Before revision:  we first specialize the optimal transport plan introduced in \mbox{%DIFAUXCMD BeiglJuil0pt%DIFAUXCMD , following the construction of \mbox{%DIFAUXCMD BrenierMartingale}0pt%DIFAUXCMD , }  as well as the one 0pt%DIFAUXCMD and further studied in \mbox{%DIFAUXCMD BrenierMartingale}\hspace{0pt}%DIFAUXCMD . We show that, } in \mbox{%DIFAUXCMD \cite{HobsonKlimmek2013}\hspace{0pt}%DIFAUXCMD , to } the case of positive martingales and a single maximizer for the difference between the c.d.f.'
After revision:  we first specialize the optimal transport plan introduced in \mbox{%DIFAUXCMD BeiglJuil0pt%DIFAUXCMD , following the construction of \mbox{%DIFAUXCMD BrenierMartingale}0pt%DIFAUXCMD , } constructed in \mbox{%DIFAUXCMD HobsonKlimmek2013 as well as the one 0pt%DIFAUXCMD and further studied in \mbox{%DIFAUXCMD BrenierMartingale}\hspace{0pt}%DIFAUXCMD . We show that, } in \mbox{%DIFAUXCMD \cite{HobsonKlimmek2013}\hspace{0pt}%DIFAUXCMD , to } the case of positive martingales and a single maximizer for the difference between the c.d.f.'