The Optimal Investment Decisions Involving The CEV Model And Personal Consumption

The optimal portfolio investment theory has become an important subject in recent years. Stochastic control as a classical tool is used in portfolio field and extended to personal investment management. The constant elasticity of variance (CEV) diffusion models with stochastic volatility can describe the realistic market better than the geometric Brownian motion that volatility is only constant. It is not easy to obtain the closed-form solution of this equation especially under the CEV risk price process. Considering four different forms of personal consumption process and using the methods of J.W.Gao (2009), J.Wu, X.Z.Hong, C.L.Qin (2006) and M.Jonsson and R. Sircar (2001), we acquire the non-linear HJB partial differential equation, which is reduced to a linear partial differential equation and the dual problems with the Legendre transform. We give the analytic solutions of the primal problem by studying the dual problem. As a consequence we find an optimal asset allocation between a risky asset and a riskless asset.