A stochastic process with a stochastic differential. More precisely, a continuous stochastic process $X_t$ 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 .
|||I.V. Girsanov, "Transforming a certain class of stochastic processes by absolutely continuous substitution of measures" Theor. Probab. Appl. , 5 : 3 (1960) pp. 285–301 Teor. Veroyatnost. i Primenen. , 5 : 3 (1960) pp. 314–330|
|||R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , 1–2 , Springer (1977–1978) (Translated from Russian)|
|||A.N. Shiryaev, "Stochastic equations of nonlinear filtering of Markovian jump processes" Probl. Inform. Transmission , 2 : 3 (1966) pp. 1–8 Probl. Peredachi Inform. , 2 : 3 (1966) pp. 3–22|
For additional references see Itô formula.
Itô process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=It%C3%B4_process&oldid=35515