Unbalanced optimal total variation transport problems and generalized Wasserstein barycenters

Abstract

In this paper, we establish a Kantorovich duality for unbalanced optimal total variation transport problems. As consequences, we recover a version of duality formula for partial optimal transports established by Caffarelli and McCann; and we also get another proof of Kantorovich–Rubinstein theorem for generalized Wasserstein distance $\widetilde {W}_1^{a,b}$ proved before by Piccoli and Rossi. Then we apply our duality formula to study generalized Wasserstein barycenters. We show the existence of these barycenters for measures with compact supports. Finally, we prove the consistency of our barycenters.

Entropy martingale optimal transport and nonlinear pricing–hedging duality

“…In the framework of Liero et al [30] (i.e., with penalisations of the marginals induced by divergence functions) and after the first version of the present work was posted on arXiv, duality results were obtained in the context of weak martingale optimal entropy transport problems by Chung and Trinh [16].…”

“…Remark 2. 16 We now provide conditions ensuring that π(0) < ∞, which by Theorem 2.4 (i) implies that π(c) ∈ R for every c ∈ B 0:T .…”