Let M be a smooth manifold endowed with a symmetric connection del. There are two important ways of lift the connection del of M to the frame bundle BM, the canonical lift del(c) and the horizontal lift del(h). The aim of this work is determine the del(c)-martingales and the del(h)-martingales on BM. Our results allow to establish new characterizations of harmonic maps from Riemannian manifolds to frame bundles.2816169