Computation of optimal transport and related hedging problems via penalization and neural networks
Eckstein, Stephan; Kupper, Michael;
Subject: Mathematics - Optimization and Control | Statistics - Machine Learning | Quantitative Finance - Mathematical Finance
This paper presents a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a finite dimensional one w... View more
 M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. Software available from tensorflow.org.
 A. Alfonsi, J. Corbetta, and B. Jourdain. Sampling of probability measures in the convex order and approximation of martingale optimal transport problems. 2017.
 M. Arjovsky, S. Chintala, and L. Bottou. Wasserstein GAN. arXiv preprint arXiv:1701.07875, 2017.
 P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk. Mathematical Finance, 9(3):203-228, 1999.
 D. Bartl, P. Cheridito, M. Kupper, and L. Tangpi. Duality for increasing convex functionals with countably many marginal constraints. Banach Journal of Mathematical Analysis, 11(1):72-89, 2017.
 D. Bartl, S. Drapeau, and L. Tangpi. Computational aspects of robust optimized certainty equivalents. arXiv preprint arXiv:1706.10186, 2017.
 D. Bartl, M. Kupper, T. Lux, and A. Papapantoleon. Sharpness of improved Fréchet-Hoeffding bounds: an optimal transport approach. arXiv preprint arXiv:1709.00641, 2017.
 M. Beiglböck, P. Henry-Labordère, and F. Penkner. Model-independent bounds for option prices: A mass transport approach. Finance and Stochastics, 17(3):477-501, 2013.
 A. Ben-Tal and M. Teboulle. An old-new concept of convex risk measures: The optimized certainty equivalent. Mathematical Finance, 17(3):449-476, 2007.
 J.-D. Benamou, G. Carlier, M. Cuturi, L. Nenna, and G. Peyré. Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2):A1111- A1138, 2015.