# Itô process

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A stochastic process with a stochastic differential. More precisely, a continuous stochastic process on a probability space with a certain non-decreasing family of -algebras of is called an Itô process with respect to if there exists processes and (called the drift coefficient and the diffusion coefficient, respectively), measurable with respect to for each , and a Wiener process with respect to , such that Such processes are called after K. Itô , . One and the same process can be an Itô process with respect to two different families . The corresponding stochastic differentials may differ substantially in this case. An Itô process is called a process of diffusion type (cf. also Diffusion process) if its drift coefficient and diffusion coefficient are, for each , measurable with respect to the -algebra Under certain, sufficiently general, conditions it is possible to represent an Itô process as a process of diffusion type, but, generally, with some new Wiener process (cf. ). If an Itô process is representable as a diffusion Itô process with some Wiener process and if the equation is satisfied, then is called the innovation process for .

Examples. Suppose that a certain Wiener process , , with respect to has been given and suppose that where is a normally-distributed random variable with mean and variance that is measurable with respect to .

The process , regarded with respect to , has stochastic differential in which the new Wiener process , defined by is an innovation process for .

How to Cite This Entry:
Itô process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=It%C3%B4_process&oldid=16443
This article was adapted from an original article by A.A. Novikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article