Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations

Communications in Mathematics and Statistics - Tập 5 Số 4 - Trang 349-380 - 2017
E Weinan1, Jiequn Han2, Arnulf Jentzen3
1Beijing Institute of Big Data Research, Beijing, China
2Princeton University, Princeton, NJ, USA
3ETH Zurich, Zurich, Switzerland

Tóm tắt

Từ khóa


Tài liệu tham khảo

Bellman, R.: Dynamic programming. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ. Reprint of the 1957 edition, with a new introduction by Stuart Dreyfus (2010)

Bender, C., Denk, R.: A forward scheme for backward SDEs. Stoch. Process. Appl. 117(12), 1793–1812 (2007)

Bender, C., Schweizer, N., Zhuo, J.: A primal-dual algorithm for BSDEs. arXiv:1310.3694 (2014)

Bergman, Y.Z.: Option pricing with differential interest rates. Rev. Financ. Stud. 8(2), 475–500 (1995)

Briand, P., Labart, C.: Simulation of BSDEs by Wiener chaos expansion. Ann. Appl. Probab. 24(3), 1129–1171 (2014)

Chassagneux, J.-F.: Linear multistep schemes for BSDEs. SIAM J. Numer. Anal. 52(6), 2815–2836 (2014)

Chassagneux, J.-F., Richou, A.: Numerical simulation of quadratic BSDEs. Ann. Appl. Probab. 26(1), 262–304 (2016)

Crisan, D., Manolarakis, K.: Solving backward stochastic differential equations using the cubature method: application to nonlinear pricing. SIAM J. Financ. Math. 3(1), 534–571 (2012)

Darbon, J., Osher, S.: Algorithms for overcoming the curse of dimensionality for certain Hamilton–Jacobi equations arising in control theory and elsewhere. Res. Math. Sci. 3(19), 26 (2016)

Debnath, L.: Nonlinear Partial Differential Equations for Scientists and Engineers, 3rd edn. Birkhäuser/Springer, New York (2012)

E, W., Han, J., Jentzen, A.: Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. arXiv:1706.04702 (2017)

E, W., Hutzenthaler, M., Jentzen, A., Kruse, T.: Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. arXiv:1607.03295 (2017)

E, W., Hutzenthaler, M., Jentzen, A., Kruse, T.: On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. arXiv:1708.03223 (2017)

Gobet, E., Lemor, J.-P., Warin, X.: A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann. Appl. Probab. 15(3), 2172–2202 (2005)

Gobet, E., Turkedjiev, P.: Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. Math. Comput. 85(299), 1359–1391 (2016)

Gobet, E., Turkedjiev, P.: Adaptive importance sampling in least-squares Monte Carlo algorithms for backward stochastic differential equations. Stoch. Process. Appl. 127(4), 1171–1203 (2017)

Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press 2016. http://www.deeplearningbook.org

Han, J., E, W.: Deep learning approximation for stochastic control problems. arXiv:1611.07422 (2016)

Han, J., Jentzen, A., E, W.: Overcoming the curse of dimensionality: solving high-dimensional partial differential equations using deep learning. arXiv:1707.02568 (2017)

Henry-Labordère, P.: Counterparty risk valuation: a marked branching diffusion approach. arXiv:1203.2369 (2012)

Henry-Labordère, P., Oudjane, N., Tan, X., Touzi, N., Warin, X.: Branching diffusion representation of semilinear PDEs and Monte Carlo approximation. arXiv:1603.01727 (2016)

Henry-Labordère, P., Tan, X., Touzi, N.: A numerical algorithm for a class of BSDEs via the branching process. Stoch. Process. Appl. 124(2), 1112–1140 (2014)

Hinton, G.E., Deng, L., Yu, D., Dahl, G., Mohamed, A., Jaitly, N., Senior, A., Vanhoucke, V., Nguyen, P., Sainath, T., Kingsbury, B.: Deep neural networks for acoustic modeling in speech recognition. Sig. Process. Mag. 29, 82–97 (2012)

Ioffe, S., Szegedy, C.: Batch normalization: accelerating deep network training by reducing internal covariate shift. In: Proceedings of the International Conference on Machine Learning (ICML) (2015)

Kingma, D., Ba, J.: Adam: a method for stochastic optimization. In: Proceedings of the International Conference on Learning Representations (ICLR) (2015)

Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. Adv. Neural Inf. Process. Syst. 25, 1097–1105 (2012)

LeCun, Y., Bengio, Y., Hinton, G.E.: Deep learning. Nature 521, 436–444 (2015)

Pardoux, É., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)

Pardoux, É., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991), vol. 176 of Lecture Notes in Control and Inform. Sci. Springer, Berlin, pp. 200–217 (1992)

Pardoux, É., Tang, S.: Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Relat. Fields 114(2), 123–150 (1999)

Peng, S.: Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep. 37(1–2), 61–74 (1991)