Suppose we have classic Ornstein-Uhlenbeck process. How can we calculate expected number (and variance too) of crossing mean value over the certain period of time?

Say, if we have discrete OU process ($x_{k+1} = \theta(\mu - x_k)\Delta t + \sigma \varepsilon_k\sqrt{\Delta t}$), then parameter $\theta$ affects the speed of mean reversion. Large $\theta$ means higher frictions around $\mu$, therefore - we have more crossings of mean value over any period of time. Small $\theta$ means the reverse, the OU process is slow.

My questions are - is there any explicit formula that links $\theta$ (or any other parameters) and the number of crossings of mean value? If we know all parameters of OU, how can we estimate expected number of crossings of mean value?


1 Answer 1


I presume you talk about local time.

I hope it can help you : https://www.cambridge.org/core/books/stochastic-analysis/statistics-of-local-time-and-excursions-for-the-ornsteinuhlenbeck-process/C69519611B7FC1430C17209B94F3224D

  • $\begingroup$ Is the paper Statistics of Local Time and Excursions of the Ornstein-Uhlenbeck Process by J. Hawkes and A. Truman you are citing? $\endgroup$
    – Hans
    Nov 15, 2016 at 7:14
  • $\begingroup$ Yes. You are right. Thanks. I edit my answer accordingly. $\endgroup$ Nov 15, 2016 at 11:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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