On the time value of ruin for insurance risk models

This thesis studies ruin probabilities and ruin related quantities, using a unified treatment of analysis through the celebrated Gerber-Shiu (G-S) penalty function. For different insurance risk models, a G-S function discounts a penalty due at ruin, which may depend on the surplus before ruin and the deficit at ruin. These insurance risk models include Sparre Andersen's risk model, both in a continuous and in a discrete time setting, diffusion perturbed Sparre Andersen models, as well as risk models with a constant dividend barrier. All these models are extensions of the classical risk model and of diffusion perturbed classical risk model. These G-S penalty functions, considered as functions of initial surplus, satisfy certain integral equations or integro-differential equations, which can be solved to yield defective renewal equations. Such defective renewal equations have a natural probabilistic interpretation, which relies heavily on the roots to a generalized Lundberg's fundamental equation that have a positive real part. These generalized Lundberg equations are from an appropriately chosen exponential martingale. The defective renewal equations (also called recursive formulas in discrete models), that the expected penalty functions satisfy, allow the use of the existing techniques in renewal theory. They can be used to analyze many quantities associated with the time of ruin, such as explicit expressions, bounds, approximations and asymptotic formulas for ruin probabilities, the Laplace transform (or gene-rating function in discrete models) of the time of ruin, the discounted joint and marginal distribution of the surplus immediately before ruin and the deficit at ruin, as well as their moments. Finally, explicit results for the G-S discounted penalty function can be solved when the initial reserve is zero and when the claim sizes are rationally distributed, i.e., the Laplace transform of the claim size density is a rational function

The Diffusion Perturbed Compound Poisson RiskModel with a Dividend Barrier

ISBN 07340 3559 4We consider a diffusion perturbed classical compound Poisson risk modelin the presence of a constant dividend barrier. Integro-differential equationswith certain boundary conditions for the expected discounted penalty(Gerber-Shiu) functions (caused by oscillations or by a claim) are derivedand solved. Their solutions can be expressed in terms of the Gerber-Shiufunctions in the corresponding perturbed risk model without a barrier. Finally,explicit results are given when the claim sizes are rationally distributed

On the Time Value of Ruin in the Discrete Time Risk Model

Using an approach similar to that of Gerber and Shiu (1998), a recursive formula is given for the expected discounted penalty due at ruin, in the discrete time risk model. With it the joint distribution of three random variables is obtained; time to ruin, the surplus just before ruin and the deficit at ruin. The time to ruin is analyzed through its probability generating function (p.g.f.). The joint distribution for the compound binomial model is derived in Cheng et al. (2000) using martingale techniques and a duality argument. Here we find a recursive formula for the p.g.f. of ruin time T; the discounted moments of the deficit at ruin and the surplus just before ruin. A detailed discussion is given in the case u = 0 and when the claim size in a unit time is geometrically distributed

On a discrete time risk model with delayed claimsand a constant dividend barrier

In this paper a compound binomial risk model with a constant dividend barrier isconsidered. Two types of individual claims, main claims and by-claims, are de¯ned,where by-claims are produced by the main claims and may be delayed for one timeperiod with a certain probability. Some prior work on these time-correlated claimshas been done by Yuen and Guo (2001) and the references therein. Formulae for theexpected present value of dividend payments up to the time of ruin are obtained fordiscrete-type individual claims, together with some other results of interest. Explicitexpressions for the corresponding results are derived in a special case, for which acomparison is also made to the original discrete model of De Finetti (1957)