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A [[Stochastic process|stochastic process]] with a [[Stochastic differential|stochastic differential]]. More precisely, a continuous stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i0530401.png" /> 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
+
{{TEX|done}}
  
<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>
+
$\newcommand{\Prob}{\mathsf{P}}$
 +
$\newcommand{\Ex}{\mathsf{E}}$
  
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
+
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  $  \Omega $
 +
is called an Itô process with respect to $  \{ {\mathcal F} _{t} \} $
 +
if there exists processes  $  a (t) $
 +
and  $  \sigma (t) $(
 +
called the drift coefficient and the diffusion coefficient, respectively), measurable with respect to  $  {\mathcal F} _{t} $
 +
for each $  t $,  
 +
and a [[Wiener process|Wiener process]]  $  W _{t} $
 +
with respect to $  \{ {\mathcal F} _{t} \} $,
 +
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/i05304020.png" /></td> </tr></table>
+
$$
 +
d X _{t} \  = \  a ( t ) \  d t + \sigma (t) \  d W _{t} .
 +
$$
  
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
+
Such processes are called after K. Itô [[#References|[1]]], [[#References|[2]]]. One and the same process  $  X _{t} $
 +
can be an Itô process with respect to two different families  $  \{ {\mathcal F} _{t} \} $.  
 +
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  $  a (t) $
 +
and diffusion coefficient  $  \sigma (t) $
 +
are, for each  $  t $,  
 +
measurable with respect to the  $  \sigma $-
 +
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/i05304029.png" /></td> </tr></table>
+
$$
 +
{\mathcal F} _{t} ^ {\  X} \  = \  \sigma \{  \omega  : {X _{s} ,\  s \leq t} \}
 +
.
 +
$$
  
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
+
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  $  X _{t} $
 +
is representable as a diffusion Itô process with some Wiener process  $  \overline{W}\; _{t} $
 +
and if the equation  $  {\mathcal F} _{t} ^ {\  \overline{W}\;} = {\mathcal F} _{t} ^ {\  X} $
 +
is satisfied, then  $  \overline{W}\; _{t} $
 +
is called the innovation process for  $  X _{t} $.
  
<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
+
Examples. Suppose that a certain Wiener process $  W _{t} $,
 +
$  t \geq 0 $,  
 +
with respect to  $  \{ {\mathcal F} _{t} \} $
 +
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/i05304038.png" /></td> </tr></table>
+
$$
 +
d X _{t} \  = \  Y \  d t + d W _{t} ,
 +
$$
 +
 
 +
 
 +
where  $  Y $
 +
is a normally-distributed random variable with mean  $  m $
 +
and variance  $  \gamma $
 +
that is measurable with respect to  $  {\mathcal F} _{0} $.
 +
 
 +
 
 +
The process  $  X _{t} $,
 +
regarded with respect to  $  {\mathcal F} _{t} ^ {\  X} $,
 +
has stochastic differential
 +
 
 +
$$
 +
d X _{t} \  = \
 +
 
 +
\frac{m + \gamma X _ t}{1 + \gamma t}
 +
\
 +
d t + d \overline{W}\; _{t} ,
 +
$$
 +
 
 +
 
 +
in which the new Wiener process  $  \overline{W}\; _{t} $,
 +
defined by
 +
 
 +
$$
 +
\overline{W}\; _{t} \  = \  {\mathsf E}
 +
( X _{t} - Y t \mid  {\mathcal F} _{t} ^ {\  X} ) ,
 +
$$
 +
 
 +
 
 +
is an innovation process for  $  X _{t} $.
  
is an innovation process for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053040/i05304039.png" />.
 
  
 
====References====
 
====References====

Latest revision as of 17:28, 28 January 2020


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

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 $ \Omega $ is called an Itô process with respect to $ \{ {\mathcal F} _{t} \} $ if there exists processes $ a (t) $ and $ \sigma (t) $( called the drift coefficient and the diffusion coefficient, respectively), measurable with respect to $ {\mathcal F} _{t} $ for each $ t $, and a Wiener process $ W _{t} $ with respect to $ \{ {\mathcal F} _{t} \} $, such that

$$ d X _{t} \ = \ a ( t ) \ d t + \sigma (t) \ d W _{t} . $$


Such processes are called after K. Itô [1], [2]. One and the same process $ X _{t} $ can be an Itô process with respect to two different families $ \{ {\mathcal F} _{t} \} $. 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 $ a (t) $ and diffusion coefficient $ \sigma (t) $ are, for each $ t $, measurable with respect to the $ \sigma $- algebra

$$ {\mathcal F} _{t} ^ {\ X} \ = \ \sigma \{ \omega : {X _{s} ,\ s \leq t} \} . $$


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 $ X _{t} $ is representable as a diffusion Itô process with some Wiener process $ \overline{W}\; _{t} $ and if the equation $ {\mathcal F} _{t} ^ {\ \overline{W}\;} = {\mathcal F} _{t} ^ {\ X} $ is satisfied, then $ \overline{W}\; _{t} $ is called the innovation process for $ X _{t} $.


Examples. Suppose that a certain Wiener process $ W _{t} $, $ t \geq 0 $, with respect to $ \{ {\mathcal F} _{t} \} $ has been given and suppose that

$$ d X _{t} \ = \ Y \ d t + d W _{t} , $$


where $ Y $ is a normally-distributed random variable with mean $ m $ and variance $ \gamma $ that is measurable with respect to $ {\mathcal F} _{0} $.


The process $ X _{t} $, regarded with respect to $ {\mathcal F} _{t} ^ {\ X} $, has stochastic differential

$$ d X _{t} \ = \ \frac{m + \gamma X _ t}{1 + \gamma t} \ d t + d \overline{W}\; _{t} , $$


in which the new Wiener process $ \overline{W}\; _{t} $, defined by

$$ \overline{W}\; _{t} \ = \ {\mathsf E} ( X _{t} - Y t \mid {\mathcal F} _{t} ^ {\ X} ) , $$


is an innovation process for $ X _{t} $.


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=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