Stochastic delay differential equations (SDDE) on a manifold $M$ depend intrinsically on a connection $\nabla$ in this space. The main geometric result in this notes 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 $BM$. We show that every horizontal semimartingale can be represented as a solution of an SDDE.
"Geometry of Stochastic Delay Differential Equations." Electron. Commun. Probab. 10 190 - 195, 2005. https://doi.org/10.1214/ECP.v10-1151