# analytic solution to Ornstein-Uhlenbeck SDE

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

$d{X}_{t}=\kappa (\theta -{X}_{t})dt+\sigma d{W}_{t},$ | (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:

$d{Y}_{t}=d{X}_{t}=-\kappa {Y}_{t}dt+\sigma d{W}_{t}.$ | (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

$${Y}_{t}={e}^{-\kappa t}{Z}_{t}\mathit{\hspace{1em}}\iff \mathit{\hspace{1em}}{Z}_{t}={e}^{\kappa t}{Y}_{t},$$ |

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

$d{Z}_{t}$ | $=\kappa {e}^{\kappa t}{Y}_{t}dt+{e}^{\kappa t}d{Y}_{t}$ | ||

$=\kappa {e}^{\kappa t}{Y}_{t}dt+{e}^{\kappa t}\left(-\kappa {Y}_{t}dt+\sigma d{W}_{t}\right)$ | |||

$=0dt+\sigma {e}^{\kappa t}d{W}_{t}.$ |

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

${Z}_{t}={Z}_{s}+\sigma {\displaystyle {\int}_{s}^{t}}{e}^{\kappa u}\mathit{d}{W}_{u}.$ |

Reversing the changes of variables, we have:

$${Y}_{t}={e}^{-\kappa t}{Z}_{t}={e}^{-\kappa (t-s)}{Y}_{s}+\sigma {e}^{-\kappa t}{\int}_{s}^{t}{e}^{\kappa u}\mathit{d}{W}_{u},$$ |

and

$${X}_{t}={Y}_{t}+\theta =\theta +{e}^{-\kappa (t-s)}({X}_{s}-\theta )+\sigma {\int}_{s}^{t}{e}^{-\kappa (t-u)}\mathit{d}{W}_{u}.$$ |

Title | analytic solution to Ornstein-Uhlenbeck SDE |
---|---|

Canonical name | AnalyticSolutionToOrnsteinUhlenbeckSDE |

Date of creation | 2013-03-22 17:19:29 |

Last modified on | 2013-03-22 17:19:29 |

Owner | stevecheng (10074) |

Last modified by | stevecheng (10074) |

Numerical id | 4 |

Author | stevecheng (10074) |

Entry type | Derivation |

Classification | msc 60H10 |

Classification | msc 60-00 |