Title:

Asymptotic analysis of dependent risks and extremes in insurance and finance

In this thesis, we are interested in the asymptotic analysis of extremes and risks. The heavytailed distribution function is used to model the extreme risks, which is widely applied in insurance and is gradually penetrating in finance as well. We also use various tools such as copula, to model dependence structures, and extreme value theorem, to model rare events. We focus on modelling and analysing of extreme risks as well as demonstrate how the derived results that can be used in practice. We start from a discretetime risk model. More concretely, consider a discretetime annuityimmediate risk model in which the insurer is allowed to invest its wealth into a riskfree or a risky portfolio under a certain regulation. Then the insurer is said to be exposed to a stochastic economic environment that contains two kinds of risk, the insurance risk and financial risk. The former is traditional liability risk caused by insurance loss while the latter is the asset risk resulting from investment. Within each period, the insurance risk is denoted by a realvalued random variable X, and the financial risk Y as a positive random variable fulfils some constraints. We are interested in the ruin probability and the tail behaviour of maximum of the stochastic present values of aggregate net loss with Sarmanov or FarlieGumbelMorgenstern (FGM) dependent insurance and financial risks. We derive asymptotic formulas for the finiteruin probability with lightedtailed or moderately heavytailed insurance risk for both riskfree investment and risky investment. As an extension, we improve the result for extreme risks arising from a rare event, combining simulation with asymptotics, to compute the ruin probability more efficiently. Next, we consider a similar risk model but a special case that insurance and financial risks following the least risky FGM dependence structure with heavytailed distribution. We follow the study of Chen (2011) that the finitetime ruin probability in a discretetime risk model in which insurance and financial risks form a sequence of independent and identically distributed random pairs following a common bivariate FGM distribution function with parameter 1 ≤ θ ≤ 1 governing the strength of dependence. For the subexponential case, when 1 < θ ≤ 1, a general asymptotic formula for the finitetime ruin probability was derived. However, the derivation there is not valid for θ = 1. In this thesis, we complete the study by extending Chen's work to θ = 1 that the insurance risk and financial risk are negatively dependent. We refer this situation as the least risky FGM dependent insurance risk and financial risk. The new formulas for θ = −1 look very different from, but are intrinsically consistent with, the existing one for 1 < θ ≤ 1, and they offer a quantitative understanding on how significantly the asymptotic ruin probability decreases when θ switches from its normal range to its negative extremum. Finally, we study a continuoustime risk model. Specifically, we consider a renewal risk model with a constant premium and a constant force of interest rate, where the claim sizes and interarrival times follow certain dependence structures via some restriction on their copula function. The infinitetime absolute ruin probabilities are studied instead of the traditional infinitetime ruin probability with lighttailed or moderately heavytailed claimsize. Under the assumption that the distribution of the claimsize belongs to the intersection of the convolutionequivalent class and the rapidvarying tailed class, or a larger intersection class of Osubexponential distribution, the generalized exponential class and the rapidvarying tailed class, the infinitetime absolute ruin probabilities are derived.
