# Computation of Optimal Transport and Related Hedging Problems via Penalization and Neural Networks

- Published: 23 Feb 2018 Journal: Applied Mathematics & Optimization (issn: 0095-4616, eissn: 1432-0606, Copyright policy)
- Publisher: Springer Science and Business Media LLC

[1] 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.

[2] A. Alfonsi, J. Corbetta, and B. Jourdain. Sampling of probability measures in the convex order and approximation of martingale optimal transport problems. 2017. [OpenAIRE]

[3] M. Arjovsky, S. Chintala, and L. Bottou. Wasserstein GAN. arXiv preprint arXiv:1701.07875, 2017.

[4] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk. Mathematical Finance, 9(3):203-228, 1999. [OpenAIRE]

[5] 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.

[6] D. Bartl, S. Drapeau, and L. Tangpi. Computational aspects of robust optimized certainty equivalents. arXiv preprint arXiv:1706.10186, 2017.

[7] 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. [OpenAIRE]

[8] 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.

[9] 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.

[10] 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.

[11] C. Bernard, X. Jiang, and R. Wang. Risk aggregation with dependence uncertainty. Insurance: Mathematics and Economics, 54:93-108, 2014.

[12] C. Bernard, L. Rüschendorf, S. Vanduffel, and J. Yao. How robust is the value-at-risk of credit risk portfolios? The European Journal of Finance, 23(6):507-534, 2017.

[13] H. Bühler, L. Gonon, J. Teichmann, and B. Wood. arXiv:1802.03042, 2018.

[14] G. Carlier, V. Duval, G. Peyré, and B. Schmitzer. Convergence of entropic schemes for optimal transport and gradient flows. SIAM Journal on Mathematical Analysis, 49(2):1385-1418, 2017.

[15] P. Cheridito, M. Kupper, and L. Tangpi. Representation of increasing convex functionals with countably additive measures. arXiv preprint arXiv:1502.05763, 2015. [OpenAIRE]

###### Related research

[1] 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.

[2] A. Alfonsi, J. Corbetta, and B. Jourdain. Sampling of probability measures in the convex order and approximation of martingale optimal transport problems. 2017. [OpenAIRE]

[3] M. Arjovsky, S. Chintala, and L. Bottou. Wasserstein GAN. arXiv preprint arXiv:1701.07875, 2017.

[4] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk. Mathematical Finance, 9(3):203-228, 1999. [OpenAIRE]

[5] 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.

[6] D. Bartl, S. Drapeau, and L. Tangpi. Computational aspects of robust optimized certainty equivalents. arXiv preprint arXiv:1706.10186, 2017.

[7] 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. [OpenAIRE]

[8] 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.

[9] 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.

[10] 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.

[11] C. Bernard, X. Jiang, and R. Wang. Risk aggregation with dependence uncertainty. Insurance: Mathematics and Economics, 54:93-108, 2014.

[12] C. Bernard, L. Rüschendorf, S. Vanduffel, and J. Yao. How robust is the value-at-risk of credit risk portfolios? The European Journal of Finance, 23(6):507-534, 2017.

[13] H. Bühler, L. Gonon, J. Teichmann, and B. Wood. arXiv:1802.03042, 2018.

[14] G. Carlier, V. Duval, G. Peyré, and B. Schmitzer. Convergence of entropic schemes for optimal transport and gradient flows. SIAM Journal on Mathematical Analysis, 49(2):1385-1418, 2017.

[15] P. Cheridito, M. Kupper, and L. Tangpi. Representation of increasing convex functionals with countably additive measures. arXiv preprint arXiv:1502.05763, 2015. [OpenAIRE]