Ornstein–Uhlenbeck integral equation

Let $(\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t{)}_{t\ge 0},P)$ carry a standard Brownian motion $B$ adapted to the filtration, with future increments independent of the present sigma-algebra. Fix constants $\gamma >0$ and $\sigma >0$, and an ${\mathcal{F}}_0$-measurable real random variable $X_0$ with finite second moment. A continuous adapted solution of the Ornstein–Uhlenbeck equation on $[0,T]$ is a continuous adapted process $X$ satisfying, almost surely for every $t\in [0,T]$, $X_t=X_0-\gamma {\int }_0^t{X}_{s}ds+\sigma B_t,$ where the time integral is the ordinary Lebesgue integral along the continuous path. In differential notation this is $d{X}_t=-\gamma X_{t}dt+\sigma d{B}_t$, with the stochastic differential interpreted through the displayed integral equation.

The definition asserts neither existence nor uniqueness. The Itô integral in the explicit representation is the square-integrable predictable-process integral of Ito integral for square-integrable predictable processes.