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Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. (English) Zbl 1442.91116

Summary: High-dimensional partial differential equations (PDEs) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work, we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (1) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (2) a merged formulation of the PDE and the 2BSDE problem, (3) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (4) a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black-Scholes-Barenblatt equation, a 100-dimensional Hamilton-Jacobi-Bellman equation, and a nonlinear expectation of a 100-dimensional \(G\)-Brownian motion.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
91G20 Derivative securities (option pricing, hedging, etc.)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
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[1] Amadori, A.L.: Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach. Differ. Integral Equ. 16(7), 787-811 (2003) · Zbl 1052.35083
[2] Avellaneda, M., Arnon, L., Parás, A.: Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2, 73-88 (1995) · Zbl 1466.91323
[3] Bally, V., Pagès, G.: A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems. Bernoulli 9(6), 1003-1049 (2003) · Zbl 1042.60021
[4] Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory 39(3), 930-945 (1993) · Zbl 0818.68126
[5] Bayraktar, E., Young, V.: Pricing options in incomplete equity markets via the instantaneous Sharpe ratio. Ann. Finance 4(4), 399-429 (2008) · Zbl 1233.91256
[6] Bayraktar, E., Milevsky, M.A., Promislow, S.D., Young, V.R.: Valuation of mortality risk via the instantaneous Sharpe ratio: applications to life annuities. J. Econ. Dyn. Control 33(3), 676-691 (2009) · Zbl 1170.91406
[7] Bender, C., Denk, R.: A forward scheme for backward SDEs. Stoch. Process. Appl. 117(12), 1793-1812 (2007) · Zbl 1131.60054
[8] Bender, C., Schweizer, N., Zhuo, J.: A primal-dual algorithm for BSDEs. Math. Finance 27(3), 866-901 (2017) · Zbl 1423.91008
[9] Bengio, Y.: Learning deep architectures for AI. Found. Trends Mach. Learn. 2(1), 1-127 (2009) · Zbl 1192.68503
[10] Bergman, Y.Z.: Option pricing with differential interest rates. Rev. Financ. Stud. 8(2), 475-500 (1995)
[11] Bismut, J.-M.: Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384-404 (1973) · Zbl 0276.93060
[12] Bouchard, B., Elie, R., Touzi, N.: Discrete-time approximation of BSDEs and probabilistic schemes for fully nonlinear PDEs. In: Advanced financial modelling, vol. 8 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter, Berlin, pp. 91-124 (2009) · Zbl 1179.65004
[13] Bouchard, B.: Lecture notes on BSDEs: main existence and stability results. Ph.D. thesis, CEREMADE-Centre de Recherches en MAthématiques de la DEcision (2015)
[14] Bouchard, B., Touzi, N.: Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stoch. Process. Appl. 111(2), 175-206 (2004) · Zbl 1071.60059
[15] Briand, P., Labart, C.: Simulation of BSDEs by Wiener chaos expansion. Ann. Appl. Probab. 24(3), 1129-1171 (2014) · Zbl 1311.60077
[16] Cai, Z.: Approximating quantum many-body wave-functions using artificial neural networks (2017). arXiv:1704.05148
[17] Carleo, G., Troyer, M.: Solving the quantum many-body problem with artificial neural networks. Science 355(6325), 602-606 (2017) · Zbl 1404.81313
[18] Chang, D., Liu, H., Xiong, J.: A branching particle system approximation for a class of FBSDEs. Probab. Uncertain. Quant. Risk 1, 9 (2016). 34 · Zbl 1440.60062
[19] Chassagneux, J.-F.: Linear multistep schemes for BSDEs. SIAM J. Numer. Anal. 52(6), 2815-2836 (2014) · Zbl 1326.65017
[20] Chassagneux, J.-F., Crisan, D.: Runge-Kutta schemes for backward stochastic differential equations. Ann. Appl. Probab. 24(2), 679-720 (2014) · Zbl 1303.60045
[21] Chassagneux, J.-F., Richou, A.: Numerical stability analysis of the Euler scheme for BSDEs. SIAM J. Numer. Anal. 53(2), 1172-1193 (2015) · Zbl 1311.93088
