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Difference between revisions of "Itô process"

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m
m (fixing \;)
 
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is called an Itô process with respect to  $  \{ {\mathcal F} _{t} \} $
 
is called an Itô process with respect to  $  \{ {\mathcal F} _{t} \} $
 
if there exists processes  $  a (t) $
 
if there exists processes  $  a (t) $
and  $  \sigma (t) $(
+
and  $  \sigma (t) $ (called the drift coefficient and the diffusion coefficient, respectively), measurable with respect to  $  {\mathcal F} _{t} $
called the drift coefficient and the diffusion coefficient, respectively), measurable with respect to  $  {\mathcal F} _{t} $
 
 
for each  $  t $,  
 
for each  $  t $,  
 
and a [[Wiener process|Wiener process]]  $  W _{t} $
 
and a [[Wiener process|Wiener process]]  $  W _{t} $
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and diffusion coefficient  $  \sigma (t) $
 
and diffusion coefficient  $  \sigma (t) $
 
are, for each  $  t $,  
 
are, for each  $  t $,  
measurable with respect to the  $  \sigma $-
+
measurable with respect to the  $  \sigma $-algebra
algebra
 
  
 
$$  
 
$$  
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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} $
 
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} $
+
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} $
+
and if the equation  $  {\mathcal F} _{t} ^ {\  \overline{W}} = {\mathcal F} _{t} ^ {\  X} $
is satisfied, then  $  \overline{W}\; _{t} $
+
is satisfied, then  $  \overline{W} _{t} $
 
is called the innovation process for  $  X _{t} $.
 
is called the innovation process for  $  X _{t} $.
  
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\frac{m + \gamma X _ t}{1 + \gamma t}
 
\frac{m + \gamma X _ t}{1 + \gamma t}
 
  \  
 
  \  
d t + d \overline{W}\; _{t} ,
+
d t + d \overline{W} _{t} ,
 
$$
 
$$
  
  
in which the new Wiener process  $  \overline{W}\; _{t} $,  
+
in which the new Wiener process  $  \overline{W} _{t} $,  
 
defined by
 
defined by
  
 
$$  
 
$$  
\overline{W}\; _{t} \  = \  {\mathsf E}
+
\overline{W} _{t} \  = \  {\mathsf E}
 
( X _{t} - Y t \mid  {\mathcal F} _{t} ^ {\  X} ) ,
 
( X _{t} - Y t \mid  {\mathcal F} _{t} ^ {\  X} ) ,
 
$$
 
$$

Latest revision as of 03:42, 4 March 2022


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

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=52161
This article was adapted from an original article by A.A. Novikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article