最佳停止問題與套利

Abstract

本計畫探討二個和財務數學有密切關係的數學問題。我們首先探討報 酬函數是二邊形式的最佳停止問題。這問題和新奇選擇權的定價問題有密 切的關聯，我們將在跳躍-擴散過程的假設下，探討這方面的相關問題。另 外一方面，我們將探討隨機資產理論-最佳套利的問題。隨機資產理論是 Fernholz 在1990, 2001 發展的一套理論。除了實務的應用外，它也提供了 很多具有挑戰性及有深度的理論問題。在本計畫裡，我們將試著解決一些 和最佳套利相關的open problems，同時也將把文獻上的相關結果應用到能 真正捕捉到市場特性的理論模型。

In this project, we consider two problems arising from mathematical finance. First we consider the optimal stopping problems for reward functions of two-sided form. This problem is related to the rational valuation of exotic options and we will work in a jump-diffusion context. Secondly, we consider the optimal arbitrage problems in the stochastic portfolio theory. Stochastic portfolio theory began with the papers of Fernholz(1999,2001) and was introduced in the monograph Fernholz(2002). Since then stochastic portfolio theory has evolved into a novel mathematical framework for analyzing portfolio behavior and equity market structure. Besides its possible applications to real equity markets, the study of stochastic portfolio theory raises many interesting and challenging open problems for the theory of stochastic processes. In this project we will focus on some open problems related to optimal arbitrage. In addition we are interested to apply recent results in the literature to some abstract stock markets (i.e., stochastic models that exhibit some of the properties of real stock markets).