# Mean Crossing for Ornstein-Uhlenbeck

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

I presume you talk about local time.

• Is the paper Statistics of Local Time and Excursions of the Ornstein-Uhlenbeck Process by J. Hawkes and A. Truman you are citing? – Hans Nov 15 '16 at 7:14
• Yes. You are right. Thanks. I edit my answer accordingly. – MJ73550 Nov 15 '16 at 11:26