Merton's Portfolio Problem under Jourdain--Sbai Model

Abstract: Portfolio selection has always been a fundamental challenge in the field of finance and captured the attention of researchers in the financial area. Merton's portfolio problem is an optimization problem in finance and aims to maximize an investor's portfolio. This thesis studies Merton's Optimal Investment-Consumption Problem under the Jourdain--Sbai stochastic volatility model and seeks to maximize the expected discounted utility of consumption and terminal wealth. The results of our study can be split into three main parts. First, we derived the Hamilton--Jacobi--Bellman equation related to our stochastic optimal control problem.  Second, we simulated the optimal controls, which are the weight of the risky asset and consumption. This has been done for all the three models within the scope of the Jourdain--Sbai model: Quadratic Gaussian, Stein & Stein, and Scott's model. Finally, we developed the system of equations after applying the Crank-Nicolson numerical scheme when solving our HJB partial differential equation.