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I have seen many Ito formulae giving dynamics for $f(X_t)$ where $f:M\to \mathbb{R}$ is a smooth function from a manifold $M$ and $X_t$ is a (continous) manifold-valued semi-martingale.

My question is, is there an Ito formula describing the dynamics of $ g(X_t), $ where $g:M\to N$ is a smooth map between manifolds? What about the case where $N=\mathbb{R}^d$?

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  • $\begingroup$ You can extract information about $g(X_t)$ by composing it with a function $h:N\to\mathbb{R}$ – would that be enough? $\endgroup$ – Mateusz Kwaśnicki Sep 6 '17 at 18:52

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