Research On Ruin Probabilities In Risk Models With Stochastic Interest Rates

Risk theory originates from the feasibility study on insurance items. The insur-ance trade of our country started later than that of other country, but it develops veryrapidly. Facing the above situation and certainty of economic environment, how an in-surer objectively assesses investment risk and estimates ruin probabilities are importantresearch contents of risk theory, which has certain theoretical significance and appli-cable value for the stable function and sustainable development of the insurance tradeand risk prevention of personal investors. However, under the complex economic envi-ronment, it is very difcult to directly compute ruin probabilities, so some researchersare devoted to asymptotic estimates for ruin probabilities.In this thesis, we mainly study the asymptotics of ruin probabilities in risk modelswith stochastic interest rates. It is composed of four chapters and the main contentsare as follows:Chapter One is a preface, which mainly introduces some heavy-tailed distributionclasses, some concepts and notations of L′evy processes and the dependence of randomvariables, summarizes the present research situation on ruin probabilities of the riskmodels related to this thesis, and shows its research motives and research contents.In Chapter Two, we consider a non-standard renewal risk model with upper-tailindependent premium sizes and claim sizes, where the price process of the investmentportfolio is modelled by a geometric L′evy process, the claim size and its correspondinginter-claim time satisfy a certain dependence structure, there is a similar dependencestructure between the premium size and the inter-arrival time before the premiumis paid, and the premiums and claims are not necessarily independent. When theclaim-size distribution belongs to the extended-regular-varying distribution class, weobtain a uniform asymptotic formula for the tail probability of stochastically discountedaggregate claims. Furthermore, assuming that the tail of the premium-size distributionis lighter than that of the claim-size distribution, we discuss the uniform estimates forthe finite-and infinite-time ruin probabilities. The conclusions show that the upper- tail independence between claim sizes, the upper-tail independence between premiumsizes, the dependence structure between the premium size and the inter-arrival timebefore the premium arrivals, and the dependence between premium sizes and claim sizesdon’t play any roles in the asymptotics of the ruin probabilities, but the dependentstructure between the claim size and its corresponding inter-claim time takes efect onthe uniform asymptotics of the ruin probabilities.In Chapter Three, we discuss another non-standard renewal risk model withstochastic interest rates, where some insurer invests a part of surplus into a Black-Scholes market whose price process is modelled by a geometric Brownian motion, claimsizes form a sequence of not necessarily identically distributed and upper-tail indepen-dent random variables. When the distributions of claims have dominated-varying tails,we obtain a weakly asymptotic formula for the finite-time and infinite-time ruin proba-bilities. In particular, according to the above formula, when the distributions of claimsare consistently-varying tailed, we can derive an asymptotic formula of the finite-timeand infinite-time ruin probabilities. The conclusions show that the weighted renewalfunction has an important status in ruin theory and the behavior modelled by a geo-metric Brownian motion takes efect on the asymptotics of the ruin probabilities.In Chapter Four, we consider a discrete Markov-modulated risk model, where theinterest rates follow a homogeneous Markov chain with a finite state space, there isan autoregressive structure between premium sizes, and the claim sizes are controlledby another homogeneous Markov chain. We present recursive integral equations forthe finite-time and infinite-time ruin probabilities. When the interest rates are non-negative, generalized Lundberg inequalities for the infinite-time ruin probability arederived. When the distributions of claims have regular-varying tails, we obtain asymp-totic formulas for the finite-time ruin probability.