An Optimal Investment Problem Under Correlated Noises

This paper is concerned with a risk-sensitive stochastic control problem,moti-vated by an optimal invest,ment problem under correlated noises in the financial market.First of all,we consider a financial market where there is only one risky asset and one non-risky asset.Non-risky assets are represented by bonds,the rate of return is fixed,and risky assets are represented by stocks.Its price is af-fected by two correlative factors,the correlation coefficient is a constant p.The investor's goal is to rationally allocate the proportion of wealth invested in risky assets to maximize the investment utility at the terminal moment.In this paper,the HARA type utility function is selected for the description of the investment utility.A new stochastic maximum principle for this kind of problem is obtained by dimension expansion,where the adjoint equations and maximum condition heavily depend on the risk-sensitive parameter and the correlation coefficient.In the derivation process,we used the relationship between the maximum princi-ple and the dynamic programming principle to transform the adjoint variables,eliminating the amount of calculation caused by unnecessary extended state com-ponents,which also made us need to make the value function absolutely smooth.Then we applied the obtained maximum principle to an optimal investment prob-lem with correlated noise,and proved that if the risk-sensitive parameters and correlation coefficients meet a certain condition,we can get an optimal invest-ment in the form of feedback strategy.There are various factors affecting the stock price in the financial market,and they often show a certain correlation,so this is of great guiding significance for anlyzing investors' decision when the stock price in the financial market is affected by two related factors.At the end of this paper,numerical simulation simulation and figures are given to explicit-ly illustrate the changes and sensitivity for optimal investment strategies with respect to the risk-sensitive parameter and the correlation coefficient.