Stochastic differential equation

The basic form is $dX_t = \mu_t\,dt + \sigma_t\,dW_t$, where $\mu_t$ is the drift and $\sigma_t$ is the coefficient of the $dW_t$ term. If $\mu_t$ and $\sigma_t$ are constants, X is Brownian motion with drift, $X_t = \mu t + \sigma W_t$. Geometric Brownian motion corresponds to $\mu_t = \mu S_t$ and $\sigma_t = \sigma S_t$.

The equation takes on its meaning when it is written in integral form. The integral of the $dW_t$ term is called the Ito integral, and it requires the integrand to be determined by information up to that time. Ito's lemma is used to transform such equations.