Research On Ruin Probabilities Of Insurance Risk Models With Investment Returns And Dependence Structures

In modern insurance risk theory, one of the core problems actuaries have to face ishow to measure the risks of insurance businesses with stochastic investment returns anddependence structures. The thesis focuses on the estimation problems of ruin probabili-ties under several important risk models. For various financial investment environmentsand different dependence structures of risks, the thesis establishes asymptotic estimatesor inequalities for ruin probabilities of initial capital, and discusses their applications toinsurance and finance. The results in this thesis enrich the contents of risk theory and areimportant from the viewpoint of insurance practice.First, the thesis introduces the background of insurance risk theory and recalls someresults which are closely related to the focuses in the thesis, and for later use, gives severalclasses of heavy-tailed distributions which are widely used in insurance risk theory.Second, the thesis studies the estimation problems on ruin probabilities under renewalrisk models with exponential Lévy process investment returns and one-sided linear claims.Suppose insurers can make risky and risk-free investments for their insurance surplus anduse constant investment strategy to allocate the surplus among risky and risk-free assets.The price process of the risky assets is an exponential Lévy process. Claim sizes are as-sumed to follow a one-sided linear process with independent and identically distributedsteps. When the step-size distribution has a subexponential and dominated tail, by us-ing the large derivation theory of randomly weighted sums, the thesis establishes uniformasymptotic estimates for ruin probabilities under the risk models of initial capital over cer-tain time regions. Furthermore, when the step-size distribution has a regularly varying tail,the thesis derives some asymptotic estimates for ruin probabilities under the risk modelsof initial capital which hold uniformly over certain infinite-time regions.Third,thethesisinvestigatestheasymptoticbehaviorsofruinprobabilitiesunderPois-son risk models of initial capital with general process investment returns and bivariate up-pertailindependentclaims. Investmentreturnsareassumedtobeageneraladaptedprocesswith cádlág paths, while claim sizes are assumed to be identically distributed and bivari-ate upper tail independent. For the case that the claim-size distribution has a consistently varying tail, by expressing the surplus process as randomly weighted sums and applyingthe large derivation theory of randomly weighted sums, the thesis obtains the asymptoticformulas for finite-and infinite-time ruin probabilities of initial capital. As applications,the thesis discusses the estimation problems of the ruin probabilities when investment re-turns are modelled by a geometric fractional Brownian motion, an exponential integratedVacisek or Cox-Ingersoll-Ross model, or the Heston model.Fourth, the thesis considers the ruin probabilities under two discrete-time risk models.In both models, premiums sequence is assumed to be a nonnegative Markov chain. Forthe case that claim-size sequence is independent and identically distributed, while invest-ment returns are a nonnegative Markov chain, or the case that claim-size sequence andinvestment incomes are both modelled by first-order autoregressive processes, the thesisobtains recursive and integral equations for finite-and infinite-time ruin probabilities re-spectively by using renewal recursive technique. General Lundberg-type inequalities forultimate ruin probabilities are derived by using induction methods.Finally, the thesis concludes the studies and points out future research directions.