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Has anyone ever applied the Ito formula on $|X^+|^2$ for $X^+ = \max(X,0)$ with

$X(t) = X(0) + M(t) + V(t)$, where $M(t)$ is a local martingale and $V(t)$ is bounded variation process. I found it in a lemma for $M$ continuous local martingale but the proof was not provided either.

Thanks for any hint.

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  • $\begingroup$ Yes, it's called Tanaka's formula. $\endgroup$
    – Martin Hairer
    Commented Jan 31, 2014 at 18:38
  • $\begingroup$ Thanks, yes I found the Tanaka formula so I was able to do it. $\endgroup$
    – Max
    Commented Jan 31, 2014 at 20:27

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Use the fact that $max(x,y)=\frac{x+y+|x-y|}{2}$ and Tanaka's formula.

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