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Sampling of probability measures in the convex order and approximation of Martingale Optimal Transport problems

  • Alfonsi, Aurélien
  • Corbetta, Jacopo
  • Jourdain, Benjamin
Publication Date
Sep 18, 2017
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Motivated by the approximation of Martingale Optimal Transport problems, westudy sampling methods preserving the convex order for two probability measures$\mu$ and $\nu$ on $\mathbb{R}^d$, with $\nu$ dominating $\mu$. When$(X_i)_{1\le i\le I}$ (resp. $(Y_j)_{1\le j\le J}$) are i.i.d. according $\mu$(resp. $\nu$), the empirical measures $\mu_I$ and $\nu_J$ are not in the convexorder. We investigate modifications of $\mu_I$ (resp. $\nu_J$) smaller than$\nu_J$ (resp. greater than $\mu_I$) in the convex order and weakly convergingto $\mu$ (resp. $\nu$) as $I,J\to\infty$. In dimension 1, according to Kertzand R\"osler (1992), the set of probability measures with a finite first ordermoment is a lattice for the increasing and the decreasing convex orders. Fromthis result, we can define $\mu\vee\nu$ (resp. $\mu\wedge\nu$) that is greaterthan $\mu$ (resp. smaller than $\nu$) in the convex order. We give efficientalgorithms permitting to compute $\mu\vee\nu$ and $\mu\wedge\nu$ when $\mu$ and$\nu$ are convex combinations of Dirac masses. In general dimension, when $\mu$and $\nu$ have finite moments of order $\rho\ge 1$, we define the projection$\mu\curlywedge_\rho \nu$ (resp. $\mu\curlyvee_\rho\nu$) 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$\rho=2$, $\mu_I\curlywedge_2 \nu_J$ can be computed efficiently by solving aquadratic optimization problem with linear constraints. It turns out that, indimension 1, the projections do not depend on $\rho$ and their quantilefunctions are explicit, which leads to efficient algorithms for convexcombinations of Dirac masses. Last, we illustrate by numerical experiments theresulting sampling methods that preserve the convex order and their applicationto approximate Martingale Optimal Transport problems.

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