Section: New Results

Optimal transport and applications

Martingale Optimal Transport.

B. Jourdain and W. Margheriti exhibit a new family of martingale couplings between two one-dimensional probability measures $\mu$ and $\nu$ in the convex order. This family is parametrised by two dimensional probability measures on the unit square with respective marginal densities proportional to the positive and negative parts of the difference between the quantile functions of $\mu$ and $\nu$. It contains the inverse transform martingale coupling which is explicit in terms of the cumulative distribution functions of these marginal densities. The integral of $|x-y|$ with respect to each of these couplings is smaller than twice the $W^1$ distance between $\mu$ and $\nu$. When the comonotoneous coupling between $\mu$ and $\nu$ is given by a map $T$, the elements of the family minimize ${\int }_{𝐑}|y-T\left(x\right)|M(dx,dy)$ among all martingale couplings $M$ between $\mu$ and $\nu$. When $\mu$ and $\nu$ are in the decreasing (resp. increasing) convex order, the construction can be generalized to exhibit super (resp. sub) martingale couplings.

A. Alfonsi and B. Jourdain show that any optimal coupling for the quadratic Wasserstein distance $W_2^2(\mu ,\nu )$ between two probability measures $\mu$ and $\nu$ with finite second order moments on ${𝐑}^d$ is the composition of a martingale coupling with an optimal transport map $𝒯$. They check the existence of optimal couplings in which this map gives the unique optimal coupling between $\mu$ and $𝒯\#\mu$. Next, they prove that $\sigma \mapsto W_2^2(\sigma ,\nu )$ is differentiable at $\mu$ in both Lions and the geometric senses iff there is a unique optimal coupling between $\mu$ and $\nu$ and this coupling is given by a map.

Numerical methods for optimal transport.

Optimal transport problems have got a recent attention in many different fields including physics, quantum chemistry and finance, where Martingale Optimal Transport problems allow to quantify the model risk. In practice, few numerical methods exist to approximate the optimal coupling measure and/or the optimal transport. In particular, to deal with large dimensions or with the optimal transport problems with many marginal laws, a natural direction is to develop Monte-Carlo methods.

A. Alfonsi, V. Ehrlacher (CERMICS, Inria Project-team MATHERIALS), D. Lombardi (Inria Project-team Reo) and R. Coyaud (PhD student of A. Alfonsi) are working on numerical approximations of the optimal transport between two (or more) probability measures.