Noncausal calculus approach to the comparison theorem for solutions of SDEs

We are concerned with a generalization of the comparison theorem for solutions of two SDEs that are driven by the same Brownian motion. The purpose of the present note is two folded: first we aim to show that in the framework of the noncausal stochastic calculus (cf. Ogawa,S.,“Noncausal Stochastic Calculus”, 2017 Springer) the subject can be treated in a much simpler way, second we show that this approach permits us to establish similar comparison theorems for more general case where the diffusion coefficients of SDEs are different. Moreover we show as one of the merits of our noncausal approach that we can extend our main results to a case of genuine noncausal SDEs, that is the SDE with noncausal initial data.