A Result And The Application In Investment Decision-making Of Optimal Stopping

The optimal stopping theory is one of the issues that the people study in the probability theory, and a relatively independent method system has formed. It develops and improves continuously the theory with the actual problem solving. Accordingly, it is also one of the hot research directions to determine the optimal investment timing and maximum return for the investment decision-making problem by using the optimal stopping theory. So the three problems are studied in this thesis. The first problem is the optimal stopping time for a particular process, and the next two problems are how the optimal stopping theory is implicated into investment decision-making problems. The work is as follows.Firstly, an optimal stopping problem about special maximum process is given. In the process for solving the problem, it proves that the stopping in the diagonal is not optimal, and the original problem is associated with the Stefan problem in together. The optimal stopping boundary must admit a first-order nonlinear differential equation, furthermore, it is the solution that satisfies the maximality principle to the nonlinear differential equation, and the original problem’s maximum expected function and optimal stopping time can be expressed visibly.Secondly, a mathematical model is established about the best investment decision-making timing problem of the enterprise expansion project. It requires investors to decide when to make investment according to the current observed information for maximum discounted income, and use the optimal stopping theory to solve the model. So the best mvestment timing is found, and the investment expansion’s average waiting time and the results are analyzed in detail.Finally, a mathematical model is established about profit for enterprise to sell, which is studied in-depth by using optimal stopping theory again. Under given different conditions, best time is determined for enterprise to sell. When the discounted rate is greater than the growth rate, the best timing is found for enterprise to sell, namely the discounted value is the greatest at this moment. Conversely, the best timing for enterprise to sell does not exist; however, the maximum expected value can be given with the help of the Dynkin formula.In a word, the above every problem associates with Stefan problem or free boundary problem, and the result can be given directly or a possible solution can be guessed and verified with the aid of them which play a role as a bridge for the solution of the above problems.