# Ito formula between manifolds

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 $N=\mathbb{R}^d$?

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