# Asymptotics of Variable Drift Ornstein–Uhlenbeck Process

The Ornstein–Uhlenbeck process is defined as the stochastic process that solves the following SDE:

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

where $\theta>0$, $\mu$ and $\sigma>0$ are parameters and $W_t$ is Brownian motion. It is well known the solution to this equation. In particular, it is known that

$E(x_t)=x_0 e^{-\theta t}+\mu(1-e^{-\theta t})$

and

$\operatorname{cov}(x_s,x_t) = \frac{\sigma^2}{2\theta}\left( e^{-\theta(t-s)} - e^{-\theta(t+s)} \right).$

It can be easily seen that $\lim_{t\to+\infty}E(x_t)=\mu$ and that $\lim_{t\to+\infty}Var(x_t)=\frac{\sigma^2}{2\theta}$. Assume that $f(t)$ is a well behaved function. What is it known about the process

$dx_t = \theta (f(t)-x_t)\,dt + \sigma\, dW_t$?

In particular, assume that $f(t)$ is periodic with certain period $\tau$. What is the limit of $E(x_t)$?