Asymptotic Estimates For The Ruin Probabilities Of Two-Dimensional Renewal Risk Models With Stochastic Returns

With the change of the economy and finance, the financial insurance industry has been put into a huge risk whirlpool. The financial securities of many insurance companies have been threatened by various risk factors which from the system, internal or external, so. it is essential to establish risk model for the establishment of an internal risk management system. Since Lundberg proposed the classical risk model named Lundberg-Cramer(L-C model for short) in 1903, the asymptotic esti-mates for risk models have been studied extensively. Owing to its nice tractability in mathematics, the one dimensional risk models have been deeply investigated and extensively applied in the early stage. However, it is worth saying that insurance companies will run several kinds of insurance business in real life. Hence, it’s more practical to study multidimensional risk models. In addition, it is found that there are many similarities between the two-dimensional and multidimensional risk mod-els, so in this paper we study the asymptotic estimates of the ruin probabilities for two kinds of two-dimensional risk models with stochastic returns.Firstly, we consider a two-dimensional risk model in which one insurance com-pany operates two kinds of insurance business. Suppose that the insurer will invest its wealth in risk-free and risky assets, and the price process of the investment port-folio is described as a geometric Levy process. By supposing that the two kinds of claim-size distributions belong to the class of consistently varying, the asymptotic estimate for the finite-time ruin probability of this two-dimensional renewal risk model with stochastic returns is obtained.In real life, in order to diversify risk the insurance company tends to associate with the reinsurance to take on part of risk. Assuming that the insure and reinsure share the claim size together, and the proportion is δ1,δ2, which satisfy δ1+δ2= 1-Note that in reality, if the deductible retained to insureds is raised, then the inter- arrival time will increase because small claims will be ruled out. So we quote a type of time-dependent structure. Suppose that both insurer and reinsure will in-vest their wealth into risk-free and risky assets, and then the asymptotic estimate for the finite-time joint ruin probability of the two companies is derived, with the distributions of the claim amount belong to the intersection of the long-tailed dis-tributions class and the dominated varying distributions class.