On Computations in Renewal Risk Models—Analytical and Statistical Aspects

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.

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

“…For the path-differentiability, we follow the line of arguments as given in Strini and Thonhauser (2020). Define for some deterministic r > 0 the bounded stopping time = r ∧ T 1 , where T 1 denotes the first jump-time of the PDMP.…”

Nonparametric estimation of some dividend problems in the perturbed compound Poisson model

“…In insurance risk theory, Chan et al [5,6] applied the COS method to approximate the ruin probability and the expected discounted penalty function, respectively; Zhang used the COS method to approximate the density function of the time to ruin; Yang et al [28] applied a two-dimensional COS method to estimate the discounted density function of the deficit at ruin; Lee et al [14] studied the finite time ruin probabilities by the COS method. On the estimation of the expected discounted penalty function with dividend payments, we also would like to point out the work Strini and Thonhauser [22]. In their paper, ruin problems in a renewal risk model with a general surplus dependent premium rate are studied.…”