Stochastic differential equation

A stochastic differential equation (SDE) is a differential equation in which one or more terms is a stochastic process, typically a Wiener process. It extends the framework of ordinary differential equations to systems that are subject to random perturbations: the rate of change of a variable depends not only on its current state but on a random noise term.

The canonical form is:

dX_t = a(X_t, t)dt + b(X_t, t)dW_t

where a is the drift term, b is the diffusion term, and dW_t is the increment of a Wiener process. The theory was developed by Kiyoshi Itô, whose Itô calculus provides the rules for manipulating such equations.

SDEs are the workhorse of modern mathematical finance, population biology, and statistical physics. They are not a generalization of ODEs but a different kind of object entirely, requiring a distinct calculus and a distinct intuition.