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Suppose that we are given the following processes:

$B=(B_t)_{t\geq0}\ $ a standard Brownian motion starting at zero, $I=I_t=\int_0^t|B_s|^2ds,\ S=S_t=\sup_{0\leq s\leq t} B_s$ for $t\geq0$ and a $C^{1,1,2,1}$ function $F:\mathbb{R}_+^2\times\mathbb{R}\times\mathbb{R}_+\rightarrow\mathbb{R}$ .

Can we apply Ito's formula to $F(t,I_t,B_t,S_t)$ (i.e. do we know that $I$ and $S$ are continuous semimartingales) and in this case how can we tell which is the continuous local martingale part and the bounded variation part?

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both I and $S_t$ are continuous bounded variation processes, in the case of S because it is increasing – mike Dec 6 '12 at 11:58
the process $I$ is also increasing by the way. – TheBridge Dec 7 '12 at 20:08

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