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| The \emph{Wiener space} $W(\mathbb{R})$ is just the set of all continuous paths $\omega : [0, \infty) \to \mathbb{R}$ satisfying $\omega(0)=0$. Thus, an $\mathbb{R}$-valued continuous-time stochastic process $X_t$ with continuous sample paths can be thought of as a random variable taking its values in $W(\mathbb{R})$. |
The \emph{Wiener space} $W(\mathbb{R})$ is just the set of all continuous paths $\omega : [0, \infty) \to \mathbb{R}$ satisfying $\omega(0)=0$. Thus, an $\mathbb{R}$-valued continuous-time stochastic process $X_t$ with continuous sample paths can be thought of as a random variable taking its values in $W(\mathbb{R})$. |
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| In the case where $X_t = W_t$ is Brownian motion, the distribution measure $P$ induced on $W(\mathbb{R})$ is called the \emph{Wiener measure}. That is, $P$ is the unique probability measure on $W(\mathbb{R})$ such that for any finite sequence of times $0<t_1<\ldots<t_n$ and Borel sets $A_1,\ldots,A_n \subset \mathbb{R}$ |
In the case where $X_t = B_t$ is Brownian motion, the distribution measure $P$ induced on $W(\mathbb{R})$ is called the \emph{Wiener measure}. That is, $P$ is the unique probability measure on $W(\mathbb{R})$ such that: |
| \begin{eqnarray} |
\begin{enumerate} |
| P(\{\omega : \omega(t_1)\in A_1,\ldots,\omega(t_n) \in A_n\}) &=& \int_{A_1}\cdots\int_{A_n} p(t_1,0,x_1)p(t_2-t_1,x_1,x_2)\cdots \\ |
\item For any $t \in [0,\infty)$ and $A \subset \mathbb{R}$ a Borel set, we have $$P(\{\omega : \omega(t) \in A\}) = \frac{1}{\sqrt{2 \pi t}} \int_A e^{-x^2/2t}\,dx.$$ That is, $\omega(t)$ has a normal distribution with mean $0$ and variance $t$. |
| && \cdots p(t_n-t_{n-1},x_{n-1},x_n) \; dx_1 \cdots \; dx_n, |
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| \end{eqnarray} |
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| where $p(t,x,y) = \frac{1}{\sqrt{2\pi t}}\exp(-\frac{(x-y)^2}{2t})$ defined for any $x,y\in\mathbb{R}$ and $t>0$. |
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| This of course corresponds to the defining property of Brownian motion. The other properties carry over as well; for instance, the set of paths in $W(\mathbb{R})$ which are nowhere differentiable is of $P$-measure $1$. |
\item For any $s \le t \in [0, \infty)$ and $A, B \subset \mathbb{R}$ Borel sets, we have $$P(\{\omega : \omega(s) \in A, \omega(t)-\omega(s) \in B\}) = P(\{\omega : \omega(s) \in A\}) P(\{\omega : \omega(t)-\omega(s) \in B\}).$$ That is, $\omega(s)$ and $\omega(t)-\omega(s)$ are independent $\mathbb{R}$-valued random variables. |
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\end{enumerate} |
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These of course correspond to the defining properties of Brownian motion. The other properties carry over as well, for instance, the set of paths in $W(\mathbb{R})$ which are nowhere differentiable is of $P$-measure $1$. |
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| The Wiener space $W(\mathbb{R}^d)$ and corresponding Wiener measure are defined similarly, in which case $P$ is the distribution of a $d$-dimensional Brownian motion. |
The Wiener space $W(\mathbb{R}^d)$ and corresponding Wiener measure are defined similarly, in which case $P$ is the distribution of a $d$-dimensional Brownian motion. |
