A Stochastic Maximum Principle And Its Application To Portfolio Selection And Consumption

Abstract:The stochastic maximum principle is an important method in solving the stochastic optimal control problem. It is a stochastic extension of the Pontryagin maximum principle which is used in the optimal control theory of deterministic dynamic systems. As an important instrument in mainstream economics, the methods and thoughts of games theory have been widely applied in optimal portfolio theory Therefore, it is very important to study optimal consumption and portfolio selection game problem with two or more investors.In this paper we study a sufficient stochastic maximum principle for correlative diffusion process and its application to portfolio selection and consumption. We first study stochastic maximum principle for the state process which is governed by the correlated Brownian Motions. We also establish the relationship between the stochastic maximum principle and the dynamic programming principle. Applications of the stochastic maximum principle to the nonzero-sum stochastic differential portfolio games are discussed. We obtain the closed-form expressions of the optimal portfolio strategies and the value functions. Then we study the optimal consumption strategy under the deterministic wealth process and the stochastic wealth process. We obtain the optimal consumption strategy and the value function. Comparisons on optimal consumption strategies and the value functions are made between the deterministic wealth process and the stochastic wealth process. Finally, we investigate the optimal consumption game problem that not only depends on absolute consumption but also depends on relative consumption. We obtain the solutions under the cooperative game and the non-cooperative game. Numerical analyses are also provided to illustrate the results.