[4] Brownian Motion

1. Stochastic process

(1) Stochastic process

For a probability space $(\Omega, \mathcal{F}, P)$, A stochastic process $\{X_t\}_{t\in[0,T]}$ is a parametrized collection of random variables. and it has two explantory variables which are time $t$ and individual experiment $w$.

For each $t\in[0,T]$,   $X_t : \Omega \rightarrow \R$ is a random variable. and a random variable $X_t(w)$ has its own distribution at each time $t$.

For each $w \in \Omega$,   $X_t : [0,T] \rightarrow \R^n$ is a trajectory in $\R^n$ and each $w$ produces a trajectory $X_t(w)$.

(2) Generating $\mathcal{F}_T$ & $P$

We may consider $\Omega$ as a subset of $\{$ all functions : $[0,T] \rightarrow \R^n$ $\}$.

1) Our basic sets are of the form

$(a_1<X_{t_1}<b_1,a_2<X_{t_2}<b_2,\cdots,a_k<X_{t_k}<b_k)$

$=\{w\in\Omega : (a_1<X_{t_1}(w)<b_1,a_2<X_{t_2}(w)<b_2,\cdots,a_k<X_{t_k}(w)<b_k) \}$ for some $\{t_1, t_2, \cdots, t_k\}\in[0,T]$ and open intervals $(a_1,b_1), (a_2,b_2), \cdots, (a_k,b_k)$

$=\{$ all trajectories passing through $F_j$ at $t_j$,   $j = 1, 2, \cdots, k$ $\}$

2) We can assign probability to basic sets by

$P(a_1<X_{t_1}<b_1,a_2<X_{t_2}<b_2,\cdots,a_k<X_{t_k}<b_k)$

2. Brownian motion

It is also called Wiener process which is the most important example of stochastic process.

(1) Related Notations

Suppos $p(t,x,y)$ is a pdf for $N(x,t)$.
$p(t,x,y) = \frac{1}{\sqrt{2\pi t}}e^{-\frac{(x-y)^2}{2t}}$   with $x,y \in \R, \ t>0$

We can assign probabilities to the basic sets by
$P(a_1<W_{t_1}<b_1,a_2<W_{t_2}<b_2,\cdots,a_k<W_{t_k}<b_k)$
$=\int_{a_k}^{b_k}\int_{a_{k-1}}^{b_{k-1}}\cdots\int_{a_1}^{b_1}p(t_1,0,x_1)p(t_2-t_1,x_1,x_2) \cdots p(t_k-t_{k-1},x_{k-1},x_k) dx_1dx_2\cdots dx_k$

(2) Definition of Weiner process

A (one-dimensional) stochastic process $\{W(t):\Omega\rightarrow\R\}_{t\in[0,T]}$ is called a Wiener process if

• $W(t)$ generates a continuous trajectory with $W(0) = 0$
• The process $W(t)$ has independent increments, i.e.
  for any $t_1<t_2\leq t_3 <t_4$,   $W(t_4)-W(t_3)$ and $W(t_2)-W(t_1)$ are independent.
• For $t_2>t_1$,   $W(t_2)-W(t_1) \sim N(0,t_2-t_1)$

We can prove these properties from (1) notations.