Wiener Process
Standard Wiener Process
Intuition
A Wiener process (also called Brownian motion) is a continuous time process that is like a random walk.
Imagine a discrete-time random process where you start at position $\mathbf{x_{t=0}} = \mathbf{0}$. Then, your position at time $t$ is determined by
\[\mathbf{x}_{t} = \mathbf{x}_{t-\Delta t} + \mathcal{N}(0,\Delta t I)\]Or equivalently,
\[\mathbf{x}_{t} = \mathbf{x}_{t-\Delta t} + \sqrt{\Delta t} \cdot \mathcal{N}(0,I)\]In other words, every time step, you change your position by a vector sampled from $\mathcal{N}(0,\Delta t I)$, where $\Delta t$ is how long each time step is.
A Wiener process is the continuous limit of this as $\Delta t \rightarrow 0$.
Definition
We define a set of independent random variables, or a function mapping the time $t$ to a random variable. It satisfies the property that
\[W_0 = \mathbf{0}\]And
\[W_{t_2} - W_{t_1} = \mathcal{N}(0, (t_2 - t_1)I)\]Note that this restriction is only possible because the variance of the sum of two independent random variables is the sum of the two variances. In other words, since $W_{t_1} \perp W_{t_2}$ for all $t_1 \neq t_2 $, the variance accumulates linearly over time.
Also, under this formulation, we have
\[W_{t} = \mathcal{N}(0, tI)\]linearly as $t$ increases.
In other words, at every time step, we take a infinitesimally small step in a random direction proportional to a vector sampled from the standard normal:
\[dW \sim \mathcal{N}(0, I dt)\]Last Reviewed: 2/4/25