# Covariance of mean-reverting Vasicek process?

I am dealing with a mean-reverting Vasicek process defined as:

$$\begin{equation} S_t = S_0 e^{-at} + b(1-e^{(-at)}) + \sigma e^{(-at)} \int_{0}^{t} e^{(-as)} \ W_t \end{equation}$$

I want to determine the following covariance:

$$\begin{equation} Cov[(S_{t+i}),(S_{t})] \end{equation}$$

Could someone help me with the analytical derivation? Thanks in advance!

## 1 Answer

Hint: We need to start with SDE:

$$\begin{equation} S_t = S_0 e^{-at} + b(1-{\rm e}^{-at}) + \sigma \int_0^t {\rm e}^{-a(t-u)}\; dW_u \end{equation}$$

As first two terms are deterministic, using standard properties of covariance, computation of $${\rm cov} (S_{t_1}, S_{t_2})$$ can be reduced to the computation of

$${\rm cov} (Y_{t_1}, Y_{t_2})$$ where

$$Y_t = \int_0^t {\rm e}^{au}\; dW_u.$$

Last covariance calculation can be found here.

Edit: Once $${\rm cov} (S_{t_1}, S_{t_2})$$ is available for all $$t_1$$ and $$t_2$$, covariance properties (on linear combinations) can then be used again to answer the original question.

• @MarkMarconi Yes. I added an edit (covariance behaviour on linear combinations of random variables). – ir7 Oct 5 '20 at 13:27
• I'm assuming your term is $S_{t+i+h} - {\rm e}^{-2ah}S_{t+i}$. – ir7 Oct 5 '20 at 13:29