Gerber-Shiu函數在財務及保險領域的應用

Abstract

給定一個實數的馬可夫過程和一個開集合。我們研究該過程首次離 開此開集合時間的泛函問題。自從Gerber 和Shiu(1997,1998)定義此泛 函至今，這主題一直引起眾多學者關注並也獲得相當的回響。我們主要 研究方針是考慮在某些實用的馬可夫過程之下，推導出此泛函的顯解。 在得到顯解的表示式之後，我們不但可以整合並且進而能解決更多財務 和保險方面的問題。其中，我們對多維度風險模型下的破產問題更感興 趣，因為這是一個理論尚未萌芽因而深具挑戰性的領域。

Given a real-valued Markov process, we consider a general first exit functional of the process from an open set. The definition of this functional goes back to Gerber and Shiu (1997, 1998) and, since then, the topic experienced an enormous interest and activity. Our main objective 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 mathematical finance and insurance. In particular we will consider the ruin problems in multidimensional risk models, which is a challenging area not yet far developed.