Stability of stochastic differential equations in manifolds

Abstract

We extend the so-called topology of semimartingales to continuous semimartingales with values in a manifold and with lifetime, and prove that if the manifold is endowed with a connection ° then this topology and the topology of compact convergence in probability coincide on the set of continuous °-martingales. For the topology of manifold-valued semimartingales, we give results on differentiation with respect to a parameter for second order, Stratonovich and Itô stochastic differential equations and identify the equation solved by the derivative processes. In particular, we prove that both Stratonovich and Itô equations differentiate like equations involving smooth paths (for the Itô equation the tangent bundles must be endowed with the complete lifts of the connections on the manifolds). As applications, we prove that differentiation and antidevelopment of C ¹ families of semimartingales commute, and that a semimartingale with values in a tangent bundle is a martingale for the complete lift of a connection if and only if it is the derivative of a family of martingales in the manifold.