Itô process
A stochastic process with a stochastic differential. More precisely, a continuous stochastic process $X_t$ on a probability space 
with a certain non-decreasing family 
of 
-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.
Itô process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=It%C3%B4_process&oldid=23338