A Double Cox Risk Model With Portfolio

Risk theory studies mainly the stochastic risk model of insurance Actuality. Not only it isa important embranchment of application mathematics in modern, but also a hot topic in thepresent actuarial science research. By using probability and stochastic processes theory, theclassical risk theory studies mainly the surplus process of risk model, ruin time, ruinprobability and adjustment coefficient. At present, in order to make the risk model moreaccord with actural operational situation of insurance companies, the research of insurancerisk theory is mainly to the improvement and promotion of classical risk model. This articlehas outlined the research and development of the risk theory, and summarized presentresearch situation of the classic risk model in the domestic and overseas and the main resultsof the classic continuous risk model. Then based on this, in view of the gradually complexand specific situation of the present actual insurance business, we have studied the questionsof the surplus capital being invested, the additional premium rate, fuzzy random variables andreinsurance in risk process, so that the actual operational situation of insurance companies canbe reflected more truly and accurately, and it’s helpful for the insurance companies to makethe overall plan arrangement in insurance business.This article firstly introduced a risk model with portfolio. Because the insurancedevelops rapidly and the scale of the business expands incessantly, and considering thelimitations of a single investment risk model, by the method of martingale, we present studiedthe risk model with portfolio, and obtained the formula of ultimate ruin probability and itsupper bound.Secondly, in the situation the premium and claims to the occurrence of Cox process, weset up a double Cox risk model with investment portfolios and premium rate, and discussedsome problems related to the model that the premium, the number of premium, the foreheadof claims and the number of claims are fuzzy random variables.Thirdly, we put forword a double Cox risk model of the portfolio under fuzzy randomvariables. Because random interest which is involved in insurance business is fuzzy randomvariables, the double Cox risk model of the portfolio under fuzzy random variables is fullycomply with the situation of the insurance business. By the analysis of the model, we obtainedthe formula of ruin probability and its upper bound. The results of the model are feasible andcan provide some good ideas for insurance company to prevent and control some risks.Lastly, we discussed a double Cox risk model of portfolio and reinsurance. There aresome random factors that will not regularly appear in the process of insurance business makecustomers will choose reinsurance business. Therefore we set up the double Cox risk modelof portfolio and reinsurance and studied the problems of the ultimate ruin probability and Lundberg inequality. And by the analysis of the example, reinsurance is accord with themanagement need of insurance companies.