## Ornstein – Uhlenbeck Process

Ornstein – Uhlenbeck process is a mean-reverting process, which is described by the SDE

where *α* > 0 and *W*_{t} is the Wiener process. It can easily be solved explicitly:

So we deduce that

The coefficient *α* is called the *speed of mean reversion*.

*Half-life of the mean-reversion*, *t*_{1/2}, is the average time it will take the process to get pulled half-way back to the mean. To this end, we consider the ODE , which has the solution *x*(*t*) = *μ* + *e*^{-αt} (*x*_{0} – *μ*). So we can find the half-time from the equation

,

i.e. In particular, the higher the mean-reversion speed is, the smaller is the half-life.

### Like this:

Like Loading...

*Related*

or leave a trackback:

Trackback URL.

## Trackbacks

[…] see here for some more […]