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A [[Stochastic process|stochastic process]] with a [[Stochastic differential|stochastic differential]]. More precisely, a continuous stochastic process $X_t$ on a probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530402.png" /> with a certain non-decreasing family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530403.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530404.png" />-algebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530405.png" /> is called an Itô process with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530406.png" /> if there exists processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530408.png" /> (called the drift coefficient and the diffusion coefficient, respectively), measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530409.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304010.png" />, and a [[Wiener process|Wiener process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304011.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304012.png" />, such that
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A [[Stochastic process|stochastic process]] with a [[Stochastic differential|stochastic differential]]. More precisely, a continuous stochastic process $X_t$ on a probability space $(\Omega, \mathcal{F}, \Prob)$ with a certain non-decreasing family $\{\mathcal F_t\}$ of $\sigma$-algebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530405.png" /> is called an Itô process with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530406.png" /> if there exists processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530408.png" /> (called the drift coefficient and the diffusion coefficient, respectively), measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530409.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304010.png" />, and a [[Wiener process|Wiener process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304011.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304012.png" />, such that
  
 
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304013.png" /></td> </tr></table>

Latest revision as of 06:38, 10 December 2014

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A stochastic process with a stochastic differential. More precisely, a continuous stochastic process $X_t$ on a probability space $(\Omega, \mathcal{F}, \Prob)$ with a certain non-decreasing family $\{\mathcal F_t\}$ of $\sigma$-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 [1], [2]. 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. [3]). 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 .

References

[1] 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
[2] R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , 1–2 , Springer (1977–1978) (Translated from Russian)
[3] 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


Comments

For additional references see Itô formula.

How to Cite This Entry:
Itô process. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=It%C3%B4_process&oldid=35534
This article was adapted from an original article by A.A. Novikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article