Empirical martingale projections via the adapted Wasserstein distance
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917020046458880 |
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| author | Blanchet, Jose Wiesel, Johannes Zhang, Erica Zhang, Zhenyuan |
| author_facet | Blanchet, Jose Wiesel, Johannes Zhang, Erica Zhang, Zhenyuan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_12197 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Empirical martingale projections via the adapted Wasserstein distance Blanchet, Jose Wiesel, Johannes Zhang, Erica Zhang, Zhenyuan Probability 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. |
| title | Empirical martingale projections via the adapted Wasserstein distance |
| topic | Probability |
| url | https://arxiv.org/abs/2401.12197 |