Itô process

Definition

integral

An Ito process or stochastic integral is a stochastic process on probability space $(\Omega, \mathcal{F},\mathbb{P})$ adopted to $\mathcal{F}_t$ which can be written in the form

where $U,V \in \mathcal{L}_2$.

differential

A continuous stochastic process $X_t$ on probability space $(\Omega, \mathcal{F},\mathbb{P})$ with certain non-decreasing family $\{\mathcal{F}_t\}$ of σ-algebras of $\Omega$ is called an Itô process with respect to $\{\mathcal{F}_t\}$ if there exists processes $a(t)$, drift coefficient, and $\sigma(t)$ diffusion coefficient, measurable with respect to $\mathcal{F}_t$ for each $t$, and Wiener process $W_t$ with respect to $\{\mathcal{F}_t\}$ such that

See also

• martingale
• stochastic process
• Wiener process (Brownian motion)
• Itô lemma

References

1. https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/d9d7372cbf65d56aa8aa9d59ba0ab2e8_MIT15_070JF13_Lec17.pdf
2. https://encyclopediaofmath.org/wiki/Itô_process
3. https://math.nyu.edu/~goodman/teaching/StochCalc2020/week2/Week2.pdf
4. https://chewisinho.github.io/main.pdf, Chapter 1, p. 9
5. https://www.columbia.edu/~mh2078/FoundationsFE/IntroStochCalc.pdf