2017/03/30 by Nutz, Marcel, Stebegg, Florian, Tan, Xiaowei · 2 citations
#49N05 #60G42 #FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Optimization and Control (math.OC) #Probability (math.PR)
paper · doi:10.48550/arxiv.1703.10588
Consider a multiperiod optimal transport problem where distributions μ0,…,μn are prescribed and a transport corresponds to a scalar martingale X with marginals Xt∼μt. We introduce particular couplings called left-monotone transports; they are characterized equivalently by a no-crossing property of their support, as simultaneous optimizers for a class of bivariate transport cost functions with a Spence--Mirrlees property, and by an order-theoretic minimality property. Left-monotone transports are unique if μ0 is atomless, but not in general. In the one-period case n=1, these transports reduce to the Left-Curtain coupling of Beiglböck and Juillet. In the multiperiod case, the bivariate marginals for dates (0,t) are of Left-Curtain type, if and only if μ0,…,μn have a specific order property. The general analysis of the transport problem also gives rise to a strong duality result and a description of its polar sets. Finally, we study a variant where the intermediate marginals μ1,…,μn-1 are not prescribed.