On the stability of the martingale optimal transport problem.

Summary This thesis is motivated by the study of the stability of the martingale optimal transport problem, and is naturally organized into two parts. In the first part, we exhibit a new family of martingale couplings between two one-dimensional probability measures μ and ν comparable in convex order. In particular, this family contains the inverse transform martingale coupling, which is explicit in terms of quantile functions of the marginals. The integral M_1(μ,ν) of |x-y| against each of these couplings is increased by twice the Wasserstein distance W_1(μ,ν) between μ and ν. We show a similar inequality when |x-y| and W_1 are respectively replaced by |x-y|^ρ and the product of W_ρ by the centered moment of order ρ of the second marginal raised to the exponent ρ-1, for any ρ∈[1,+∞[. We then study the generalization of this new stability inequality to the higher dimension. Finally, we establish a strong connection between our new family of martingale couplings and the projection of a coupling between two comparable given marginals in convex order onto the set of martingale couplings between these same marginals. This last projection is taken with respect to the adapted Wasserstein distance, which majors the usual Wasserstein distance and thus induces a finer topology better suited for financial modeling, since it takes into account the temporal structure of martingales. In the second part, we prove that any martingale coupling whose marginals are approximated by comparable probability measures in convex order can itself be approximated by martingale couplings in the sense of the adapted Wasserstein distance. We then discuss various applications of this result. In particular, we strengthen a stability result for the weak optimal transport problem and establish a stability result for the weak optimal martingale transport problem.