##
**Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks.**
*(English)*
Zbl 1469.60220

Summary: Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man-made complex systems. In particular, parabolic PDEs are a fundamental tool to approximately determine fair prices of financial derivatives in the financial engineering industry. The PDEs appearing in financial engineering applications are often nonlinear (e.g., in PDE models which take into account the possibility of a defaulting counterparty) and high-dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue in the scientific literature is that most approximation methods for nonlinear PDEs suffer from the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy grows exponentially in the dimension of the PDE or in the reciprocal of the prescribed approximation accuracy and nearly all approximation methods for nonlinear PDEs in the scientific literature have not been shown not to suffer from the curse of dimensionality. Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced and it was proven that MLP algorithms do overcome the curse of dimensionality for semilinear heat equations. In this paper we extend and generalize those findings to a more general class of semilinear PDEs which includes as special cases the important examples of semilinear Black-Scholes equations used in pricing models for financial derivatives with default risks. In particular, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove, under the assumption that the nonlinearity in the PDE is globally Lipschitz continuous, that the computational effort of the proposed method grows at most polynomially in both the dimension and the reciprocal of the prescribed approximation accuracy. We thereby establish, for the first time, that the numerical approximation of solutions of semilinear Black-Scholes equations is a polynomially tractable approximation problem.

### MSC:

60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |

35K58 | Semilinear parabolic equations |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

91G20 | Derivative securities (option pricing, hedging, etc.) |

### Keywords:

curse of dimensionality; high-dimensional PDEs; semilinear PDEs; semilinear Kolmogorov PDEs; multilevel Picard method### Software:

DGM
PDFBibTeX
XMLCite

\textit{M. Hutzenthaler} et al., Electron. J. Probab. 25, Paper No. 101, 73 p. (2020; Zbl 1469.60220)

### References:

[1] | Agarwal, R. P. Difference equations and inequalities: theory, methods, and applications. CRC Press, 2000. · Zbl 0952.39001 |

[2] | Bally, V., Pages, G., et al. A quantization algorithm for solving multidimensional discrete-time optimal stopping problems. Bernoulli 9, 6 (2003), 1003-1049. · Zbl 1042.60021 |

[3] | Beck, C., Becker, S., Grohs, P., Jaafari, N., and Jentzen, A. Solving stochastic differential equations and Kolmogorov equations by means of deep learning. arXiv:1806.00421 (2018), 56 pages. |

[4] | Beck, C., Gonon, L., Hutzenthaler, M., and Jentzen, A. On existence and uniqueness properties for solutions of stochastic fixed point equations. arXiv:1908.03382 (2019), 33 pages. |

[5] | Beck, C., Hutzenthaler, M., and Jentzen, A. On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations. arXiv:2004.03389 (2020), 54 pages. |

[6] | Beck, C., Weinan, E., and Jentzen, A. Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. Journal of Nonlinear Science (2017), 1-57. · Zbl 1442.91116 |

[7] | Becker, S., Cheridito, P., and Jentzen, A. Deep optimal stopping. J. Mach. Learn. Res. 20(2019), Paper No. 74, 25. · Zbl 1495.60029 |

[8] | Bellman, R. Dynamic programming. Science 153, 3731 (1966), 34-37. |

[9] | Bender, C., and Denk, R. A forward scheme for backward SDEs. Stochastic processes and their applications 117, 12 (2007), 1793-1812. · Zbl 1131.60054 |

[10] | Bender, C., Schweizer, N., and Zhuo, J. A primal-dual algorithm for BSDEs. Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics 27, 3 (2017), 866-901. · Zbl 1423.91008 |

[11] | Berg, J., and Nyström, K. A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing 317 (2018), 28-41. |

[12] | Black, F., and Scholes, M. The pricing of options and corporate liabilities. Journal of political economy 81, 3 (1973), 637-654. · Zbl 1092.91524 |

[13] | Bouchard, B., Elie, R., and Touzi, N. Discrete-time approximation of BSDEs and probabilistic schemes for fully nonlinear PDEs. Advanced financial modelling 8 (2009), 91-124. · Zbl 1179.65004 |

