# Difference between revisions of "Itô process"

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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 | + | $\newcommand{\Prob}{\mathsf{P}}$ |

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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 | ||

<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> | <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> |

## Revision as of 04:38, 10 December 2014

$\newcommand{\Prob}{\mathsf{P}}$ $\newcommand{\Ex}{\mathsf{E}}$

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://encyclopediaofmath.org/index.php?title=It%C3%B4_process&oldid=35515