Ornstein – Uhlenbeck Process

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

dX_t = \alpha\, (\mu - X_t)\, dt + \sigma\, dW_t

where α > 0 and Wt is the Wiener process. It can easily be solved explicitly:

\displaystyle X_t = e^{-\alpha t} X_0 + \mu\, (1 - e^{-\alpha t}) + \int_0^t \sigma\, e^{\alpha(s-t)}\, dW_s.

So we deduce that

\displaystyle \begin{aligned} \mathsf EX_t &= e^{-\alpha t} X_0 + \mu\, (1 - e^{-\alpha t}) = \mu + o(1), \quad \mbox{ as } t\to\infty, \\ \mathrm{Var}\, X_t &= \frac {\sigma^2}{2\alpha} \left( 1 - e^{-2\alpha t} \right) = \frac {\sigma^2}{2\alpha}, \quad \mbox{ as } t\to\infty. \end{aligned}

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 \dot x = \alpha\, (\mu - x), which has the solution x(t) = μ + e-αt (x0μ). So we can find the half-time from the equation

\displaystyle x(t_{1/2}) - \mu = \frac {x_0 - \mu}2,

i.e. \displaystyle t_{1/2} = \frac {\log 2}\alpha. In particular, the higher the mean-reversion speed is, the smaller is the half-life.

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