Itô process

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

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.