The Asymptotic Behaviors Of The Ruin Probabilities For Several Non-standard Risk Models

It is well known that the risk theory is acknowledged to an important part of the actuarial mathematics, whose investigation is not only of application prospects in insurance practice, but also is of theoretical value in probability theory. While the main problem in risk theory is the asymptotic behavior for the probability of the surplus process of an insurance company firstly becoming negative, the probability is the so-called ruin probability (including the finite-time and infinite-time ruin probability). The classical risk theory started from Filip Lundberg (1903), and thereafter a lot of researchers have improved, generalized and perfected this theory through the months and years, and then they have established the well-known-risk models and achieved gratifying results.In this paper, we will consider several generalized non-standard risk models, which is on the basis of the standard renewal risk model(i.e. Sparre-Andersen model) and closely related to insurance and finance industry, and study the asymptotic behavior of the ruin probabilities for various risk models as the insurer's initial capital tends to infinity, which means that as the initial capital tends to infinity, the ruin probabilities are equivalent or weekly equivalent to some known quantities. The paper is constructed of the following chapters:In Chapter 1, in order to state our main results better, we will introduce some notions and notation. Firstly, because of the important role of the distribution class in applied probability, we present some concepts of heavy-tailed distribution and light-tailed distribution classes. In this paper, we will mainly discuss the risk models with heavy-tailed claim sizes, and these heavy-tailed distributions chiefly refer to the subex-ponential distribution, which can explain better the big-jump principle in actuarial practice, that is that the bankruptcy of an insurance cmpany is often duo to a large claim. Secondly, we will describe the standard renewal risk model, and meanwhile in-troduce various ruin probabilities including the finite-time and infinite-time ruin proba-bility, random time ruin probability and local ruin probability, where we should remark that the terms " local ruin probability " is first proposed by Professor Wang Yuebao, and it is infinitesimal than the non-local (global) ruin probability for the heavy-tailed claim case, and thus we can spend a little to control and estimate the local ruin prob-ability; additionally, " random time ruin probability " is the probability of ruin of an insurance company occurring in a random time interval, which is a randomized ver-sion of the finite-time ruin probability and whose study is of important academic sense and practical value (see Section 1.2). Thirdly, we will present some concepts about negatively dependent random variables. This paper will replace the independent claim sizes or claim inter-arrival times by the negatively dependent claim sizes or claim inter-arrival times, and hence structure some non-standard risk models which is different from the standard renewal risk model.In Chapter 2, we will investigate the asymptotic behavior of the random time ruin probability for the non-standard renewal risk model with heavy-tailed claim sizes. Under the assumption that the claim sizes are independent and long-tailed, we will give the equivalent conditions on asymptotic behavior for the random time ruin prob-ability, where the independent or dependent structure among the inter-arrival times is not considered. While, under the assumption that the claim sizes are of some negative dependence structure and consistently varying tails, we will obtain the sufficient con-dition of asymptotic behavior for the random time ruin probability which will require some other negative dependence structure among the inter-arrival times.In Chapter 3, we will introduce a new risk model-the random multi-delayed renewal risk model, which is of practical sense in actuarial business. Based on this model, we will respectively study the asymptotic behavior of ruin probability and local ruin probability. The former involves both the heavy-tailed claim case and the light-tailed claim case, while the latter only needs to discuss the heavy-tailed claim case.In Chapter 4, we will discuss the weakly asymptotic behavior of the randomly weighted sums and their maxima with a sequence of bivariate upper tail independent and dominated varying-tailed random variables. Our results obtained can be applied to estimate the finite-time and infinite-time ruin probabilities in a discrete-time risk model with dependent insurance risk and dependent financial risk. We note particularly that the above-mentioned bivariate upper tail independence structure can not only embrace all of the familiar negative dependence structures, but also can cover many positive dependence structures.