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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?

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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

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  • $\begingroup$ 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. $\endgroup$
    – M. Jeunesse
    Nov 15, 2016 at 11:26

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