Momenta

The n-dimensional OU-process

$$\mathrm{d}\xi = (A\xi+a) \mathrm{d}t + \Sigma\,\mathrm{d}W$$ (67)

has solution

$$\xi_t = e^{t A}\int_{t_0}^t e^{-\tau A} \left(a\,\mathrm{d}... ...a\,\mathrm{d}W_\tau\right) +e^{(t-t_0) A}\,\xi_{t_0}\quad .$$ (68)

Then $\xi_t\vert\,\xi_{t^\prime}$ is normal-distributed

$$\xi_t\vert\,\xi_{t^\prime} \sim \mathcal{N}\left(\mu_{t\vert\,t^\prime}, V_{t-t^\prime}\right)$$ (69)

with

$$\mu_{t\vert\,t^\prime} = \mathsf{E}\,\xi_t\vert\,\xi_{t^\prime} = e^{t A}\int_{t^\prime}^t e^{-\tau A}\,a\,\mathrm{d}\tau + e^{(t-t^\prime) A}\,\xi_{t^\prime}$$ (70)

$$= e^{(t-t^\prime)A}\left(\xi_{t^\prime}-\mu_\infty\right) + \mu_\infty$$ (71)

$$V_\delta = \mathsf{var}(\xi_{t+\delta}\vert\,\xi_t) = \int_0^\delta \mathrm{d}\tau\, e^{\tau A}\,V\, e^{\tau A^\prime}\quad .$$ (72)

The unconditional mean is $\mu_\infty:=\lim_{t\to\infty}\mu_{t\vert\,t^\prime}=-A^- a$.