Optimal Portfolio and Consumption in Modified Black-Scholes Model

The optimal stochastic control is a hot topic among recent problems in economics and finance. In fact, many economical and financial problems present the necessity of taking decisions based on the uncertainty of random nature. This paper modifies the classical Black-Scholes model by taking the consumption into consideration. By splitting the initial endowment into two parts, we respectively find the optimal consumption and optimal portfolio. We also analyze how to get the optimal solution of the combined portfolio and consumption problem in the model.