Before revision:  More precisely, we first revisit the optimal transport plan introduced in BeiglJuil, following the construction of BrenierMartingale, as well as the one in HobsonKlimmek2013 in the case of positive martingales and a single maximizer for the difference between the c.d.f.'s of the two marginals.
After revision:  More precisely, we first revisit the optimal transport plan introduced in BeiglJuil, following the construction of BrenierMartingale, as well as the one in HobsonKlimmek2013 , to the case of positive martingales and a single maximizer for the difference between the c.d.f.'s of the two marginals.