[22] Chassagneux, J.-F., Richou, A.: Numerical simulation of quadratic BSDEs. Ann. Appl. Probab. 26(1), 262-304 (2016) · Zbl 1334.60129
[23] Cheridito, P., Soner, H.M., Touzi, N., Victoir, N.: Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60(7), 1081-1110 (2007) · Zbl 1121.60062
[24] Chiaramonte, M., Kiener, M.: Solving differential equations using neural networks. Machine Learning Project (2013)
[25] Crépey, S., Gerboud, R., Grbac, Z., Ngor, N.: Counterparty risk and funding: the four wings of the TVA. Int. J. Theor. Appl. Finance 16(2), 1350006 (2013) · Zbl 1266.91115
[26] Crisan, D., Manolarakis, K.: Probabilistic methods for semilinear partial differential equations. Applications to finance. M2AN Math. Model. Numer. Anal. 44 44(5), 1107-1133 (2010) · Zbl 1210.65011
[27] 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) · Zbl 1259.65005
[28] Crisan, D., Manolarakis, K.: Second order discretization of backward SDEs and simulation with the cubature method. Ann. Appl. Probab. 24(2), 652-678 (2014) · Zbl 1303.60046
[29] Crisan, D., Manolarakis, K., Touzi, N.: On the Monte Carlo simulation of BSDEs: an improvement on the Malliavin weights. Stoch. Process. Appl. 120(7), 1133-1158 (2010) · Zbl 1193.65005
[30] Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2(4), 303-314 (1989) · Zbl 0679.94019
[31] 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 (2016) · Zbl 1348.49026
[32] Dehghan, M., Nourian, M., Menhaj, M.B.: Numerical solution of Helmholtz equation by the modified Hopfield finite difference techniques. Numer. Methods Partial Differ. Equ. 25(3), 637-656 (2009) · Zbl 1165.65062
[33] Delarue, F., Menozzi, S.: A forward – backward stochastic algorithm for quasi-linear PDEs. Ann. Appl. Probab. 16(1), 140-184 (2006) · Zbl 1097.65011
[34] Douglas Jr., J., Ma, J., Protter, P.: Numerical methods for forward – backward stochastic differential equations. Ann. Appl. Probab. 6(3), 940-968 (1996) · Zbl 0861.65131
[35] E, W., Hutzenthaler, M., Jentzen, A., Kruse, T.: Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. (2017a). arXiv:1607.03295
[36] 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. (2017b). arXiv:1708.03223 · Zbl 1418.65149
[37] E, W., Han, J., Jentzen, A.: Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat. 5(4), 349-380 (2017c) · Zbl 1382.65016
[38] Ekren, I., Muhle-Karbe, J.: Portfolio choice with small temporary and transient price impact (2017). arXiv:1705.00672 · Zbl 1432.91102
[39] El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7(1), 1-71 (1997) · Zbl 0884.90035
[40] Fahim, A., Touzi, N., Warin, X.: A probabilistic numerical method for fully nonlinear parabolic PDEs. Ann. Appl. Probab. 21(4), 1322-1364 (2011) · Zbl 1230.65009
[41] Forsyth, P.A., Vetzal, K.R.: Implicit solution of uncertain volatility/transaction cost option pricing models with discretely observed barriers. Appl. Numer. Math. 36(4), 427-445 (2001) · Zbl 1072.91578
[42] Fu, Y., Zhao, W., Zhou, T.: Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs. Discrete Contin. Dyn. Syst. Ser. B 22(9), 3439-3458 (2017) · Zbl 1368.60071
[43] Geiss, S., Ylinen, J.: Decoupling on the Wiener space, related Besov spaces, and applications to BSDEs. (2014). arXiv:1409.5322 · Zbl 1494.60002
[44] Geiss, C., Labart, C.: Simulation of BSDEs with jumps by Wiener chaos expansion. Stoch. Process. Appl. 126(7), 2123-2162 (2016) · Zbl 1336.60138
[45] Glorot, X., Bordes, A., Bengio, Y.: Deep sparse rectifier neural networks. In: Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics (Fort Lauderdale, FL, USA, 11-13 Apr 2011), G. Gordon, D. Dunson, and M. Dudk, Eds., vol. 15 of Proceedings of Machine Learning Research, PMLR, pp. 315-323
[46] Gobet, E., Lemor, J.-P.: Numerical simulation of BSDEs using empirical regression methods: theory and practice. (2008). arXiv:0806.4447
[47] Gobet, E., Labart, C.: Solving BSDE with adaptive control variate. SIAM J. Numer. Anal. 48(1), 257-277 (2010) · Zbl 1208.60055
