stochastic differential equation

SDEs are typically written in terms of the differential $dW$ of a Weiner process (Brownian motion), e.g.,

Although Weiner processes are nowhere differential, this notation is given meaning by the Itô integral which describes a procedure to evaluate the relevant integrals.

Background: an ordinary differential equation

can be equivalently written in differential form

and also integral form

Similarly a stochastic differential equation may be specified as a differential ODE on realizations of Gaussian white noise $\xi_t(\omega) \sim \mathcal{N}(0, 1)$:

or in the equivalent integral form:

Appealing to the Itô integral formalism we recast the second integral in terms of the differential of a Wiener process $\xi_t(\omega)dt = dW_t(\omega)$, which provides formal justification for writing the SDE in the simpler form

where we view $x(t)$ as being 'driven' by an underlying Weiner process.