連續時間 filtration 的正交分解

Abstract

在此計畫中我們想考慮一組布朗運動的線性轉換的性質. 與一般考慮情形不同之處在於此組線性轉換為連續型的狀況.當布朗運動經此線性轉換後仍為布朗運動且所生成的 filtration 有變小的情形時, 能否將原本的布朗運動的 Brownian filtration化成orthogonal decomposition是我們比較感興趣的問題. 此外, 這個orthogonal decomposition是finite亦或是infinite, 遍歷性質如何, 是我們想探討的主題之一.

In this project we are mainly concerned with a group of linear transforms of Brownian motion and the related properties. If the resulting stochastic processes are again Brownian motions and whose Brownian filtration is strictly smaller than the original Brownian filtration, we aim to investigate the orthogonal decomposition of the original Brownian filtration. Moreover, the ergodic property of the resulting stochastic processes is also one of our main discussion points.