Asymptotic tail probabilities of risk processes in insurance and finance

In this thesis we are interested in the impact of economic and financial factors, such as interest rate, tax payment, reinsurance, and investment return, on insurance business. The underlying risk models of insurance business that we consider range from the classical compound Poisson risk model to the newly emerging and more general Lévy risk model. In these risk models, we assume that the claim-size distribution belongs to some distribution classes according to its asymptotic tail behavior. We consider both light-tailed and heavy-tailed cases.

Our study is through asymptotic tail probabilities. Firstly, we study the asymptotic tail probability of discounted aggregate claims in the renewal risk model by introducing a constant force of interest. In this situation we focus on claims with subexponential tails. We derive for the tail probability of discounted aggregate claims an asymptotic formula, which holds uniformly for finite time intervals. For various special cases, we extend this uniformity to be valid for all time horizons.

Then, we investigate the asymptotic tail probability of the maximum exceedance of a sequence of random variables over a renewal threshold. We derive a unified asymptotic formula for this tail probability for both light-tailed and heavy-tailed cases.

By using the previous result, we study how to capture the impact of tax payments on the ruin probability in the Lévy risk model. We introduce periodic taxation under which the company pays tax at a fixed rate on its net income during each period. Assuming the Lévy measure, representing the claim-size distribution in the Lévy risk model, has a subexponential tail, a convolution-equivalent tail, or an exponential-like tail, we derive for the ruin probability several explicit asymptotic relations, in which the prefactor varies with the tax rate, reflecting the impact of tax payments.

Finally, we consider the renewal risk model in which the surplus is invested into a portfolio consisting of both a riskless bond and a risky stock. The price process of the stock is modeled by an exponential Lévy process. We derive an asymptotic formula for the tail probability of the stochastically discounted net loss process.