這是三年期的研究計畫，我們將研究幾個隨機分析專題。第一個研究課題是最佳投資組合問題，這是我們過去幾年的研究工作的延續，在過去幾年我們在兩個最佳投資組合問題得到相當好的成果，一個是risk-sensitive最佳投資組合問題，一個是最佳消費問題，在過去的研究，我們應用動態規劃的想法得到Hamilton-Jacobi-Bellman(HJB)方程，我們解HJB方程而得到最佳投資組合，由這樣的想法而引入幾個有趣的數學問題，有幾個問題尚待研究，另外我們也將考慮部份可觀測的市場模型，這是相當重要但相當困難的問題。另外我們也將考慮Affine Diffusion市場模型，這個模型今近幾年有很多的討論，因為在這個模型下若干財務問題有解析解，我們希望了解Affine Diffusion隨機過程的行為，並考慮相關的財務數學問題。我們的第二個研究課題是橢圓型HJB方程的解的長時間的漸近行為的研究。 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. 研究期間：10008 ~ 10107
