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!


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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.