 Itô process

An Itô process has the representationwhere $W$ is a Brownian motion, $X_0$ is ${\mathcal{F}}_0$-measurable, and $b,\sigma$ are progressively measurable processes satisfying ${\int }_0^t(|b_s|+{\sigma }_s^2)ds<\infty$ almost surely on every finite horizon. A predictable version of $\sigma$ may equivalently be used for the Itô integral. The process is a continuous semimartingale; the coefficients may depend on its whole past.

An Itô diffusion usually specifies the coefficients as functions of time and the current state. A continuously differentiable deterministic function is an Itô process with zero noise coefficient. For $X_t=\mu t+\sigma W_t$, its quadratic variation is ${\sigma }^{2}t$, and the Itô formula determines how a smooth function of $X$ evolves.