Approximating Ito integrals of differential forms and geodesic deviation

Summary Suppose X is a semimartingale on a differential manifold M with a linear connection . The main purpose of this paper is to show that the Ito integral (with respect to ) of a differential form along the path of X is the limit in probability of certain Riemann sums, constructed in a natural way using the exponential map in differential geometry. For this, we study the deviation between the stochastic development of X in the tangent space at some point, and the image of X under the inverse of the exponential map at the point.