# Looking for a version of Itô's Lemma

Hi Everyone

I have some difficulties deriving the Stochastic Differential Equation for the following problem, any help or reference would be appreciated.

We are given a Brownian Motion $B_t$ and we note $M_t=\sup_{s\le t}B_s$. Moreover we have a smooth real valued function $F(t,x,y)$ (for example a $C^{1,2,1}$) over $\mathbb{R}^+\times \mathbb{R} \times \mathbb{R}^+$ and we are looking for the SDE followed by $F(t,M_t-B_t,M_t)$

I have a hard time trying to express $dF$.

Best regards

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After two seconds of reflection the answer is simple. It suffices to view $F$ as a function of $B_t$ and of the two increasing functions $M_t$ and $t$ and to apply Itô's lemma (Revuz and Yor bok's version for example). Sorry for this post, I first thought some Itô-Tanaka-Meyer formula were involved because of the fact that the processes $(L_t,|W_t|)$ (L is the local time of W at 0) and $(M_t-B_t,M_t)$ have the same law. Regards –  The Bridge Apr 22 '10 at 8:43
The Bridge: The MO bot bumped your question since in the system it remains "unanswered". Since zhoraster did provide an answer, it would be polite to accept it and award him or her the points. Plus, this will be useful for future generations of students who may ask the same question as you did. –  Tom LaGatta Sep 8 '10 at 15:31
Let $F(t,m-b,m)=G(t,m,b)$ (this way it's easier to write). As long as $M$ has bounded variation, we can happily write (I skip arguments, which are $t,M_t,B_t$) $$dG(t,M_t,B_t) = G'_t dt + G'_m dM_t + G'_b dB_t + \frac12 G''_{bb} dt.$$