Risk models with capital injections

Abstract

© 2016 Dr. Marjan QazviniOne of the main issues in ruin theory is that existing formulae for continuous time models can only be applied to some special claim size distributions and the analytical expressions for other claim size distributions do not exist. This thesis addresses this issue by considering discrete time models as approximations to continuous time models, including the classical risk model, the Markov-modulated risk model and the classical risk model with dividends. It also shows that how these models are affected by the introduction of capital injections. In Chapters 3 and 4 we construct a Gerber-Shiu function and use this to analyse the classical risk model with capital injections both analytically and probabilistically. Quantities such as the ultimate ruin probability and the joint density of the time of ruin and the number of claims until ruin are obtained by the inversion of the Laplace transform of our Gerber-Shiu function. In Chapter 5 we develop a discrete time model to approximate the probability of ruin in infinite and finite time under the classical risk model with capital injections, and show that capital injections can lead to a reduction in the probability of ruin even when claim amounts follow a heavy-tailed distribution. In Chapter 6 we extend our numerical algorithm from Chapter 5 to approximate the ultimate probability of ruin under a two-state Markov-modulated risk model with and without capital injections, and the density of the time of ruin under the same model with more than two states. The final chapter investigates dividend strategies with capital injections. We examine the effect of capital injections on the barrier and threshold strategies and consider a reinsurance arrangement that covers any fall below a positive pre-determined surplus level, so that the insurance company may operate indefinitely