View source for Itô process

{{TEX|done}}

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

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

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


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

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


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. [[#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} $.


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====
<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]].