-Shiu function g as given in (4) is of adequate regularity, so that it can be represented as a solution to a partial-integro-differential equation
Summary
In this article we study the computation of Gerber-Shiu functions. These functions—or more precisely functionals—have been established in Gerber and Shiu (1998) in the context of the classical risk model, with the goal to study ruin relevant quantities in an universal manner. In the last references the approximation is used for theoretical distribution functions, but certainly one can approximate empirical distribution functions based on a sample by means of the EM-algorithm; see Asmussen et al (1996) Another non-parametric approach in the classical risk model works on the level of Laplace or Fourier transforms of the ruin probability or even the Gerber-Shiu function. Several results in this direction are presented by Shimizu (2012) and Shimizu and Zhang (2017), who apply kernel estimators for the claims’ distribution and investigate their properties on the level of the transforms
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