# Derivation of the transition probability for Ornstein–Uhlenbeck process

According to Wikipedia, the transitional probability of the Ornstein–Uhlenbeck process

$$\mathrm dx_t = -\theta(\mu-x_t)\,\mathrm dt + \sigma\,\mathrm dW_t$$

is

$$P(x, t\,|\,x_0, t_0) = \sqrt{-\frac{\theta}{\sigma^2(1-e^{-2\theta(t-t_0)})}}\exp\left(-\frac{\theta}{\sigma^2}\frac{(x-x_0e^{-\theta(t-t_0)})^2}{\sigma^2(1-e^{-2\theta(t-t_0)})}\right)$$

i.e.

$$x_t\,|\,x_0 = N\left(x_0e^{-\theta(t-t_0)}, -\frac{1}{\theta}\sigma^2(1-e^{-2\theta(t-t_0)})\right)$$

without any proof. It mentions that this is derived from the Fokker–Planck representation, but would there be a way to derive this directly? Perhaps from the formal solution?

$$x_t - x_0 e^{-\theta(t-t_0)} = \mu(1-e^{-\theta(t-t_0)}) + \sigma\int_{t_0}^t e^{-\theta(t-s)}\,\mathrm dW_s$$

Any references to literature with details would also be appreciated.

First, note that for the OU process with drift (note the sign change to agree with the usual definition!) $$d X_t= \theta(\mu−X_t) dt + \sigma dW_t$$ one has the solution $$X_t = x_0 e^{-\theta (t - t_0)} + \mu (1 - e^{-\theta (t - t_0)}) + \sigma \int_{t_0}^t e^{-\theta(t-r)}\,dW_r$$
Now, since $$\int_{t_0}^t e^{-2\theta(t-r)}\,dr = \frac{1 - e^{-2\theta (t-t_0)}}{2\theta}$$ it follows the Itô integral process is $$\mathcal N\left(0, \frac{1 - e^{-2\theta (t-t_0)}}{2\theta}\right)$$. Hence by a scaling argument $$X_t \mid X_0 \sim N\left(x_0 e^{-\theta (t - t_0)} + \mu (1 - e^{-\theta (t - t_0)}),\ \frac{\sigma^2}{2\theta}(1 - e^{-2\theta (t-t_0)})\right)$$ from which the transition density can be easily deduced (thus there is a typo in the proposed solution in the question).