[14] | Bouchard, B., Elie, R., and Touzi, N. Discrete-time approximation of BSDEs and probabilistic schemes for fully nonlinear PDEs. In Advanced financial modelling, vol. 8 of Radon Ser. Comput. Appl. Math. Walter de Gruyter, Berlin, 2009, pp. 91-124. · Zbl 1179.65004 |

[15] | Bouchard, B., and Touzi, N. Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Processes and their Applications 111, 2 (2004), 175-206. · Zbl 1071.60059 |

[16] | Briand, P., and Labart, C. Simulation of BSDEs by Wiener chaos expansion. The Annals of Applied Probability 24, 3 (2014), 1129-1171. · Zbl 1311.60077 |

[17] | Burgard, C., and Kjaer, M. Partial differential equation representations of derivatives with bilateral counterparty risk and funding costs. The Journal of Credit Risk 7, 3 (2011), 1-19. |

[18] | Chan-Wai-Nam, Q., Mikael, J., and Warin, X. Machine learning for semi linear PDEs. arXiv:1809.07609 (2018). · Zbl 1433.68332 |

[19] | Chang, D., Liu, H., and Xiong, J. A branching particle system approximation for a class of FBSDEs. Probability, Uncertainty and Quantitative Risk 1 (2016), Paper No. 9, 34. · Zbl 1440.60062 |

[20] | Chassagneux, J.-F. Linear multistep schemes for BSDEs. SIAM Journal on Numerical Analysis 52, 6 (2014), 2815-2836. · Zbl 1326.65017 |

[21] | Chassagneux, J.-F., and Crisan, D. Runge-Kutta schemes for backward stochastic differential equations. The Annals of Applied Probability 24, 2 (2014), 679-720. · Zbl 1303.60045 |

[22] | Chassagneux, J.-F., and Richou, A. Numerical stability analysis of the Euler scheme for BSDEs. SIAM Journal on Numerical Analysis 53, 2 (2015), 1172-1193. · Zbl 1311.93088 |

[23] | Chassagneux, J.-F., and Richou, A. Numerical simulation of quadratic BSDEs. The Annals of Applied Probability 26, 1 (2016), 262-304. · Zbl 1334.60129 |

[24] | Cheridito, P., Soner, H. M., Touzi, N., and Victoir, N. Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences 60, 7 (2007), 1081-1110. · Zbl 1121.60062 |

[25] | Cox, S., Hutzenthaler, M., and Jentzen, A. Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations. arXiv:1309.5595 (2013). |

[26] | Cox, S., Hutzenthaler, M., Jentzen, A., van Neerven, J., and Welti, T. Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions. To appear in IMA J. Numer. Anal. arXiv:1605.00856 (2016), 48 pages. |

[27] | Crandall, M. G., Ishii, H., and Lions, P.-L. User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the American mathematical society 27, 1 (1992), 1-67. · Zbl 0755.35015 |

[28] | Crépey, S., Gerboud, R., Grbac, Z., and Ngor, N. Counterparty risk and funding: The four wings of the TVA. International Journal of Theoretical and Applied Finance 16, 02 (2013), 1350006. · Zbl 1266.91115 |

[29] | Crisan, D., and Manolarakis, K. Probabilistic methods for semilinear partial differential equations. Applications to finance. M2AN. Mathematical Modelling and Numerical Analysis 44, 5 (2010), 1107-1133. · Zbl 1210.65011 |

[30] | Crisan, D., and Manolarakis, K. Solving backward stochastic differential equations using the cubature method: application to nonlinear pricing. SIAM Journal on Financial Mathematics 3, 1 (2012), 534-571. · Zbl 1259.65005 |

[31] | Crisan, D., and Manolarakis, K. Second order discretization of backward SDEs and simulation with the cubature method. The Annals of Applied Probability 24, 2 (2014), 652-678. · Zbl 1303.60046 |

[32] | Crisan, D., Manolarakis, K., and Touzi, N. On the Monte Carlo simulation of BSDEs: an improvement on the Malliavin weights. Stochastic Processes and their Applications 120, 7 (2010), 1133-1158. · Zbl 1193.65005 |

