View source for Itô process

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

<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>

Such processes are called after K. Itô [[#References|[1]]], [[#References|[2]]]. One and the same process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304014.png" /> can be an Itô process with respect to two different families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304015.png" />. The corresponding stochastic differentials may differ substantially in this case. An Itô process is called a process of diffusion type (cf. also [[Diffusion process|Diffusion process]]) if its drift coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304016.png" /> and diffusion coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304017.png" /> are, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304018.png" />, measurable with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304019.png" />-algebra

<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/i05304020.png" /></td> </tr></table>

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. [[#References|[3]]]). If an Itô process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304021.png" /> is representable as a diffusion Itô process with some Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304022.png" /> and if the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304023.png" /> is satisfied, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304024.png" /> is called the innovation process for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304025.png" />.

Examples. Suppose that a certain Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304027.png" />, with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304028.png" /> has been given and suppose 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/i05304029.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304030.png" /> is a normally-distributed random variable with mean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304031.png" /> and variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304032.png" /> that is measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304033.png" />.

The process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304034.png" />, regarded with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304035.png" />, has stochastic differential

<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/i05304036.png" /></td> </tr></table>

in which the new Wiener process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304037.png" />, defined by

<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/i05304038.png" /></td> </tr></table>

is an innovation process for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304039.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.S. Liptser,   A.N. Shiryaev,   "Statistics of random processes" , '''1–2''' , Springer  (1977–1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====
For additional references see [[Itô formula|Itô formula]].