Brownian motion

Idea

Think of Brownian motion $W_{t}$ as the limit of a random walk when:

The step size becomes infinitesimally small, and
The time between steps shrinks to zero, but
The variance scales properly with time.

Definition

Brownian motion $W (t)$ , also called Wiener process, is a fundamental stochastic process with the following properties:

$W (0) = 0$
Independent increments
$W (t + Δ t) - W (t) \sim N (0, Δ t)$
Continuous paths, but nowhere differentiable

Heuristic Interpretation of $\frac{d W}{d t}$

Although $\frac{d W}{d t}$ does not exist in the classical sense (because Brownian paths are too irregular), we can informally treat it as a kind of random increment:

\frac{d W}{d t} \sim Gaussian noise \sim N (0, \infty)

So over a small interval $d t$ , we write:

d W (t) = \frac{d W}{d t} \cdot d t \sim N (0, d t)

This forms the basis of numerical simulation of Brownian motion and stochastic differential equations (SDEs).

Simulation Heuristic

To simulate $W (t)$ over small time steps $d t$ , we generate:

d W \sim N (0, d t) or d W = \sqrt{d t} \cdot ξ, ξ \sim N (0, 1)