On the surplus process of ruin theory when perturbed by a diffusion

Abstract

This thesis studies in detail the expected discounted function of a penalty at ruin which involves the time of ruin, the surplus immediately prior to the time of ruin, and the deficit at the time of ruin, based on the surplus process of ruin theory containing an independent Wiener (diffusion) process.

First, main background for this thesis is reviewed in chapter 1, which contains the surplus process of ruin theory with and without a Wiener process, the defective renewal equations for some expected (discounted) functions, reliability-based classification and equilibrium distribution.

In chapter 2, we will derive the defective renewal equation and the asymptotic formula for the expected discounted function of a penalty at time of ruin, and propose the Tijms-type approximation for and an upper and a lower bounds on a compound geometric distribution function. Moreover, the reliability-based class implications for the associated claim size distribution are also given. When the claim size distribution is a combination of exponentials or a mixture of Erlangs, explicit analytical solutions to the compound geometric distribution function and to the expected discounted probability of ruin due to oscillation and a claim, respectively can be obtained.

Moments are studied in chapter 3 include the (discounted) moment of the deficit at the time of ruin, the joint moment of the deficit at ruin and the time of ruin, and the moments of the time of ruin due to oscillation and caused by a claim, respectively.

In chapter 4, we give the explicit expressions for the (discounted) joint and marginal distribution functions of the surplus immediately before the time of ruin and the deficit at the time of ruin, and for the (discounted) distribution function of the amount of the claim causing ruin, and for the (discounted) distribution function of the amount of the claim causing ruin, Then the (discounted) probability density functions are obtained by differentiating the corresponding (discounted) distribution functions. In addition, the defective renewal equations for these (discounted) distribution functions and probability density functions, respectively, are also derived.

Finally, summary and future research are presented in chapter 5.