Optimal Investment With Some Risky Assets

In this paper,we study optimal investment in a risk model with one risky asset.The main model is: dR (t)=cdt-dS(t),R0=u. and c is the premium rate, Nt is a claim number, and obeys poisson distri-bution with parameters λ, we assume{Yi} are independent identically distributed random variables.Insurance companies divide the surplus U (t) into two parts, suppose there are only one risk free asset (bond or bank account) and one risky asset (stock).We invest (1-b) U (t) into risk free market at the t time. The price of stock at time t is Pt, and Pt satisfy the following stochastic differential equation dPt=μPtdt+σPtdW (t). where μ and σ are positive constant, μ is expected rate of stock return, σ is the volatility of the stock price,{Wt,t≥0} is a standard Brownian motion defined in a complete proba-bility space(Ω,F, P). The price of the bond at the t time satisfies the following stochastic differential equation dB(t)=r0B(t)dt, r0is interest rates,and r0is non-negative constant. This return on investment process {I(t),t＞0} satisfies: dI(t)=((1-bt) r0dt+btμdt+btσdW (t))U(t). we get the differential expression of investment risk model as following dU (t)=cdt-dS (t)+((1-bt) r0U(t)dt+bt+btμU(t)dt+btσU(t)dW (t))={c+(1-bt)r0U(t)+btμU(t)}dt+btσU(t)dW (t)-dS (t), then we make use of stochastic differential to get the following HJB equation is obtained by the relevant theory: max{λE[δ(u-Y)-δ(u)]+δ’(u)[c+(μ-ro)bu+rou]+(?)δ"(u)(σbu)2}=0.While Gerber et al get the HJB equation by the common stochastic dynamic ideas of cy-bernetics.We solve the corresponding HJB equation by two method, one is numerical simula-tion,the other method is finite difference method. We find the relation of the optimal ratio and initial surplus. These results are heuristic to operate the insurance business. When the surplus was very small, insurance companies would like to invest the surplus into stockmarket, and accept high risk returns to prevent bankruptcy and loss. When the surplusincreases slowly, the proportion of the investment to the stock market is reduced, that is,the greater the surplus, insurance company are willing to hold the conservative investmentstrategy.In Chapter One: We introduce the relevant knowledge, and research background.Firstwe introduce the risk model, risk theory and the theory of investment, then introducemartingale theory and related stochastic analysis. In Chapter Two: We solve the relation-ship between the optimal investment ratio and the initial surplus. In Chapter Three: Wesummarize of the paper and state for the future work in this area.