[48] Gobet, E., Turkedjiev, P.: Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression. Bernoulli 22(1), 530-562 (2016a) · Zbl 1339.60094
[49] Gobet, E., Turkedjiev, P.: Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. Math. Comp. 85(299), 1359-1391 (2016b) · Zbl 1344.60067
[50] 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) · Zbl 1083.60047
[51] Gobet, E., López-Salas, J.G., Turkedjiev, P., Vázquez, C.: Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs. SIAM J. Sci. Comput. 38(6), C652-C677 (2016) · Zbl 1352.65008
[52] Grohs, P., Hornung, F., Jentzen, A., von Wurstemberger, P.: A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations (2018) · Zbl 07679303
[53] Guo, W., Zhang, J., Zhuo, J.: A monotone scheme for high-dimensional fully nonlinear PDEs. Ann. Appl. Probab. 25(3), 1540-1580 (2015) · Zbl 1321.65158
[54] Guyon, J., Henry-Labordère, P.: The uncertain volatility model: a Monte Carlo approach. J. Comput. Finance 14(3), 37-61 (2011)
[55] Han, J., Jentzen, A., E, W.: Overcoming the curse of dimensionality: solving high-dimensional partial differential equations using deep learning (2017). arXiv:1707.02568
[56] Han, J., E, W.: Deep learning approximation for stochastic control problems (2016). arXiv:1611.07422
[57] Henry-Labordère, P., Oudjane, N., Tan, X., Touzi, N., Warin, X.: Branching diffusion representation of semilinear PDEs and Monte Carlo approximation (2016). arXiv:1603.01727 · Zbl 1467.60067
[58] Henry-Labordère, P.: Counterparty risk valuation: a marked branching diffusion approach (2012). arXiv:1203.2369
[59] 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) · Zbl 1301.60084
[60] Hinton, G., Deng, L., Yu, D., Dahl, G.E., Mohamed, A-r, Jaitly, N., Senior, A., Vanhoucke, V., Nguyen, P., Sainath, T.N., Kingsbury, B.: Deep neural networks for acoustic modeling in speech recognition: the shared views of four research groups. IEEE Signal Process. Mag. 29(6), 82-97 (2012)
[61] Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximators. Neural Netw. 2(5), 359-366 (1989) · Zbl 1383.92015
[62] Huijskens, T.P., Ruijter, M.J., Oosterlee, C.W.: Efficient numerical Fourier methods for coupled forward – backward SDEs. J. Comput. Appl. Math. 296, 593-612 (2016) · Zbl 1336.65010
[63] Ioffe, S., Szegedy, C.: Batch normalization: accelerating deep network training by reducing internal covariate shift. In: Proceedings of The 32nd International Conference on Machine Learning (ICML) (2015)
[64] Jentzen, A., Kuckuck, B., Neufeld, A., von Wurstemberger, P.: Strong error analysis for stochastic gradient descent optimization algorithms (2018). arXiv:1801.09324 · Zbl 1460.65071
[65] Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus, 2nd edn, vol. 113 of Graduate Texts in Mathematics. Springer, New York, (1991) · Zbl 0734.60060
[66] Khoo, Y., Lu, J., Ying, L.: Solving parametric PDE problems with artificial neural networks (2017). arXiv:1707.03351 · Zbl 1501.65154
[67] Kingma, D., Ba, J.: Adam: a method for stochastic optimization. In: Proceedings of the International Conference on Learning Representations (ICLR) (2015)
[68] Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics (New York). Springer, Berlin (1992) · Zbl 0752.60043
[69] Kong, T., Zhao, W., Zhou, T.: Probabilistic high order numerical schemes for fully nonlinear parabolic PDEs. Commun. Comput. Phys. 18(5), 1482-1503 (2015) · Zbl 1388.65016
[70] Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. Adv. Neural Inf. Process. Syst. 25, 1097-1105 (2012)
[71] Labart, C., Lelong, J.: A parallel algorithm for solving BSDEs. Monte Carlo Methods Appl. 19(1), 11-39 (2013) · Zbl 1263.65005
[72] Lagaris, I.E., Likas, A., Fotiadis, D.I.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw. 9(5), 987-1000 (1998)
[73] Laurent, J.-P., Amzelek, P., Bonnaud, J.: An overview of the valuation of collateralized derivative contracts. Rev. Deriv. Res. 17(3), 261-286 (2014) · Zbl 1300.91050
