# Itô process

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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=44352