弱布朗運動及其應用

Abstract

弱布朗運動為一其k 次邊際分佈與布朗運動之k 次邊際分佈相同之隨機過程. Stoyanov 於 1987 年提出此隨機過程並不需為布朗運動的猜測, 而Föllmer-Wu-Yor 則於2000 年解決 了這個問題. 在此研究計畫中, 我們主要是想解決一些弱布朗運動的open problems, 例如二次與三次弱布朗運動的quadratic variation 的值的問題. 另外, 弱布朗運動在財務數 學上的應用亦是我們有興趣的問題.

A weak Brownian motion of order k is a stochastic process whose k-dimensional marginal is identical with that of a standard Brownian motion. A weak Brownian motion is not necessary to be a Brownian motion. This was a conjecture appeared in Stoyanov (1987) and solved in Föllmer-Wu-Yor (2000). In this project we aim to solve some open problems in the weak Brownian motion, for example, the value of the quadratic variation of weak Brownian motion of order 2 and 3. Furthermore, we want to discuss the applications of the weak Brownian motion in finance.