Itô process ($X_t=X_0+{\int }_0^t{b}_{s}ds+{\int }_0^t{\sigma }_{s}d{W}_s$)

An Itô process is a continuous semimartingale expressible as the displayed sum, with adapted coefficients locally integrable for the time integral and locally square integrable for the Itô integral. The coefficients may depend on the whole past; an Itô diffusion usually specifies them as functions of the current state and time. A continuously differentiable deterministic process is an Itô process with zero Brownian coefficient. The Itô formula and Itô product rule apply to these processes.