Non-symmetric approximations for manifold-valued semimartingales

Abstract We study general approximations of continuous semimartingales in a manifold. Classically the limits of integrals with respect to the approximated semimartingales yield Stratonovich integrals. Nevertheless several authors have remarked that a skew-symmetric extra-term may appear for specific approximations when the manifold is a vector space. We give the geometric meaning of the skew-symmetric term and an interpretation in term of a “second order non-symmetric intrinsic calculus”. This stochastic non-symmetric calculus is further extended to stochastic differential equations between manifolds. A particular emphasis is pointed on the role of interpolators in approximations.