When developing stochastic calculus in a Euclidean space, we start with Brownian motion, move on to martingales, and eventually reach the class of general semi-martingales. In a general differential manifold we must proceed in the opposite direction, because already the definitions of Brownian motion or more general martingales require some additional geometrical structure. In fact, martingales can only be defined intrinsically in manifolds with a connection, and for Brownian motion we need a Riemannian structure.

But even then, the classical semi-martingale decomposition fails, simply because there is no notion of addition in a general manifold S. My aim is to show how we can still define the local characteristics of a general semi-martingale in S, providing the drift and diffusion rates of the process. (A notion of local characteristics appearing in Meyer (1981) seems to be totally unrelated to mine.)