Quantization and martingale couplings

Abstract

Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly supported, it is smaller than any of its dual quantizations while the dominated original measure is greater than any of its stationary (and therefore any of its optimal) quadratic primal quantization. Moreover, the quantization errors then correspond to martingale couplings between each original probability measure and its quantization. This permits to prove that any martingale coupling between the original probability measures can be approximated by a martingale coupling between their quantizations in Wassertein distance with a rate given by the quantization errors but also in the much finer adapted Wassertein distance. As a consequence, while the stability of (Weak) Martingale Optimal Transport problems with respect to the marginal distributions has only been established in dimension 1 so far, their value function computed numerically for the quantized marginals converges in any dimension to the value for the original probability measures as the numbers of quantization points go to ∞.

Optimal dual quantizers of 1D log-concave distributions: Uniqueness and Lloyd like algorithm

“…which emphasizes the connections with martingale optimal transport explored in other papers [9,10] on the one hand and with Voronoi/primal quantization. Indeed if one replaces the above Delaunay projection by a (Borel) nearest neighbour projection on the grid Γ, denoted Proj vor Γ and if we set if X Γ,vor = Proj vor Γ (X) for some L r -integrable random vector, then e r,N (X…”

“…Its dual behaviour with respect to convex order provides an informal way to provide lower and upper-bounds in various stochastic control problems. More recently, with the development of martingale optimal transport problems in finance, both Voronoi and Delaunay quantization methods have been shown as a systematic tool to design time discretization schemes that preserve convex order (see [10]) and more generally to solve numerically discrete time martingale optimal transport problems (see [9]) which turns out to be a quite challenging problem (see [1], [3], [4], [7], [8]).…”

Instability of Martingale optimal transport in dimension d $\ge$ 2

“…Firstly, they enable the strategy of approximating the problem by a discretized problem or by any other that can rapidly be solved computationally (cf. [1,12]). Secondly, any application to noisy data would require stability for the results to be meaningful.…”