Optimal Investment Strategy Analysis Under Stochastic Environment

The optimal investment and consumption problem is the problem that under certain wealth level,how to arrange the proportion of investment and consumption in fixed time domail,to maximizes the expected-utility about the consumption and terminal wealth extreme.The earliest research about this aspect started with Harry Markowit's investment portfolio theory in 1952.After that,Merton reasearched the continuous-time's investment portfolio theory(1969-1971),investment portfolio theory obtained unprecedented development,but for the investment objects in Merton's research problem are only risky assets and risk-free assets,with the persistent and high speed development of Chinese economic,the most activitily investors in financial market of china have not only satisfied at risky assets and risk-free assets,they also focusing on the derivertive securities which are more complex and have higher risk, so in the situation that the securitive derivatives become more popular investment object today, how to obtain the optimal solution about the expected-utility of consumption and final wealth is very urgently problem now, but so far there are not too much research in this aspect.In this article, according to the indifference curve of exponential utility function and the efficient frontiers of the finite assets investment portfolio, we first proved the exsistence of optimal strategy in incomeplete arbitrage-free market; then the derivative security is made as investment object, with indifference price and relative entropy,in the condition of negative exponential utility,by using duality principle and stirict concavity of the indifference price function, we proved the exsistence of the optimal investment strategy in derivative assets in a incomplete and arbitrage-free financial market;For the situation that there is only a risky asset and a call option of the risky asset,using optimal control theory,I abtained the explicit solution of optimal investment and consumption strategy in the condition of negative exponential utility.