Sampling of Probability Measures in the Convex Order and Approximation of Martingale Optimal Transport Problems

Weak monotone rearrangement on the line

“…A particular consequence of Theorem 1.2 is that the optimizer of (1.2) does not depend on the choice of the convex function θ. We find this fact non-trivial as well as remarkable and highlight that it is not new: different independent proofs were given by Gozlan et al [15], Alfonsi, Corbetta, Jourdain [2] and Shu [20]. Alfonsi, Corbetta and Jourdain [3,Example 2.4] notice that this does not pertain in higher dimensions.…”

“…Theorem 1.3 represents a necessary and sufficient condition for the optimality of the measure T (µ * ) in (1.2). We note that the 'necessary' part was first obtained (using somewhat different phrasing) by Alfonsi, Corbetta, and Jourdain [2,Proposition 3.12]. We also refer the reader to the semi-explicit representation of T and T (µ * ) given in [2].…”

Existence, duality, and cyclical monotonicity for weak transport costs

“…By contradiction, suppose there exists a coupling χ with the same marginals asπ such that C(x, χ x )μ(dx) < C(x,π x )μ(dx). 1 In a preliminary version of this article the restriction property Proposition 4.2 was used to derive Theorem 1.4 from the compact version given by Gozlan and Juillet [23]. Following the insightful suggestion of the anonymous referee, we now give a more self contained argument that does not require Proposition 4.2 / [23].…”

Computation of Optimal Transport and Related Hedging Problems via Penalization and Neural Networks

“…Here, we consider a simple example with d = 2, where an analytical solution is known. This example is taken from [2]. Let X := [−1, 1] × [−2, 2], θ := U(X ) and set…”

“…More precisely: The choice of reference measure θ always has the implicit objective to lead to narrow bounds in Equation (2.5) in Theorem 2.2. In this example, if one presumes that an optimal measure ν * ∈ Q has mass near the perfectly correlated diagonal, it makes sense to choose a reference measure which puts mass in this region, as does θ (2) . Figure 4: Portfolio optimization under dependence uncertainty: As reference measure we take either the product measure or a positively correlated measure.…”