Brownian Motion Simulator

Understanding Brownian Motion

Brownian motion—also known in mathematical contexts as the Wiener process—describes a continuous-time stochastic process that models random movement. These terms are often used interchangeably, although "Brownian motion" is the more general term used in physics and applied sciences, while "Wiener process" refers specifically to the standard mathematical formulation with mean zero and variance growing linearly in time.

A common generalization of Brownian motion includes both a deterministic trend and a random component. This is captured by a stochastic differential equation (SDE) for a one-dimensional process $X(t)$:

In this equation, $dX(t)$ is the infinitesimal change in the process $X(t)$ over a small time interval $dt$. The term $\mu\,dt$ represents the drift—a deterministic component where $\mu$ is the drift coefficient indicating the average rate of change. Positive drift ($\mu > 0$) implies an upward trend over time, while negative drift implies a downward trend.

The term $\sigma\,dW(t)$ captures the diffusion, where $dW(t)$ is an increment of the standard Wiener process. This component introduces randomness, modeling unpredictable fluctuations. The coefficient $\sigma$ controls the magnitude of this noise: higher values of $\sigma$ lead to more volatile trajectories.

The Wiener process $W(t)$ is a mathematical idealization of Brownian motion: it starts at zero, has continuous paths, independent increments, and for any time $t$, the increment $W(t + \Delta t) - W(t)$ is normally distributed with mean 0 and variance $\Delta t$. In symbols: $dW(t) \sim \mathcal{N}(0,\,dt)$.

When $\mu = 0$ and $\sigma = 1$, the SDE simplifies to $dX(t) = dW(t)$, and the process $X(t)$ is just the standard Wiener process itself—that is, standard Brownian motion without drift or scaling.

This framework is foundational in modeling systems that combine predictable trends with random variation, including applications in physics, biology, and especially finance (e.g., the Black–Scholes model). The parameters $\mu$ and $\sigma$ together determine the relative strength of deterministic versus stochastic forces.

For more background, visit the Wikipedia pages on Brownian motion and Wiener process.