Stochastic covariant calculus of order two

Abstract

We study continuous semimartingales in a vector fibre bundle E over a differentiable manifold M. Following Meyer and Schwartz's principle, we show that stochastic covariant integration on a manifold involves second order differential operators, according to the Itô formula integrating not only the differential but the 2-jet of a function, and using a notion of connection of order 2. A fundamental example of such a connection is given by the 2-jet of the parallel transport along geodesics on M. To cite this article: L. Maillard-Teyssier, C. R. Acad. Sci. Paris, Ser. I 341 (2005).