analytic solution to Ornstein-Uhlenbeck SDE

This entry derivesthe analytical solutionto the stochastic differential equationfor 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 $\sigma >0$ areconstants.

Motivated by the observationthat $\theta$ is supposed to be the long-term meanof 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 drifttowards the value zero, at an exponential rate $\kappa$. This motivatesthe 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 integralsshows 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 obtainedby Itô-integrating both sides from $s$ to $t$:

$Z_t=Z_s+\sigma {\displaystyle {\int }_s^t}{e}^{\kappa u}𝑑W_u.$

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

$$X_t=Y_t+\theta =\theta +e^{-\kappa (t-s)}(X_s-\theta )+\sigma {\int }_s^t{e}^{-\kappa (t-u)}𝑑W_u.$$