The Ruin Theory Of A Dependent Discrete-time Risk Model With Subexponential Insurance Risk And Related Problems

Risk theory is one of the important study theme in actuarial science and its core is ruin theory. Because of its wide application in risk management, this topic has attracted much attention recently. In this paper, we consider a discrete-time risk model, in which we assume a company lies in a stochas-tic economic environment facing insurance and financial risks simultaneously. Insurance risks include the potential reliability or claim risks resulting from policy of insurance. Financial risks are those of risks resulting from the fluctu-ation of stock price because we always assume that the company will invest its capital into the risky market. Intuitively, both of these two risks will impact the insolvency.The discrete-time risk model with insurance and financial risks was first-ly proposed by Nyrhinen[22,23]. Subsequently, Tang[30,31]has done some basic work in this direction. However, all the literature mentioned above assumed that the insurance and financial risks are independent, which is far from in-surance reality. To dismiss this problem, recently, the study of discrete-time risk model with insurance and financial risks under some dependent structure becomes more and more popular. Many applied probability researchers de-vote themselves to characterize impact of the dependence structure to the ruin probability. Clearly, this study will bring much importantly theoretical and practical value. Based on some existing literature, in this paper, we assume that the insurance and financial risks follow a wide type of dependence struc-ture and the insurance risks are subexponential. Then the asymptotic for the finite-time ruin probability of such a risk model is derived. The paper mainly includes the following two aspects of contents:(1) It is well known that, when taking account of the finite-time ruin probability of a discrete-time risk model with subexponential insurance and financial risks under some dependence structure, the most important problem is the subexponentiality of the product of dependent random variables. In doing so, in the second chapter of this paper, let X be a real-valued random variable, Y be a positive random variable, and they follow some given depen-dence structure. If X is subexponential, under some conditions, we prove that XY is also subexponential. The obtained result successfully extends the one of Tang[30]some dependent cases.(2) Consider a discrete-time risk model with subexponential insurance and financial risks under a specific dependence structure. The asymptotic for finite-time ruin probability is derived and the obtained result explicitly cap-tures the impact of dependence structure to the ruin probability. Particularly, if we further assume that the insurance risks belong to the regularly-varying class, the asymptotic formula becomes more transparent. Under some oth-er conditions, we prove that the uniformity of the asymptotic which can be applied to estimate infinite-time ruin probability.