Optimal Reinsurance And Investment Strategies Research For Insurer

In recent years, with the rapid development of economy and market com-petition intensifying, reinsurance and investment problem is more and more attention by insurance company, and also become the important topic of the academic research.In this paper, we consider the optimal reinsurance and investment strate-gies of insurance company. Assuming that the market is no arbitrage, the rein-surance of the insurance company is the proportional reinsurance, nominal interest rate satisfaction the CIR model, real interest rate for a fixed function of time, the inflation index obtained by Fisher Equation expansion. Assume that the insurance company invests in four assets, risk-free assets, zero-coupon bonds, Treasure Inflation Protected Securities and stocks. The surplus process and the asset pricing equation of the insurance company is given, after the transformation, we obtain the assets process of insurance company.Theorem 1 Consider the case of reinsurance and investment for insurance company, the assets process X(t) has the following formdX(t)=λμ1(η-θ)dt+u(t)Tσ[∧dt+dW(t)]+rn(t)X(t)dt, where θO(t),θB(t), θP(t), θS(t)are the money invested in the cash,zero coupon bond,TIPS and the stock,respectively.The wealth of our model is X(t) θO(t)+θB(t)+θP(t)+θs(t),and u(t)=(p(t),θB(t),θP(t),θs(t))T is called a strategy.u(t)is a combination of the reinsurance strategy and the investment strategy.We consider the CRRA utility function maximization problem under the influence of interest rate and inflation rateTo solve the problem,we add auxiliary items Where Further,by the theorem of backward stochastic differential equation,we obtain the following theorem of F(t).Theorem 2 IfthendF(t)=-λμ1(η-θ)dt+F(t)rn(t)dt+(H(t)z(t)+F(t)∧T)(∧dt+dW(t)).By the theorem 3.2.2,the original problem is transformed into the follow-ing self-financing problemd M(t)=dX(t)+dF(t)=rn(t)M(t)dt+u1(t)σ[∧dt+dW(t)],then get the relationship between the original optimal strategy and the optimal self-financing strategy u1(t)=u(t)+(H(t)Z(t)+F(t)∧T)σ-1.Wc apply the dynamic programming mcthod to solve the self-financing problem,theorem 3.4.1 gives the Hamilton-Jacobi-Bellman equation of self-financing problem.Firstly,we defineTheorem 3 The optimal strategy u1(t)of self-financing problem satisfy the following Hamilton-Jacobi-Bellman equation 0=sup{Vt+Vm(rnm+h1t(t)σ∧)+Vrna(b-rn)+VII(rn-rr+σI1λrn+σI2λI)The Vt,Vm,Vrn,VI,Vmm,Vrnrn,VII,Vmrn,VmI,VIrn respectively V(t,rn,I,m) on t,m,rn, I first and second order partial derivative,and σr=(0,σrn,0,0)T, σI,=(0,σI1,σI2,0)T.The solution of equation is the optimal strategy of the self-financing prob-lem,by transforming we can obtain the optimal strategy of original problem.