Use Ito's formula for calculate the dynamics of Ornstein-Uhlenbeck process

Question

Consider this stochastic differential equation: $$()=−a()+σ(),\quad (0)=_0∈ \Bbb R$$ where $a$ and $σ$ are constants and $(t)$ is a Brownian motion.

Can someone show me how to use the Ito's formula for the dynamics of $e^{at}X(t)$?

Answer 1

You have a stochastic process described by $$ \mathrm{d}X_t = \mu_t(X_t)\mathrm{d}t + \sigma_t(X_t)\mathrm{d}W_t, \quad\mathrm{where}\quad \begin{cases} \mu_t(x) = -ax \\ \sigma_t(x) = \sigma = const \end{cases}, $$ to which you want to apply the transformation $\varphi(x,t) = e^{at}x$, hence thanks to Itô's lemma : $$ \begin{array}{rcl} \mathrm{d}\varphi(X_t,t) &=&\displaystyle \left(\dot{\varphi}(X_t,t) + \mu_t(X_t)\varphi'(X_t,t) + \frac{1}{2}\sigma_t(x_t)^2\varphi''(X_t,t)\right)\mathrm{d}t + \sigma_t(X_t)\varphi'(X_t,t)\mathrm{d}W_t \\ \mathrm{d}(e^{at}X_t) &=&\displaystyle \left(ae^{at}X_t - aX_t\cdot e^{at} + \frac{1}{2}\sigma^2\cdot0\right)\mathrm{d}t + \sigma\cdot e^{at}\mathrm{d}W_t \\ &=&\displaystyle \sigma e^{at}\mathrm{d}W_t \end{array} $$ where $\dot{\varphi} = \frac{\partial\varphi}{\partial t}$ and $\varphi' = \frac{\partial\varphi}{\partial x}$. In consequence, one finds : $$ X_t = e^{-at}\left(x_0 + \sigma\int_0^t e^{as} \,\mathrm{d}W_s\right) $$