| With the development of the insurance in the world, people are realizing increas-ingly that actuary which is core technology of insurance is playing an important role.The thesis consists of five chapter. In chapter one, the history of insurance and the development of the actuary are introduced, as well the theme. In chapter two, there is an introduction of basic theories of life insurance actuarial, probability and option pricing.In chapter three, we consider minimum equity-linked life annuity insurance con-tract, and we obtain a investment strategy that minimize the discounted value of liabil-ity risk by the method of stochastic dynamic programming. Equity-linked life annuity insurance means insurance firm paying amount to insurant by contract engagement which is relevant with stocks that invests by insurance company, moreover, insurance company pay amount to insurant by the installment payment. However, the formers researche mainly focus on one-time payment. In this chapter, author considers the model is meaningful to practical application. Nowadays, dividend pension insurance can be included by equity-linked life annuity insurance in insurance market. Further-more, the investment strategy model has a reference value for insurance company in trems of how to avoid debt and make a reasonable investment.The main results we obtained are as follows:Theorem 0.1 We suppose mortality risk and financial risk are independent in the risk-neutral world. at timet0=0, ncustomers of age x buy the same equity-linked life annuity insurance contracts with the maturity time T. This contract specifies that annuity f(Sti) is paid to a customer at timeti, i=1,2 if he or she is still alive at this time, f(Sti)is linked to the stock, which the insurance company bought atti, and we have f(Sti)=max(Sti,G). Sti is satisfied the stochastic differential equation dSti/Sti=λdt+σdWti. Then there exists the optimalξt1,ξt2, to minimize the risk of insurer's loss L, such that and wherethe constantGis the minimum guaranteed benefit,ξt1 is the number of stocks which insurer bought in the first period,ξt2 is the number of stocks which insurer bought in the second period,δis the risk free interest rate,f*(Sti)=e-δti f(Sti)is the discounted value of f(Sti),constantkis level premium,Sti*=e-δti Sti is the discounted stock price,Ytin is denoted the number of policyholders who survive through time ti,tiPx=Pr{T(x)>ti}, is denoted the probability that an insured agex will be alive ti years later.In chapter four, we consider that insurance company bought the stocks, and the the price of stocks is changed by the non-market risk (natural calamities, technical innovation etc). We describe the change of stock price by Poisson jump, and we obtain the optimal investment strategy of insurance by stochastic dynamic programming. The optimal investment strategy is as follows: Theorem 0.2 We suppose mortality risk and financial risk are independent in the risk-neutral world. at timet0=0, ncustomers of age x bought the same equity-linked life annuity insurance contracts with the maturity time T. The contract specifies that annuity f(Sti) is paid to a customer at timeti,i=1,2. if he or she is still alive at this time, f(Sti)is linked to the stock, which the insurance company bought atti, and we have f(Sti)=max(Sti,G), Sti is satisfied the stochastic differential equation dSti/Sti=(δ-λp)dti+σdWti+d(Σj=1Ntiφj). Lis denoted the discounted value of insurer's total loss, there exists the optimalξt1,ξt2, to minimize the risk of insurer's loss L, such that whereInsurance company can analyze unpredictable accidents to reduce risk investment and avoid debt because of Poisson process.
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