Title: Linear Constraints in Optimal Transport
Description: This thesis studies the problem of optimal mass transportation with linear constraints -- supermartingale and martingale transport in discrete and continuous time. Appropriate versions of corresponding dual problems are introduced and shown to satisfy fundamental properties: weak duality, absence of a duality gap, and the existence of a dual optimal element. We show how the existence of a dual optimizer implies that primal optimizers can be characterized geometrically through their support -- an infinite dimensional analogue of complementary slackness. In discrete time martingale and supermartingale transport problems, we utilize this result to establish the existence of canonical transport plans, that is joint optimizers for large families of reward functions. To this end, we show that the optimal support coincides for these families. We additionally characterize these transport plans through order-theoretic minimality properties, with respect to second stochastic order and convex order, respectively, in the supermartingale and the martingale case. This characterization further shows that the canonical transport plan is unique.