[74] LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proc. IEEE 86(11), 2278-2324 (1998)
[75] LeCun, Y., Bottou, L., Orr, G.B., Müller, K.R.: Efficient BackProp, pp. 9-50. Springer, Berlin, Heidelberg (1998)
[76] LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521, 436-444 (2015)
[77] Lee, H., Kang, I.S.: Neural algorithm for solving differential equations. J. Comput. Phys. 91(1), 110-131 (1990) · Zbl 0717.65062
[78] Leland, H.E.: Option pricing and replication with transactions costs. J. Finance 40(5), 1283-1301 (1985)
[79] Lemor, J.-P., Gobet, E., Warin, X.: Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli 12(5), 889-916 (2006) · Zbl 1136.60351
[80] Lionnet, A., dos Reis, G., Szpruch, L.: Time discretization of FBSDE with polynomial growth drivers and reaction-diffusion PDEs. Ann. Appl. Probab. 25(5), 2563-2625 (2015) · Zbl 1342.65011
[81] Ma, J., Yong, J.: Forward – backward stochastic differential equations and their applications, vol. 1702 of Lecture Notes in Mathematics. Springer, Berlin (1999) · Zbl 0927.60004
[82] Ma, J., Protter, P., Yong, J.M.: Solving forward-backward stochastic differential equations explicitly—a four step scheme. Probab. Theory Relat. Fields 98(3), 339-359 (1994) · Zbl 0794.60056
[83] Ma, J., Protter, P., San Martín, J., Torres, S.: Numerical method for backward stochastic differential equations. Ann. Appl. Probab. 12(1), 302-316 (2002) · Zbl 1017.60074
[84] Maruyama, G.: Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo (2) 4, 48-90 (1955) · Zbl 0053.40901
[85] McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Commun. Pure Appl. Math. 28(3), 323-331 (1975) · Zbl 0316.35053
[86] Meade Jr., A.J., Fernández, A.A.: The numerical solution of linear ordinary differential equations by feedforward neural networks. Math. Comput. Model. 19(12), 1-25 (1994) · Zbl 0807.65079
[87] Mehrkanoon, S., Suykens, J.A.: Learning solutions to partial differential equations using LS-SVM. Neurocomputing 159, 105-116 (2015)
[88] Milstein, G.N.: Approximate integration of stochastic differential equations. Teor. Verojatnost. i Primenen. 19, 583-588 (1974) · Zbl 0314.60039
[89] Milstein, G.N., Tretyakov, M.V.: Numerical algorithms for forward – backward stochastic differential equations. SIAM J. Sci. Comput. 28(2), 561-582 (2006) · Zbl 1114.60054
[90] Milstein, G.N., Tretyakov, M.V.: Discretization of forward – backward stochastic differential equations and related quasi-linear parabolic equations. IMA J. Numer. Anal. 27(1), 24-44 (2007) · Zbl 1109.65009
[91] Moreau, L., Muhle-Karbe, J., Soner, H.M.: Trading with small price impact. Math. Finance 27(2), 350-400 (2017)
[92] Øksendal, B.: Stochastic differential equations. An introduction with applications. Universitext. Springer, Berlin (1985) · Zbl 0567.60055
[93] Pardoux, E., 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 Information Sciences, pp. 200-217. Springer, Berlin (1992) · Zbl 0766.60079
[94] Pardoux, É., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55-61 (1990) · Zbl 0692.93064
[95] Pardoux, E., Tang, S.: Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Relat. Fields 114(2), 123-150 (1999) · Zbl 0943.60057
[96] Peng, \[S.: G\] G-expectation, \[G\] G-Brownian motion and related stochastic calculus of Itô type. In: Stochastic Analysis and Applications, vol. 2 of Abel Symposium, pp. 541-567. Springer, Berlin (2007) · Zbl 1131.60057
[97] Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty (2010). arXiv:1002.4546
[98] Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures. In: Stochastic Methodsin Finance, vol. 1856 of Lecture Notes in Mathematics, pp. 165-253. Springer, Berlin (2004) · Zbl 1127.91032
[99] Peng, S.: Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep. 37(1-2), 61-74 (1991) · Zbl 0739.60060
[100] Peng, S.: Nonlinear expectations and nonlinear Markov chains. Chin. Ann. Math. Ser. B 26(2), 159-184 (2005) · Zbl 1077.60045
