Abstract. In this paper we prove that there is a bijective correspondence between connections of $H^2M$, the principal bundle of the second order frames of $M$, and stochastic parallel transport in the tangent space of $M$. We construct in a direct geometric way a prolongation of connections without torsion of $M$ to connections of $H^2M$. We interpret such prolongation in terms of stochastic calculus.
AMSclassification. 58G32, 53B15
Keywords. Second order geometry, stochastic calculus, connections, parallel transport