# analytic solution to Ornstein-Uhlenbeck SDE

This entry derives the analytical solution to the stochastic differential equation for the Ornstein-Uhlenbeck process:

 $\displaystyle dX_{t}=\kappa(\theta-X_{t})\,dt+\sigma\,dW_{t}\,,$ (1)

where $W_{t}$ is a standard Brownian motion, and $\kappa>0$, $\theta$, and $\sigma>0$ are constants.

Motivated by the observation that $\theta$ is supposed to be the long-term mean of the process $X_{t}$, we can simplify the SDE (1) by introducing the change of variable

 $Y_{t}=X_{t}-\theta$

that subtracts off the mean. Then $Y_{t}$ satisfies the SDE:

 $\displaystyle dY_{t}=dX_{t}=-\kappa Y_{t}\,dt+\sigma\,dW_{t}\,.$ (2)

In SDE (2), the process $Y_{t}$ is seen to have a drift towards the value zero, at an exponential rate $\kappa$. This motivates the change of variables

 $Y_{t}=e^{-\kappa t}Z_{t}\quad\Leftrightarrow\quad Z_{t}=e^{\kappa t}Y_{t}\,,$

which should remove the drift. A calculation with the product rule for Itô integrals shows that this is so:

 $\displaystyle dZ_{t}$ $\displaystyle=\kappa e^{\kappa t}Y_{t}\,dt+e^{\kappa t}\,dY_{t}$ $\displaystyle=\kappa e^{\kappa t}Y_{t}\,dt+e^{\kappa t}\bigl{(}-\kappa Y_{t}\,% dt+\sigma\,dW_{t}\bigr{)}$ $\displaystyle=0\,dt+\sigma e^{\kappa t}\,dW_{t}\,.$

The solution for $Z_{t}$ is immediately obtained by Itô-integrating both sides from $s$ to $t$:

 $\displaystyle Z_{t}=Z_{s}+\sigma\int_{s}^{t}e^{\kappa u}\,dW_{u}\,.$

Reversing the changes of variables, we have:

 $Y_{t}=e^{-\kappa t}Z_{t}=e^{-\kappa(t-s)}Y_{s}+\sigma e^{-\kappa t}\int_{s}^{t% }e^{\kappa u}\,dW_{u}\,,$

and

 $X_{t}=Y_{t}+\theta=\theta+e^{-\kappa(t-s)}(X_{s}-\theta)+\sigma\int_{s}^{t}e^{% -\kappa(t-u)}\,dW_{u}\,.$
Title analytic solution to Ornstein-Uhlenbeck SDE AnalyticSolutionToOrnsteinUhlenbeckSDE 2013-03-22 17:19:29 2013-03-22 17:19:29 stevecheng (10074) stevecheng (10074) 4 stevecheng (10074) Derivation msc 60H10 msc 60-00