本論文在討論一個投資模型，此模型中有兩個風險，分別是股票市場風險及系統性風險，在不完全市場(incomplete market)中，我們無法藉由股票的操作將系統性風險消除。我們將探討在此模型下的各種最佳化問題，並找出對應的最佳策略，問題包括極小化破產機率、極小化折扣罰款的期望值、極大化資產在中止時間作用在效用函數上的期望值。
我們主要藉由解HJB方程的方法來找出最佳策略， 另外我們還會引進一些方法當HJB 方程不好解的時候。

In this paper, we study the investment models that have two risk processes. One is the risk process from the stock market and another is an external risk process. Also, we are only interested in the incomplete market model, that is, we cannot eliminate the external risk process by investing in the stock market. We will consider several optimization problems, including minimizing the ruin probability, minimizing the expected discounted penalty and maximizing the expected utility at terminal time.
Our main approach is to solve the HJB equation to find an optimal strategy and the value function. In addition, we will use other approaches to find an optimal strategy when the HJB equation is hard to solve.

致謝 i
摘要 ii
Abstract iii
1. Introduction 1
2. A Brief Review to HJB Equation 6
3. Minimize the Ruin Probability 8
3.1. Minimize the Ruin Probability with No Constraint 8
3.2. Minimize the Ruin Probability with Constraint 12
3.3. An Alternative Approach 14
4. Minimize the Expected Discounted Penalty 22
4.1. Minimize the Expected Discounted Penalty with no Constraint 22
4.2. Minimize the Discounted Penalty with Constraint 25
5. Maximize the Expected Exponential Utility at Terminal Time 36
Appendix A. 42
References 56

[1] Durrett, R.(1995). Probability: Theory and Examples, Second edition. Duxbury Press, Belmont, CA.
[2] Durrett, R. (1996) Stochastic Calculus: A Practical Introduction. CRC Press.
[3] Pestein, V.C., W.D. Sudderth (1985). Continuous-time red and black: How to control a diffusion to a goal. Math. Oper. Res. 10 599-611.
[4] Browne, S.(1995) Optimal investment policies for a rm with a random risk process: exponential utility and minimizing the probability of ruin. Math. Oper. Res. 20 937-957.
[5] Chicone, C.(1999). Ordinary Differential Equations with Applications. Springer. New York.
[6] Friedman, A. (1975) Stochastic Differential Equations and Applications, Volume 1. Academic Press, New York.