# Difference between revisions of "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