

In [1]:

    
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

Let $N(\mu,\sigma^2)$ be a normal distribution. A Wiener process $W(t)$ is the limit of a discrete stocastic process. To construct this process, split the total time $t$ into $n$ small intervals of length $h$: $t=nh$. Then, at each time $t_j=jh$, the change in $W$ will just be normally distributed:

$$ W_{j+1}-W_j = \sqrt{h}Z_j $$

where $Z_j \sim N(0,1)$. Finally take $h\rightarrow0$ to make it a continuous process.

Here is a Python function that implements one realization of a Wiener process:



In [31]:

    
def wiener(maxt, n):
    """Return one realization of a Wiener process with n steps and a max time of t."""
    t = np.linspace(0.0,maxt,n)
    h = t[1]-t[0]
    Z = np.random.normal(0.0,1.0,n)
    dW = np.sqrt(h)*Z
    W = dW.cumsum()
    return t, W



In [48]:

    
t, W = wiener(1.0,500)



In [49]:

    
plt.plot(t,W)









    Out[49]:





[<matplotlib.lines.Line2D at 0x109023510>]



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