[33] | Da Prato, G., Jentzen, A., and Röckner, M. A mild Itô formula for SPDEs. Trans. Amer. Math. Soc. 372, 6 (2019), 3755-3807. · Zbl 1423.35472 |

[34] | Dehghan, M., Nourian, M., and Menhaj, M. B. Numerical solution of Helmholtz equation by the modified Hopfield finite difference techniques. Numerical Methods for Partial Differential Equations: An International Journal 25, 3 (2009), 637-656. · Zbl 1165.65062 |

[35] | Delarue, F., and Menozzi, S. A forward-backward stochastic algorithm for quasi-linear PDEs. The Annals of Applied Probability 16, 1 (2006), 140-184. · Zbl 1097.65011 |

[36] | Douglas, Jr., J., Ma, J., and Protter, P. Numerical methods for forward-backward stochastic differential equations. The Annals of Applied Probability 6, 3 (1996), 940-968. · Zbl 0861.65131 |

[37] | Duffie, D., Schroder, M., Skiadas, C., et al. Recursive valuation of defaultable securities and the timing of resolution of uncertainty. The Annals of Applied Probability 6, 4 (1996), 1075-1090. · Zbl 0868.90008 |

[38] | E, W., Han, J., and Jentzen, A. Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Communications in Mathematics and Statistics 5, 4 (2017), 349-380. · Zbl 1382.65016 |

[39] | E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. arXiv:1607.03295 (2016), 18 pages. |

[40] | E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations. Journal of Scientific Computing (Mar 2019). · Zbl 1418.65149 |

[41] | Fahim, A., Touzi, N., and Warin, X. A probabilistic numerical method for fully nonlinear parabolic PDEs. The Annals of Applied Probability 21, 4 (2011), 1322-1364. · Zbl 1230.65009 |

[42] | Fu, Y., Zhao, W., and Zhou, T. Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs. Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences 22, 9 (2017), 3439-3458. · Zbl 1368.60071 |

[43] | Geiss, C., and Labart, C. Simulation of BSDEs with jumps by Wiener chaos expansion. Stochastic Processes and their Applications 126, 7 (2016), 2123-2162. · Zbl 1336.60138 |

[44] | Geiss, S., and Ylinen, J. Decoupling on the Wiener space, related Besov spaces, and applications to BSDEs. arXiv:1409.5322 (2014). |

[45] | Gobet, E., and Labart, C. Solving BSDE with adaptive control variate. SIAM Journal on Numerical Analysis 48, 1 (2010), 257-277. · Zbl 1208.60055 |

[46] | Gobet, E., and Lemor, J.-P. Numerical simulation of BSDEs using empirical regression methods: theory and practice. arXiv:0806.4447 (2008). |

[47] | Gobet, E., Lemor, J.-P., Warin, X., et al. A regression-based Monte Carlo method to solve backward stochastic differential equations. The Annals of Applied Probability 15, 3 (2005), 2172-2202. · Zbl 1083.60047 |

[48] | Gobet, E., López-Salas, J. G., Turkedjiev, P., and Vàzquez, C. Stratified regression Monte-Carl scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs. SIAM Journal on Scientific Computing 38, 6 (2016), C652-C677. · Zbl 1352.65008 |

[49] | Gobet, E., and Turkedjiev, P. Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. Mathematics of Computation 85, 299 (2016), 1359-1391. · Zbl 1344.60067 |

[50] | Gobet, E., Turkedjiev, P., et al. Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression. Bernoulli 22, 1 (2016), 530-562. · Zbl 1339.60094 |

[51] | Graham, C., and Talay, D. Stochastic simulation and Monte Carlo methods: mathematical foundations of stochastic simulation, vol. 68. Springer Science & Business Media, 2013. · Zbl 1281.65003 |

[52] | Grohs, P., Hornung, F., Jentzen, A., and von Wurstemberger, P. A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. arXiv:1809.02362 (2018), 124 pages. |

[53] | Guo, W., Zhang, J., and Zhuo, J. A monotone scheme for high-dimensional fully nonlinear PDEs. The Annals of Applied Probability 25, 3 (2015), 1540-1580. · Zbl 1321.65158 |

