Wiener Process

The Wiener process (also called Brownian motion) $W(t)$ is a continuous-time stochastic process with the following properties:

1. $W(0)=0$
2. Increments are independent: $W(t)-W(s)$ is independent of $W(s)-W(r)$ for $r<s<t$
3. Increments are normally distributed: $W(t)-W(s)\sim N(0,t-s)$

Numerical Simulation

Discretise time with step $dt$. At each step draw $Z\sim N(0,1)$ and update:

$$W(t+dt)=W(t)+\sqrt{dt}\cdot Z$$

This is the Euler-Maruyama scheme applied to $dW=dW$ (trivially).

Connection to Diffusion

The probability density function of $W(t)$ is Gaussian with variance $t$:

$$p(w,t)=\frac{1}{\sqrt{2\pi t}}\exp \left(-\frac{w^2}{2t}\right)$$

This is the solution to the Heat equation with diffusion coefficient $\alpha =1/2$ and a delta-function initial condition $p(w,0)=\delta (w)$.

Python

Python

import numpy as np import matplotlib.pyplot as plt def simulate_wiener_process( time_end: float, number_of_steps: int, seed: int = 0, ) -> tuple[np.ndarray, np.ndarray]: """Simulate a single Wiener process path. Args: time_end: End time T. number_of_steps: Number of time steps. seed: Random seed for reproducibility. Returns: Tuple of (time_array, W_array). """ rng = np.random.default_rng(seed) dt = time_end / number_of_steps time_array = np.linspace(0, time_end, number_of_steps + 1) increments = np.sqrt(dt) * rng.standard_normal(number_of_steps) W_array = np.concatenate([[0.0], np.cumsum(increments)]) return time_array, W_array

Output

Random walks | Euler-Maruyama scheme | Ornstein-Uhlenbeck process | Heat equation | Stochastic differential equation