Section: New Results

Approximation of Martingale Optimal Transport problems

With J. Corbetta, A. Alfonsi and B. Jourdain study sampling methods preserving the convex order for two probability measures $\mu$ and $\nu$ on ${𝐑}^d$, with $\nu$ dominating $\mu$. When ${\left(X_i\right)}_{1\le i\le I}$ (resp. ${\left(Y_j\right)}_{1\le j\le J}$) are independent and identically distributed according $\mu$ (resp. $\nu$), in general ${\mu }_I=\frac{1}{I}{\sum }_{i=1}^I{\delta }_{X_i}$ and ${\nu }_J=\frac{1}{J}{\sum }_{j=1}^J{\delta }_{Y_j}$ are not rankable for the convex order. They investigate modifications of ${\mu }_I$ (resp. ${\nu }_J$) smaller than ${\nu }_J$ (resp. greater than ${\mu }_I$) in the convex order and weakly converging to $\mu$ (resp. $\nu$) as $I,J\to \infty$. They first consider the one dimensional case $d=1$, where, according to Kertz and Rösler, the set of probability measures with a finite first order moment is a lattice for the increasing and the decreasing convex orders. Given $\mu$ and $\nu$ in this set, they define $\mu \vee \nu$ (resp. $\mu \wedge \nu )$ as the supremum (resp. infimum) of $\mu$ and $\nu$ for the decreasing convex order when ${\int }_{𝐑}x\mu \left(dx\right)\le {\int }_{𝐑}x\nu \left(dx\right)$ and for the increasing convex order otherwise. This way, $\mu \vee \nu$ (resp. $\mu \wedge \nu$) is greater than $\mu$ (resp. smaller than $\nu$) in the convex order. They give efficient algorithms permitting to compute $\mu \vee \nu$ and $\mu \wedge \nu$ (and therefore ${\mu }_I\vee {\nu }_J$ and ${\mu }_I\wedge {\nu }_J$) when $\mu$ and $\nu$ are convex combinations of Dirac masses. In general dimension, when $\mu$ and $\nu$ have finite moments of order $\rho \ge 1$, they define the projection $\mu {⋏}_{\rho }\nu$ (resp. $\mu {⋎}_{\rho }\nu$) of $\mu$ (resp. $\nu$) on the set of probability measures dominated by $\nu$ (resp. larger than $\mu$) in the convex order for the Wasserstein distance with index $\rho$. When $\rho =2$, ${\mu }_I{⋏}_2{\nu }_J$ can be computed efficiently by solving a quadratic optimization problem with linear constraints. It turns out that, in dimension $d=1$, the projections do not depend on $\rho$ and their quantile functions are explicit in terms of those of $\mu$ and $\nu$, which leads to efficient algorithms for convex combinations of Dirac masses. Last, they illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate Martingale Optimal Transport problems.

With V. Ehrlacher, D. Lombardi and R. Coyaud, A. Alfonsi has started to develop and analyze numerical methods to approximate the optimal transport between two probability measures.