# Apply Ito's lemma to the absolute value of a process

Let $(X_t)_{t\geq 0}$ be a mean reverting stochastic process with $X_0=0$ and dynamics \begin{align*} dX_t &= -X_t + dW_t,\quad t\geq 0, \end{align*} where $(W_t)_{t\geq 0}$ is a Brownian motion. Can we use Ito's lemma to find the dynamics of the process $(Y_t)_{t\geq 0}$ defined by $Y_t=|X_t|$?

Since $f(x)=|x|$ is not differentiable at the origin, we cannot directly apply it, but I know there is an extention (Ito-Tanaka) that I am struggling to apply to this problem.

• You wrote exactly what is needed in this case (Ito-Tanaka), so what is that you struggle with? – zhoraster Dec 7 '17 at 5:06