Ornstein-Uhlenbeck process with random long run mean

Question

I am considering an OU process of the form
$$ dx_t = \theta(\mu-x_t)dt + \sigma dW_t $$
where $\kappa, \sigma>0$ and $W_t$ is a Brownian motion. I know that $x_t$ has expectation given by:
$$ \mathbb{E}[x_t]=x_0 e^{-\theta t}+𝜇(1−e^{-\theta t})\ . $$
I read here what happens when $\mu$ is a function of time but I was wondering what happens when $\mu$ is a stochastic process. For instance, what happens when $\mu$ satisfies the SDE:
$$d\mu_t = \gamma dt + \beta dB_t\ ,$$
where $B_t$ is a Brownian motion?

Answer 1

Using the standard strategy for explicitly solving the SDE of an OU process we can solve $$ dx_t = \theta \left(\mu_t - x_t\right)dt + \sigma dW_t \quad\quad \mu_t = \gamma t + \beta B_t,$$ with $W$ and $B$ independent Brownian motions. As user6247850 suggested, this then gives a nice expression for $\mathbb{E}\left[x_t\right]$. We have $$d\left(e^{\theta t} x_t\right) = \theta e^{\theta t}x_t dt + e^{\theta t}\left(\theta \left(\mu_t - x_t\right)dt + \sigma dW_t\right) = e^{\theta t} \left(\theta \mu_t dt + \sigma dW_t\right). $$ Then this means that we get $$ e^{\theta t} x_t = x_0 + \theta\int_0^te^{\theta s} \left(\gamma s + \beta B_s\right) ds + \int_0^t e^{\theta s}dW_s. $$ Bring rearranging and using integration by parts, $$x_t = e^{-\theta t}\left(x_0 + \frac{\gamma}{\theta}\right) + \gamma t - \frac{\gamma}{\theta} + \beta B_t - \beta \int_0^t e^{-\theta\left(t-s\right)}dB_s + \sigma \int_0^t e^{-\theta(t-s)}dW_s.$$ The last three terms being a martingale starting at zero, the expectation is given by $$ \mathbb{E}\left[x_t\right] = e^{-\theta t}\left(x_0 + \frac{\gamma}{\theta}\right) + \gamma t - \frac{\gamma}{\theta}.$$