Geometric aspects of stochastic delay differential equations on manifolds

We show some analytical properties of SDDE including a closed formula for a strong solution of $\dot{x}(t) = x(t-1) \circ dB_t$ with an initial adapted process $\varphi_t$ in the delay interval $[-1,0]$. SDDE's on a manifold $M$ depend intrinsically on a connection $\nabla$. The main geometric result in this article concerns the horizontal lift of solutions of SDDE on a manifold $M$ to an SDDE in the frame bundle $BM$, hence the lifted equation should come together with the prolonged horizontal connection $\nabla^H$ on $BM$.