Ornstein – Uhlenbeck process is a mean-reverting process, which is described by the SDE
where α > 0 and Wt 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, t1/2, 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 (x0 – 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.
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