研究期間：10108~10207;This is a proposal for three years research project on some topics of probability theory. The first part is concerning the portfolio optimization problems. This is a continuation of our previous research on risk-sensitive portfolio optimization problems as well as optimal consumption problems. In our previous study, we consider the factor model. We apply the dynamic programming approach to study the problems. Using this approach, the HJB (Hamilton-Jacobi-Bellma) equations can be derived. These are nonlinear partial differential equations with different nonlinearity for different portfolio optimization problems. We study the solutions of each HJB equation. A candidate of optimal portfolio can be derived from each solution. We also prove the Verification Theorem that the candidate of optimal portfolio mentioned above is indeed optimal. There are some interesting open questions that we will continue to study. We will also consider the model with partial information that we can only observe stock prices but not factor process. We also plan to study the affine diffusion models. The affine diffusion models attract many attentions in recent years because of the possibility to obtain analytical solution under such models. We will study the property of affine diffusion process. We will also consider the portfolio optimization problems for such models. In the second part we propose to study a parabolic HJB equation. We want to study the large time asymptotics of the solution of the equation. This is motivated by our previous study on the large time asymptotics of some expectations of diffusion process.
