6.1 Wiener Process

We begin with a simple symmetric random walk $ \{ X_n; \, \, n \ge
0 \} $ starting in 0 ($ X_0 = 0$). The increments $ Z_n = X_n -
X_{n-1}$ are i.i.d. with :

$\displaystyle \P(Z_n = 1) = \P(Z_n = -1) = \frac{1}{2} \, . $

By shortening the period of time of two successive observations we accelerate the process. Simultaneously, the increments of the process become smaller during the shorter period of time. More precisely, we consider a stochastic process $ \{ X_t^{\Delta} ; \,
\, t \ge 0\}$ in continuous time which increases or decreases in a time step $ \Delta t$ with probability $ \frac{1}{2} $ by $ \Delta
x.$ Between these jumps the process is constant (alternatively we could interpolate linearly). At time $ t = n \cdot \Delta t$ the process is:

$\displaystyle X_t^\Delta = \sum^n_{k=1} \, Z_k \cdot \Delta x = X_n \cdot \Delta x $

where the increments $ Z_1 \Delta x, Z_2 \Delta x, \ldots$ are mutually independent and take the values $ \Delta x$ or $ - \Delta
x$ with probability $ \frac{1}{2} $ respectively. From Section 4.1 we know:

$\displaystyle \mathop{\text{\rm\sf E}}[X_t^\Delta] = 0\ , \qquad \mathop{\text{...
...ar}}(X_n) = (\Delta x) ^2 \cdot n = t \cdot \frac{(\Delta x)^2}{\Delta t}
\, . $

Now, we let $ \Delta t$ and $ \Delta x$ become smaller. For the process in the limit to exist in a reasonable sense, $ \mathop{\text{\rm Var}}
(X_t^{\Delta})$ must be finite. On the other hand, $ \mathop{\text{\rm Var}}
(X_t^{\Delta})$ should not converge to 0, since the process would then not be random any more. Hence, we must choose:

$\displaystyle \Delta t \rightarrow 0, \, \, \Delta x = c \cdot \sqrt{\Delta t}\,
, \, \,$   such that $\displaystyle \, \mathop{\text{\rm Var}}(X_t^\Delta) \rightarrow c^2 t \, . $

If $ \Delta t$ is small, then $ n = t/\Delta t$ is large. Thus, the random variable $ X_n$ of the ordinary symmetric random walk is approximately N$ (0,n)$ distributed, and therefore for all $ t$ (not only for $ t$ such that $ t = n\ \Delta t$):

$\displaystyle {\cal L} (X_t^\Delta) \approx$   N$\displaystyle (0,\ n(\Delta x)^2)
\approx$   N$\displaystyle (0,\ c^2 t)\ . $

Thus the limiting process $ \{ X_t; \, \, t \ge 0 \}$ which we obtain from $ \{ X_t^\Delta ; \, \, t \ge 0 \}$ for $ \Delta t
\rightarrow 0, \, \, \Delta x = c\ \sqrt{\Delta t}$ has the following properties:

(i)
$ X_t$ is    N$ (0, c^2t)$ distributed for all $ t \ge 0.$
(ii)
$ \{ X_t; \, \, t \ge 0 \}$ has independent increments, i.e. for $ 0 \le s < t,$ $ X_t - X_s$ is independent of $ X_s$ (since the random walk $ \{ X_n; \, \, n \ge
0 \} $ defining $ \{ X_t^\Delta ; \, \, t \ge 0 \}$ has independent increments).
(iii)
For $ 0 \le s < t$ the increment $ (X_t - X_s)$ is N$ (0,\
c^2 \cdot (t-s))$ distributed, i.e. its distribution depends exclusively on the length $ t-s$ of the time interval in which the increment is observed (this follows from (i) and (ii) and the properties of the normal distribution).
A stochastic process $ \{ X_t; \, \, t \ge 0 \}$ in continuous time satisfying (i)-(iii) is called Wiener process or Brownian motion starting in 0 ($ X_0 = 0$). The standard Wiener process resulting from $ c=1$ will be denoted by $ \{W _t; \, \, t \ge 0 \}.$ For this process it holds for all $ 0 \le s < t\:$

