馬可夫過程離開問題及應用

Abstract

我們考慮馬可夫過程由離開時間及離開點所決定的一般函數。這個函 數在理論機率及應用機率(如隨機最佳停止時間，隨機最佳控制，保險風 險管理，選擇權理論，信用風險理論)都扮演非常重要的角色。本計畫主 要探討在適當的馬可夫過程下，這個函數的封密解及其在上述領域的應 用。

Given a real-valued Markov process, we consider a general first exit functional of the process from an open set. These functionals are not only of interest in the theory of probability, but also find many interesting applications in the theory of optimal stopping, in the theory of optimal control, in the theory of insurance ruin and risk, in the theory of credit risk, in the theory of branching processes and in the theory of fragmentation. Our main objective in this project is to derive explicit solutions for the functionals when the processes are in a suitable class of Markov processes. Moreover with the advent of these explicit formulae we will revisit and solve a number of problems from aforementioned applied probability.