随机金融环境下Cox风险模型三个重要精算诊断量的矩

N (t) X i=1
Zi ;
t
0;
where N (t) counts the number of claims on the company up to time t and Zi ; i 1 th represents the i claim amounts. The net pro…ts of the risk business over the time interval (0; t] is the risk process de…ned by X (t) = Q(t) S (t); t 0:
The expected net pro…ts E [X (t)] is called “safety loading". One classical problem in collective risk theory is to investigate the “ruin probability", i.e., the probability that the risk reserve ever becomes negative. The study of the probability of ruin has been the center of interest of many papers treating actuarial risk theory. In order to character the severity of ruin more precisely. During recent years, more and more literature began to exploit other risk measures of the risk business, such as the surplus immediately before ruin and the de…cit at ruin. Gerber and Shiu (1998) embedded the study of three most important actuarial diagnostics: the time of ruin, the surplus immediately before ruin, the de…cit at ruin in the study of an expected discounted penalty, which is due at ruin and depends on the surplus immediately before ruin and on the de…cit at ruin. We consider the Cox risk model in a Markovian environment in this paper. The so called Cox risk model is that the occurrence of the claims is described by a Cox process. Cox processes are also called doubly stochastic Poisson processes or conditioned Poisson processes, which were …rstly introduced and discussed by D. Cox in 1955. A Cox process 2
U fU (t); t 0g is called the surplus process (or risk reserve process ), where Q = fQ(t); t 0g is the income process (the simplest assumption is to regard Q(t) as the premiums collected up to time t) and S = fS (t); t 0g is the loss process, S (t) measures all losses paid up to time t, in general, it is the aggregate claims up to time t, i.e., S (t) =
is a generalization in the sense that stochastic variation in the intensity is allowed. We, here, only make an intuitive explanation on a Cox process N with an intensity process (i.e., a nonnegative random process) process (t) is generated and, conditioned upon its realization, N is a non-homogeneous Poisson process with that realization as its intensity. We introduce the idea of Gerber and Shiu (1998) to the Cox risk model for the …rst time. The expected discounted penalty of a Cox risk model, with a homogeneous markovian jump process (i.e., a Markov process with piecewise constant realizations) as its intensity, is considered …rstly. Then if we assume that the moments of the individual claim for any order exist, thus the moments of the surplus immediately before ruin and of the de…cit at ruin, for any order, can be obtained and the Laplace transform of the ruin time can be obtained by appropriate choices of the penalty function. The model and mathematical descriptions are established as follows. Let ( ; F; P) be a complete probability space containing all the objects de…ned below. The Cox risk model is de…ned by U (t) = u + ct U = fU (t); t
Moments of three important actuarial diagnostics in the Cox risk model under stochastic …nancial environment
Abstract: In this paper we consider the Cox risk model in a stochastic environment. Cox risk model belongs to the category of collective risk theory. Collective risk theory, as an important part of insurance –or actuarial –mathematics, deals with stochastic models of an insurance business. In such a model the occurrence of the claims is described by a point process and the amounts of money to be paid by the company at each claim by a sequence of random variables. The company receives a certain amount of premium to cover its liability and to pay overhead expenses. The company is furthermore assumed to have a certain initial capital at its disposal. Cox risk model is a model where the occurrence of the claims is described by a Cox process. Cox processes are very natural as models for “risk ‡ uctuation". That is, the ‡ uctuation in the underlying risk. Typical examples are automobile insurance and …re insurance. The moments of three important actuarial diagnostics: the surplus immediately before ruin, the de…cit at ruin and the time of ruin in the Cox risk model are investigated. The Gerber-Shiu discounted penalty function is considered and an integral equation is established. Some explicit expressions for the moments when the claim intensity is an n-state Markov process are obtained. An expression for the joint probability density function of the surplus immediately before ruin and the de…cit at ruin is also obtained when the initial capital is zero. Finally, an example and some numerical results are presented to illustrate the main results. The results can be used as indicators on the probability and size of potential losses. Key words: Cox risk model, moments, surplus immediately before ruin, de…cit at ruin, time of ruin.