Ornstein-Uhlenbeck Process

In this post we consider the Ornstein-Uhlenbeck process, described by the SDE

\[\begin{aligned} dX(t) & = \kappa (\theta - X(t)) dt + \sigma dW(t) \\ % X(0) & = X_0 \end{aligned}\]

with $\kappa, \theta, \sigma > 0$ three parameters. $\kappa$ defines the mean reversion coefficient, $\theta$ is the long-term mean and $\sigma$ is the volatility coefficient.

The above SDE can be solved exactly after rearranging the terms and multiplying by $e^{\kappa t}$:

\[\begin{aligned} dX(t) + \kappa X(t) dt & = \kappa \theta dt + \sigma dW(t) \\ % d(e^{\kappa t} X(t)) & = \kappa \theta e^{\kappa t} dt + \sigma e^{\kappa t} dW(t) \\ % e^{\kappa t}X(t) - X_0 & = \theta \int_0^t e^{\kappa s} \kappa ds + \sigma \int_0^t e^{\kappa s} dW(s) \\ % e^{\kappa t}X(t) - X_0 & = \theta(e^{\kappa t} - 1) + \sigma \int_0^t e^{\kappa s} dW(s) \\ % X(t) & = \theta + e^{-\kappa t} (X_0 - \theta) + \sigma \int_0^t e^{\kappa (s - t)} dW(s). \end{aligned}\]

It is easy to see that the mean is

\[\mathbb{E}[X(t)] = \theta + (X_0 - \theta) e^{-\kappa t}.\]

The covariance can be computed as follows, for $s < t$:

\[\begin{aligned} \mathbb{C}[X(s), X(t)] & = \mathbb{C}\left[ \sigma e^{-\kappa s} \int_0^s e^{\kappa s'} dW(s'), \sigma e^{-\kappa t} \int_0^t e^{\kappa t'} dW(t') \right] \\ % & = \sigma^2 e^{-\kappa(s + t)} \mathbb{E} \left[ \int_0^s e^{\kappa s'} dW(s') \,\, \int_0^t e^{\kappa t'} dW(t') \right] \\ % & = \sigma^2 e^{-\kappa(s + t)} \mathbb{E} \left[ \int_0^s e^{\kappa s'} dW(s') \left( \int_0^s e^{\kappa s'} dW(s') + \int_s^t e^{\kappa t'} dW(t') \right) \right] \\ % & = \sigma^2 e^{-\kappa(s + t)} \mathbb{E} \left[ \int_0^s e^{\kappa s'} dW(s') \int_0^s e^{\kappa s'} dW(s') \right] \\ % & = \sigma^2 e^{-\kappa(s + t)} \int_0^s e^{2 \kappa s'} ds' \\ % & = \sigma^2 e^{-\kappa(s + t)} \frac{1}{2 \kappa}(e^{2\kappa s} - 1). \end{aligned}\]

The variance is therefore

\[\mathbb{V}[X(t)] = \frac{\sigma^2}{2 \kappa}(1 - e^{-2\kappa t}).\]