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 heavy-tailed 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 discrete-time risk model. More concretely, consider a discrete-time annuity-immediate risk model in which the insurer is allowed to invest its wealth into a risk-free 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 later is the asset risk resulting from investment. Within each period, the insurance risk is denoted by a real-valued 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 Farlie-Gumbel-Morgenstern (FGM) dependent insurance and financial risks. We derive asymptotic formulas for the finite-ruin probability with lighted-tailed or moderately heavy-tailed insurance risk for both risk-free 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 heavy-tailed distribution. We follow the study of Chen (2011) that the finite-time ruin probability in a discrete-time 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 $-1\leq \theta \leq 1$ governing the strength of dependence. For the subexponential case, when $-1<\theta \leq 1$, a general asymptotic formula for the finite-time ruin probability was derived. However, the derivation there is not valid for $\theta =-1$. In this thesis, we complete the study by extending Chen's work to $\theta =-1$ 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 $\theta = −1$ look very different from, but are intrinsically consistent with, the existing one for $-1<\theta\leq 1$, and they offer a quantitative understanding on how significantly the asymptotic ruin probability decreases when $\theta$ switches from its normal range to its negative extremum. Finally, we study a continuous-time 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 inter-arrival times follow certain dependence structures via some restriction on their copula function. The infinite-time absolute ruin probabilities are studied instead of the traditional infinite-time ruin probability with light-tailed or moderately heavy-tailed claim-size. Under the assumption that the distribution of the claim-size belongs to the intersection of the convolution-equivalent class and the rapid-varying tailed class, or a larger intersection class of O-subexponential distribution, the generalized exponential class and the rapid-varying tailed class, the infinite-time absolute ruin probabilities are derived

Ruin models with investment income

This survey treats the problem of ruin in a risk model when assets earn investment income. In addition to a general presentation of the problem, topics covered are a presentation of the relevant integro-differential equations, exact and numerical solutions, asymptotic results, bounds on the ruin probability and also the possibility of minimizing the ruin probability by investment and possibly reinsurance control. The main emphasis is on continuous time models, but discrete time models are also covered. A fairly extensive list of references is provided, particularly of papers published after 1998. For more references to papers published before that, the reader can consult [47].Comment: Published in at http://dx.doi.org/10.1214/08-PS134 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org