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Empirical martingale projections via the adapted Wasserstein distance

José Blanchet et al.

Annals of Applied Probability2026https://doi.org/10.1214/25-aap2239article
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Abstract

Given a collection of multidimensional pairs {(Xi,Yi)}1≤i≤n, we study the problem of projecting the associated suitably smoothed empirical measure onto the space of martingale couplings (i.e., distributions satisfying E[Y|X]=X) using the adapted Wasserstein distance. We call the resulting distance the smoothed empirical martingale projection distance (SE-MPD), for which we obtain an explicit characterization. We also show that the space of martingale couplings remains invariant under the smoothing operation. We study the asymptotic limit of the SE-MPD, which converges at a parametric rate as the sample size increases, if the pairs are either i.i.d. or satisfy appropriate mixing assumptions. Additional finite-sample results are also investigated. Using these results, we introduce a novel consistent martingale coupling hypothesis test, which we apply to test the existence of arbitrage opportunities in recently introduced neural network-based generative models for asset pricing calibration.

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https://doi.org/https://doi.org/10.1214/25-aap2239

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@article{josé2026,
  title        = {{Empirical martingale projections via the adapted Wasserstein distance}},
  author       = {José Blanchet et al.},
  journal      = {Annals of Applied Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/25-aap2239},
}

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Empirical martingale projections via the adapted Wasserstein distance

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