Conditional Expectation of White Noise Functionals

• Main Authors: Yu-Chun Lin, 林于鈞
• Other Authors: Yuh-Jia Lee
• Format: Others
• Language: en_US
• Published: 2006
• Online Access: http://ndltd.ncl.edu.tw/handle/65832047378238199074

碩士 === 國立高雄大學 === 統計學研究所 === 94 === In this paper it is show that the conditional expectation of a white noise functional $\varphi$ given the the Brownian motion $B(t)$ is represented by $$E[\varphi|\mathcal{B}_{t}]=\int_{S^*} \varphi[\Theta_tx+(1-\Theta_t)y] \mu(dy)\ ,$$ where $\Theta_t$ is the Heaviside function $$\Theta_t(s)\equiv \left\{ \begin{array}{ll} \mbox{I}& ,s\leq t ,\cr 0 &,s>t. \end{array}\right.$$ and $\Theta_t$ is the Heaviside operator defined by $\Theta_{t}x(s)=\Theta_{t}(s)x(s)$. Note that the Brownian motion $B(t)$ can be represented by $$B_{t}(x)=\left\{ \begin{array}{rr} \langle x,1_{[0,t]}\rangle & , t\geq 0, x\in S^*(\mathbb{R}^1) ;\cr -\langle x,1_{[t,0]}\rangle & , t< 0, x\in S^*(\mathbb{R}^1). \end{array}\right.$$ If $\{e_{j}:1\leq j\leq n \}$ be an orthonormal set in $L^2(\mathbb{R}^1)$ and $\mathcal{B}_{n}=\sigma \{\langle x,e_{j}\rangle:1\leq j \leq n\}$ and if $P_{n}$ denotes the orthogonal projection of $L^2(\mathbb{R}^1)$ onto the space spanned by $\{e_{j}:1\leq j\leq n \}$, then it is shown that conditional expectation enjoy the integral representation $$E[\varphi|\mathcal{B}_{n}]=\int_{S^*}\varphi[P_{n}x+(1-P_{n})y]\mu(dy)$$ Using the above integral representation we are able to investigate the regularity properties of the conditional expectation and compute the conditional expectation easily. Moreover, we can extend the concept of conditional expectation to generalized white noise functionals. As applications, we give some examples.