Optimal consumption and portfolio decision with stochastic covariance in incomplete markets

Abstract

Abstract This paper investigates finite and infinite horizon optimal consumption and portfolio choice problems in incomplete markets with stochastic covariance among stock returns. The stock price dynamics are assumed to follow a multivariate stochastic volatility process. The Hamilton-Jacobi-Bellman (HJB) equation is derived by using dynamic programming method and the closed-form expressions for portfolio–consumption strategies are derived in the case of logarithmic utility. In the case of infinite time horizon, we prove the existence of a classical solution and provide a verification theorem. The closed form approximate solutions to the optimal consumption and portfolio polices for a power utility investor are obtained by solving the same utility maximization problem in two fictitious complete markets. Moreover, we verify the existence and optimality of the solution of the original incomplete market and also provide the upper bound of the utility loss of approximate rules. Finally, a numerical application to an incomplete market with stochastic factors show that the utility loss of the approximation rules is small and the performance of the approximate rules is close to optimal.