Empirical martingale projections via the adapted Wasserstein distance

TL;DR

The paper develops the empirical martingale projection distance (MPD) by projecting onto the set of martingale couplings with the adapted Wasserstein distance, and introduces a smoothed version SE-MPD to address empirical non-martingale behavior. It derives a closed-form MPD and shows SE-MPD retains the martingale structure under smoothing by martingality-preserving kernels, enabling consistent estimation. The authors establish parametric-rate $\sqrt{n}$-type limits and Gaussian functionals for i.i.d. data and extend to $\alpha$-mixing sequences, then build a consistent martingale-pair hypothesis test with practical implementation guidance and bandwidth considerations. They validate the approach through finite-sample analyses, finite-sample bounds, and power studies, and demonstrate a concrete application to testing arbitrage in neural SDE-based European option calibration, highlighting the method’s practical impact for finance and reinforcement learning contexts.

Abstract

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 empirical measure onto the space of martingale couplings (i.e. distributions satisfying $\mathbb{E}[Y|X]=X$) 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.

Figures (8)

• Figure 1: Histogram (with 1000 samples) and approximating density of the asymptotic distribution $\int_{\mathbb{R}^d}\left|G_x\right|_2\mathrm{d} x$ of the rescaled MPD for $d=1$, where the martingale coupling $(X,Y)$ is given by $X \sim \mathcal{N}(0,I_d), \ Z \sim \mathcal{N}(0,I_d),\ Y = X+Z$, and $X,Z$ independent.
• Figure 2: Histogram (with 1000 samples) and approximating density of the asymptotic distribution $\int_{\mathbb{R}^d}\left|G_x\right|_2\mathrm{d} x$ of the rescaled MPD for $d=2$, where the martingale coupling $(X,Y)$ is given by $X \sim \mathcal{N}(0,I_d), \ Z \sim \mathcal{N}(0,I_d),\ Y = X+Z$, and $X,Z$ independent.
• Figure 3: Probability densities of smoothed empirical measures with various $\sigma$. A large $\sigma$ leads to a smooth density, whereas spikes emerge for small $\sigma$.
• Figure 4: Effects of $\sigma$ in reducing projection error
• Figure 5: Effects of $\sigma$ on the power of test. Plot (a) shows the rejection rate (the proportion of tests that reject the null) generally increases in $\sigma$; plot (b) shows that the critical value with significance level $\alpha=0.05$ generally decreases in $\sigma$. The plots show that a smaller $\sigma$ induces more Type I error and a larger $\sigma$ induces more Type II error.

Theorems & Definitions (78)

• Definition 1: see e.g., Lemma 2.2 of bartl2021wasserstein
• Definition 2
• Definition 3
• Theorem 1: Computing the martingale projection distance
• Remark 1
• Definition 4
• Definition 5
• Proposition 1
• Example 1
• Definition 6