Assuming a variable $v$ follows some Stochastic Process as below -

$dv=\mu v dt + \sigma v dW_t, v \in \left( -\infty, \infty \right) $

I want to get the process of $|v|$

How can I use the ito's lemma in this context given that the function $|v|$ is not smooth?

  • $\begingroup$ Can the GBM take negative values? You can use Ito's lemma with confidence - please see here quant.stackexchange.com/questions/55028/… $\endgroup$ – Magic is in the chain Jul 22 '20 at 18:26
  • $\begingroup$ Sorry, my fault. I removed that. But my question is to apply Ito's lemma, how can I calculate $\frac{d|v|}{dv}$ given it doesnt exist at $v=0$? $\endgroup$ – Bogaso Jul 22 '20 at 18:49
  • 3
    $\begingroup$ Your question doesn't make sense, I think. Firstly, if $v=v(t,\omega)$ is a stochastic process, you can't compute $\frac{d|v|}{dv}$. SDEs may look like differential equations but are actually integral equations. Secondly, if $dv=\mu vdt+\sigma vdW_t$, then $v$ is a geometric Brownian motion and thus positive, i.e. $|v(t,\omega)|=v(t,\omega)$ for all $t,\omega$. So, if $v$ is supposed to be a general stochastic process, you ought to delete this equation. $\endgroup$ – Alex Jul 22 '20 at 19:38

I think you are looking for extended Ito formula (based on Tanaka's formula).

Bjork's The Pedestrian’s Guide to Local Time should be useful.


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