[101] Petersen, P., Voigtlaender, F.: Optimal approximation of piecewise smooth functions using deep relu neural networks (2017). arXiv:1709.05289 · Zbl 1434.68516
[102] Pham, H.: Feynman-Kac representation of fully nonlinear PDEs and applications. Acta Math. Vietnam. 40(2), 255-269 (2015) · Zbl 1322.60133
[103] Possamaï, D., Mete Soner, H., Touzi, N.: Homogenization and asymptotics for small transaction costs: the multidimensional case. Commun. Partial Differ. Equ. 40(11), 2005-2046 (2015) · Zbl 1366.91144
[104] Ramuhalli, P., Udpa, L., Udpa, S.S.: Finite-element neural networks for solving differential equations. IEEE Trans. Neural Netw. 16(6), 1381-1392 (2005)
[105] Rasulov, A., Raimova, G., Mascagni, M.: Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation. Math. Comput. Simulation 80(6), 1118-1123 (2010) · Zbl 1198.65023
[106] Ruder, S.: An overview of gradient descent optimization algorithms (2016). arXiv:1609.04747
[107] Ruijter, M.J., Oosterlee, C.W.: A Fourier cosine method for an efficient computation of solutions to BSDEs. SIAM J. Sci. Comput. 37(2), A859-A889 (2015) · Zbl 1314.65011
[108] Ruijter, M.J., Oosterlee, C.W.: Numerical Fourier method and second-order Taylor scheme for backward SDEs in finance. Appl. Numer. Math. 103, 1-26 (2016) · Zbl 1386.91166
[109] Rumelhart, D.E., Hinton, G.E., Williams, R.J.: Learning internal representations by error propagation. Technical report, California University of San Diego La Jolla, Institute for Cognitive Science (1985)
[110] Ruszczynski, A., Yao, J.: A dual method for backward stochastic differential equations with application to risk valuation (2017). arXiv:1701.06234
[111] Silver, D., Huang, A., Maddison, C.J., Guez, A., Sifre, L., Van Den Driessche, G., Schrittwieser, J., Antonoglou, I., Panneershelvam, V., Lanctot, M., et al.: Mastering the game of go with deep neural networks and tree search. Nature 529(7587), 484-489 (2016)
[112] Sirignano, J., Spiliopoulos, K.: DGM: a deep learning algorithm for solving partial differential equations (2017). arXiv:1708.07469 · Zbl 1416.65394
[113] Skorohod, A.V.: Branching diffusion processes. Teor. Verojatnost. i Primenen. 9, 492-497 (1964) · Zbl 0264.60058
[114] Tadmor, E.: A review of numerical methods for nonlinear partial differential equations. Bull. Am. Math. Soc. (N.S.) 49(4), 507-554 (2012) · Zbl 1258.65073
[115] Thomée, V.: Galerkin finite element methods for parabolic problems, vol. 25 of Springer Series in Computational Mathematics. Springer, Berlin (1997) · Zbl 0884.65097
[116] Turkedjiev, P.: Two algorithms for the discrete time approximation of Markovian backward stochastic differential equations under local conditions. Electron. J. Probab. 20(50), 49 (2015) · Zbl 1322.60139
[117] von Petersdorff, T., Schwab, C.: Numerical solution of parabolic equations in high dimensions. M2AN Math. Model. Numer. Anal. 38(1), 93-127 (2004) · Zbl 1083.65095
[118] Warin, X.: Variations on branching methods for non linear PDEs (2017). arXiv:1701.07660 · Zbl 07008421
[119] Watanabe, S.: On the branching process for Brownian particles with an absorbing boundary. J. Math. Kyoto Univ. 4, 385-398 (1965) · Zbl 0134.34401
[120] Windcliff, H., Wang, J., Forsyth, P.A., Vetzal, K.R.: Hedging with a correlated asset: solution of a nonlinear pricing PDE. J. Comput. Appl. Math. 200(1), 86-115 (2007) · Zbl 1152.91033
[121] Zhang, J.: A numerical scheme for BSDEs. Ann. Appl. Probab. 14(1), 459-488 (2004) · Zbl 1056.60067
[122] Zhang, G., Gunzburger, M., Zhao, W.: A sparse-grid method for multi-dimensional backward stochastic differential equations. J. Comput. Math. 31(3), 221-248 (2013) · Zbl 1289.65011
[123] Zhao, W., Zhou, T., Kong, T.: High order numerical schemes for second-order FBSDEs with applications to stochastic optimal control. Commun. Comput. Phys. 21(3), 808-834 (2017) · Zbl 1499.65027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.