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Given a collection of multidimensional pairs $\{(X_i,Y_i):1 \leq i\leq n\}$,we study the problem of projecting the associated suitably smoothed empiricalmeasure onto the space of martingale couplings (i.e. distributions satisfying$\mathbb{E}[Y|X]=X$) using the adapted Wasserstein distance. We call theresulting distance the smoothed empirical martingale projection distance(SE-MPD), for which we obtain an explicit characterization. We also show thatthe space of martingale couplings remains invariant under the smoothingoperation. We study the asymptotic limit of the SE-MPD, which converges at aparametric rate as the sample size increases if the pairs are either i.i.d. orsatisfy appropriate mixing assumptions. Additional finite-sample results arealso investigated. Using these results, we introduce a novel consistentmartingale coupling hypothesis test, which we apply to test the existence ofarbitrage opportunities in recently introduced neural network-based generativemodels for asset pricing calibration.

Empirical martingale projections via the adapted Wasserstein distance