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



Looking for a version of Itô's Lemma
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 $t,M\_t,B\_t$)
$$
dG(t,M\_t,B\_t) = G'\_t dt + G'\_m dM\_t + G'\_b dB\_t + \frac12 G''\_{bb} dt. 
$$


