Study On Optimal Strategy For Consumption And Investment With Stochastic Volatility And Poisson Jump

In recent years scholars from mathematical finance have been strongly interested in the investment portfolio. In the basic model for deciding how to invest in a portfolio, general investors will invest their own assets partly into risk products for the high profit, such as stocks and a variety of financial derivatives, and partly into risk-free products for the stability in profit, for example bonds and bank deposit. The key point of this kind problem lies in deciding an optimal investment ratio, so as to maximizing the investor’s wealth.In this context, firstly, we extend the fundamental model for optimal investment strategy of an insurance company to a model that suits the wealth process of a general investor, whose wealth process resembles that of an insurance company, that is to say the wealth process in our model incorporates a Poisson jump to describe one’s several wealth loss in a given amount of time. Secondly we proceed forward to consider the consumption behavior of the investor, which allows us to incorporate consumption strategy into the optimal investment strategy. Then we consider the factor of inflation rate, for it is essential when you consider the behavior of consumption. We suppose the inflation is completely observable and we use the diffusion process to describe the inflation rate. Now we can establish the new wealth model in the following, X(t)denotes the wealth process:Profit on risk-free product follows:Price of risk product follows the Geometric Brownian motion: Based on the hypothesis in economy, we introduce consumption basket price: And it satisfies that I>0denotes the expected instantaneous inflation rate ζ>0denotes the inflation volatility. So we have And we have (4) Filtration Ft is as follows:At last we introduce stochastic volatility, it is as follows: Then we have Risk product: The wealth process goes as: Consider the following optimal problem; its cost function is defined as follows: The value function goes: The HJB equation The verification theorem: Suppose HJB equation have a classical solution f(t,x,z,L), and followingconditions are satisfied:The solution with the utility function of CRRAthen we get the HJB equation with the CRRA utility function. We can derive the optimal consumption strategy by maximizing{U1(Ce-L)-CVx}, we set the first order derivative of{U1Ce-L)-CVx}to be0, that is, Terminal condition Then we haveAt the last part of this context, we use the numerical technique to analyze the PDE; we deduce the explicit scheme of the solution. And we apply MATLAB2014to draw graphs of the scheme.