Stochastic differential equation

Summary

A stochastic differential equation describes a quantity whose change in a short time is a drift plus a random kick. The kick is proportional to the increment of a Wiener process. It is written $dX_t = b(t, X_t) \, dt + \sigma(t, X_t) \, dW_t$ and is made precise as an integral equation with an Itô integral from Itô calculus. Such equations model stock prices, noisy physical systems, and population sizes under random effects, and their solutions are diffusion processes.