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'Brownian motion'
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| Title of object: |
Brownian motion |
| Canonical Name: |
BrownianMotion |
| Type: |
Definition |
| Created on: |
2005-04-27 16:48:14 |
| Modified on: |
2006-09-11 21:49:37 |
| Classification: |
msc:60J65 |
| Keywords: |
Levy characterization |
| Synonyms: |
Brownian motion=wiener process |
Preamble:
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Content:
\theoremstyle{definition}
\newtheorem*{mydefn}{Definition}
\begin{mydefn}
One-dimensional \emph{Brownian motion} is a stochastic process $W(t)$ or $W_t$, defined for $t\in [0,\infty)$ such that
\begin{enumerate}
\item $W(0) = 0$ a.s. (almost surely)
\item The sample paths $t\mapsto W(t)$ are almost surely continuous.
\item For any finite sequence of times $0<t_1<\cdots<t_n$ and Borel sets $A_1,\ldots,A_n \subset \mathbb{R}$,
\begin{eqnarray*}
&& \mathbb{P}(\{W(t_1)\in A_1,\ldots,W(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 p(t_n-t_{n-1},x_{n-1},x_n) \; dx_1 \cdots \; dx_n,
\end{eqnarray*}
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 $t>0$.
\end{enumerate}
A $d$-dimensional Brownian motion is a stochastic process $W(t) = (W_1(t), \dots, W_d(t))$ in $\mathbb{R}^d$ whose coordinate processes $W_i(t)$ are independent one-dimensional Brownian motions.
\end{mydefn} |
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