  We investigate duality and existence of dual optimizers for several adapted
optimal transport problems under minimal assumptions. This includes the causal
and bicausal transport, the barycenter problem, and a general multimarginal
problem incorporating causality constraints. Moreover, we discuss applications
of our results in robust finance. We consider a non-dominated model of several
financial markets where stocks are traded dynamically, but the joint stock
dynamics are unknown. We show that a no-arbitrage assumption in a quasi-sure
sense naturally leads to sets of multicausal couplings. Consequently, computing
the robust superhedging price is equivalent to solving an adapted transport
problem, and finding a superhedging strategy means solving the corresponding
dual.
