Numerical Ruin Probability in the Dual Risk Model with Risk-Free Investments

Sooie-Hoe Loke,Enrique Thomann

doi:10.3390/risks6040110

Abstract

In this paper, a dual risk model under constant force of interest is considered. The ruin probability in this model is shown to satisfy an integro-differential equation, which can then be written as an integral equation. Using the collocation method, the ruin probability can be well approximated for any gain distributions. Examples involving exponential, uniform, Pareto and discrete gains are considered. Finally, the same numerical method is applied to the Laplace transform of the time of ruin.

Highlights

The simplest surplus model in non-life insurance is known as the Cramér–Lundberg model or the classical risk model
All assumptions in the classical dual model (2) are retained, and it is further assumed that the company invests all of its surplus in a risk-free asset with constant force of interest a > 0
One can verify that the same is true for the dual model considered in this paper

Summary

Introduction

The simplest surplus model in non-life insurance is known as the Cramér–Lundberg model or the classical risk model. Many models incorporated investments with constant force of interest, for example, investing all (or part) of the surplus in bonds or time accounts The study of these risk models dated back to Segerdahl (1942), who considered the constant interest risk model and provided an explicit expression for the ruin probability when the claims are exponentially distributed. Dong and Wang (2008) studied the renewal risk model with constant force of interest and obtained an explicit expression for the ruin probability in terms of infinite series of iterated integrals. Their renewal model is more general than the Poisson process considered here, the objective and approach of this paper differ from.

The Model

Integral Equation Approach

F1 a a a

Numerical Scheme

The Laplace Transform of the Time of Ruin

Conclusions and Future Research