Ruin Probability In Discrete-time Dependent Risk Model With Stochastic Interest

Risk theory is used for quantitative analysis, forecast, management, control and decision risk. Risk theory have important applications in the field of finance and insurance, and the bankruptcy theory of insurance risk models is the main research object of Risk theory. Ruin theory researches mainly focused on the accumulation problems in the surplus process of insurance companies and ruin probabilities of the insurance company. Ruin probability is one of the important topics in actuarial science, it provides direct reference indexes for setting premium, using reserve fund and planning reinsurance. Due to the influence of the national policy changes, the development of economy and the premium reinvestment of insurance companies, interest rate appears more random than before, and so stochastic interest rates have attracted more attentions from experts and researchers. Influences from various risks, such as catastrophic risk, market risk, operating risk of insurance companies, make the premium process and the claim process show a certain of dependence, and so the importance of dependent risk model has become more prominent. Previous studies mainly focused on constant interest, and premium process or claim process are independent and identically distributed(i.i.d.) random variables(r.v.) order or first-order autoregressive structure under the constant interest rate. In this paper, considering the actual economic environment, we builded a stochastic risk model to accord with the actual situation on the basis of previous researches. In this new model the three random variables, interest rate, premium process and claims process, were changed, and the main results we got are as follows:1 In this paper, stochastic interest rate which more conforms to the actual demand, was introduced in the model. The premium process and claim process in our new model were extended to higher order autoregressive structure, therefore the corresponding risk is more closer to the actual needs, and causing prove process become difficult and building skills get complicated. In addition, the higher order autoregressive structure is more universal than first-order autoregressive structure in time series analysis. In real life, claim amount is not only related to the previous stage, but also associated with previous stages. In this paper, considering the different discrete-time risk model, (1) the integral equation and the upper and lower bounds of the equation in infinite time ruin probabilities of the discrete time risk model were got through a renewal recursive technique. (2) a martingale was constructed under suitable condition, upper bound for ruin probabilities of an infinite time horizon is derived by using the martingale method.2 Due to the influence of diversification development of economic and the premium reinvestment of insurance companies, interest rate appears a certain of markov property than before, so the importance of Markov chain in risk model is becoming more and more predominant. In this paper, we expanded the rate process to Markov chain with discrete parameter, and both the premiums process and the claims process have the higher-order autoregressive structure, higher-order autoregressive structure possess several initial values which bring corresponding difficulty of mathematical proof and complexity of skills. Considering the discrete-time risk model, we derived the upper bound of infinite time ruin probability by using martingale method.