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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!

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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.

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  • $\begingroup$ @MarkMarconi Yes. I added an edit (covariance behaviour on linear combinations of random variables). $\endgroup$
    – ir7
    Commented Oct 5, 2020 at 13:27
  • $\begingroup$ I'm assuming your term is $S_{t+i+h} - {\rm e}^{-2ah}S_{t+i}$. $\endgroup$
    – ir7
    Commented Oct 5, 2020 at 13:29

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