Ornstein–Uhlenbeck Process

Write down the SDE, give intuition for the parameters, and the solution.

Answer

The SDE of the Ornstein–Uhlenbeck (OU) process is

dX_t = \theta(\mu - X_t)\,dt + dB_t, \qquad X_0 \in \mathbb{R}.

Intuition for the parameters:

$\mu$ is the long-run mean. If $X_t < \mu$ , the drift $\theta(\mu - X_t)$ is positive and pushes $$X_t$$ upward; if $X_t > \mu$ , the drift is negative and pulls it downward.
$\theta$ is the speed of reversion. The larger the $\theta$ , the faster the mean reversion.
$\sigma$ controls the size of the random fluctuations (volatility).

The explicit solution is

X_t = \mu + (X_0 - \mu)e^{-\theta t} + \int_0^t e^{-\theta (t-s)}\,dB_s.

Hence we see that the OU process is mean-reverting and Gaussian. The later holds because the stochastic integral is a Gaussian process.