$\displaystyle \mathop{\text{\rm\sf E}}[W_t] = 0 , \, \, \mathop{\text{\rm Var}}(W_t) = t $


$\displaystyle \mathop{\text{\rm Cov}}(W_t, W_s)$ $\displaystyle =$ $\displaystyle \mathop{\text{\rm Cov}}(W_t - W_s + W_s , W_ s )$  
  $\displaystyle =$ $\displaystyle \mathop{\text{\rm Cov}}(W_t - W_s , W_ s ) + \mathop{\text{\rm Cov}}(W_s , W_ s )$  
  $\displaystyle =$ $\displaystyle 0 + \mathop{\text{\rm Var}}(W_s) = s$  

Fig.: Typical paths of a Wiener process.
6979 SFEWienerProcess.xpl
\includegraphics[width=1\defpicwidth]{wiener.ps}

As for every stochastic process in continuous time, we can consider a path or realization of the Wiener process as a randomly chosen function of time. With some major mathematical instruments it is possible to show that the paths of a Wiener process are continuous with probability 1:

$\displaystyle \P(W_t \, \,$   is continuous as a function of $\displaystyle \, t) = 1 \, .$

That is to say, the Wiener process has no jumps. But $ W_t$ fluctuates heavily: the paths are continuous but highly erratic. In fact, it is possible to show that the paths are not differentiable with probability 1.

Being a process with independent increments the Wiener process is markovian. For $ 0 \le s < t$ it holds $ W_t = W_s + ( W_t
- W_s ),$ i.e. $ W_t$ depends only on $ W_s$ and on the increment from time $ s$ to time $ t$:

    $\displaystyle \P(a < W_t < b\ \vert W_s = x\ , \, \,$   information about $\displaystyle \, \, W_{t^\prime} , \, \, 0 \le t^ \prime < s )$  
  $\displaystyle =$ $\displaystyle \P(a < W_t < b\ \vert W_s = x\ )$  

Using properties (i)-(iii), the distribution of $ W_t$ given the outcome $ W_s=x$ can be formulated explicitly. Since the increment $ (W_t - W_s)$ is N$ (0, t-s)$ distributed, $ W_t$ is N$ (x, t-s)$ distributed given $ W_s=x$ :

$\displaystyle \P( a < W_t < b\ \vert W_s = x ) = \int^b_a \frac{1}{\sqrt{t-s}}
\varphi \big( \frac{y-x}{\sqrt{t-s}}\big) dy. $

Proceeding from the assumption of a random walk $ \{ X_n; \, n \ge
0 \}$ with drift $ \mathop{\text{\rm\sf E}}[X_n] = n (2p-1)$ instead of a symmetric random walk results in a process $ X_t^\Delta$ which is no more zero on average, but

$\displaystyle \mathop{\text{\rm\sf E}}[X_t^\Delta]$ $\displaystyle =$ $\displaystyle n \cdot (2 p-1) \cdot \Delta x = (2 p -1) \cdot
t \, \frac{\Delta x}{\Delta t}$  
$\displaystyle \mathop{\text{\rm Var}}(X_t^\Delta)$ $\displaystyle =$ $\displaystyle n\ 4 p (1-p) \cdot (\Delta x)^2 = 4p (1-p) \cdot t
\cdot \frac{(\Delta x)^2}{\Delta t}.$  

For $ \Delta t \rightarrow 0, \, \Delta x = \sqrt{\Delta t}, \, p
= \frac{1}{2} (1 + \mu \sqrt{\Delta t} )$ we obtain for all $ t$:

$\displaystyle \mathop{\text{\rm\sf E}}[X_t^\Delta] \rightarrow \mu t \, , \, \, \mathop{\text{\rm Var}}(X_t^\Delta) \rightarrow t. $

The limiting process is a Wiener process $ \{ X_t; \, \, t \ge 0 \}$ with drift or trend $ \mu t .$ It results from the standard Wiener process:

$\displaystyle X_t = \mu t + W_t .$

Hence, it behaves in the same way as the standard Wiener process but it fluctuates on average around $ \mu$ instead of 0. If $ (\mu >
0)$ the process is increasing linearly on average, and if $ (\mu <
0)$ it is decreasing linearly on average.