  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 $\infty$.
