A Geometric Itô Formula

Resumo: Let $M$ and $N$ be manifolds equipped with connections $\Gamma^M$ and $\Gamma^N$ respectively and $F :M\rightarrow N$ be a smooth map. Let $X$ be an $M$-valued semimartingale and $\Theta$ be an 1-form on $N$. We prove the following It\^{o} formula in the context of Schwartz (second order) geometry,\[\int \Theta~ d^{\Gamma^N}F(X)=\int F^*\Theta~d^{\Gamma^M}X+\frac{1}{2}\int \beta_F^*\Theta(dX,dX)\]where the integrals are in the It\^{o} sense, and $\beta_F$ is the fundamental form of $F$. Some applications are discussed.Mathematics Subject Classifications (2000): Primary: 60H05 ; Secondary:58J65.Keywords: Stochastic Calculus. Schwartz Geometry. It\^{o}formula.