[54] | Han, J., Jentzen, A., and E, W. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences 115, 34 (2018), 8505-8510. · Zbl 1416.35137 |

[55] | He, J., Li, L., Xu, J., and Zheng, C. Relu deep neural networks and linear finite elements. arXiv:1807.03973 (2018). |

[56] | Henry-Labordere, P. Counterparty Risk Valuation: A Marked Branching Diffusion Approach. arXiv:1203.2369 (2012). |

[57] | Henry-Labordere, P. Deep primal-dual algorithm for BSDEs: Applications of machine learning to CVA and IM. Available at SSRN 3071506 (2017). |

[58] | Henry-Labordere, P., Oudjane, N., Tan, X., Touzi, N., Warin, X., et al. Branching diffusion representation of semilinear pdes and monte carlo approximation. In Annales de l’Institut Henri Poincaré, Probabilités et Statistiques (2019), vol. 55, Institut Henri Poincaré, pp. 184-210. · Zbl 1467.60067 |

[59] | Henry-Labordere, P., Tan, X., and Touzi, N. A numerical algorithm for a class of BSDEs via the branching process. Stochastic Processes and their Applications 124, 2 (2014), 1112-1140. · Zbl 1301.60084 |

[60] | Huijskens, T., Ruijter, M. J., and Oosterlee, C. W. Efficient numerical Fourier methods for coupled forward-backward SDEs. Journal of Computational and Applied Mathematics 296 (2016), 593-612. · Zbl 1336.65010 |

[61] | Huré, C., Pham, H., and Warin, X. Some machine learning schemes for high-dimensional nonlinear PDEs. arXiv:1902.01599 (2019). · Zbl 1440.60063 |

[62] | Hutzenthaler, M., Jentzen, A., Kruse, T., and Nguyen, T. A. A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations. SN Partial Differ. Equ. Appl. 1, 10 (2020). |

[63] | Hutzenthaler, M., Jentzen, A., Kruse, T., Nguyen, T. A., and von Wurstemberger, P. Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations. arXiv:1807.01212 (2018), 27 pages. |

[64] | Hutzenthaler, M., and Kruse, T. Multilevel Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities. SIAM Journal on Numerical Analysis 58, 2 (2020), 929-961. |

[65] | Jentzen, A., and Kloeden, P. E. Taylor approximations for stochastic partial differential equations, vol. 83. SIAM, 2011. · Zbl 1240.35001 |

[66] | Jentzen, A., and von Wurstemberger, P. Lower error bounds for the stochastic gradient descent optimization algorithm: Sharp convergence rates for slowly and fast decaying learning rates. arXiv:1803.08600 (2018), 42 pages. · Zbl 1433.68353 |

[67] | Jentzen, A., Welti, T., and Salimova, D. Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations. J. Math. Anal. Appl. 469, 2 (2019), 661-704. · Zbl 1403.60059 |

[68] | Karatzas, I., and Shreve, S. Brownian motion and stochastic calculus, vol. 113. Springer Science & Business Media, 2012. · Zbl 0638.60065 |

[69] | Khoo, Y., Lu, J., and Ying, L. Solving parametric PDE problems with artificial neural networks. arXiv:1707.03351 (2017). |

[70] | Klenke, A. Probabilitly Theory, 2 ed. Universitext. Springer-Verlag London Ltd., 2014. · Zbl 1295.60001 |

[71] | Kloeden, P. E., and Platen, E. Numerical solution of stochastic differential equations, vol. 23. Springer Science & Business Media, 2013. · Zbl 0752.60043 |

[72] | Kong, T., Zhao, W., and Zhou, T. Probabilistic high order numerical schemes for fully nonlinear parabolic PDEs. Communications in Computational Physics 18, 5 (2015), 1482-1503. · Zbl 1388.65016 |

[73] | Labart, C., and Lelong, J. A parallel algorithm for solving BSDEs. Monte Carlo Methods and Applications 19, 1 (2013), 11-39. · Zbl 1263.65005 |

[74] | Lemor, J.-P., Gobet, E., Warin, X., et al. Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli 12, 5 (2006), 889-916. · Zbl 1136.60351 |

[75] | Lionnet, A., Dos Reis, G., Szpruch, L., et al. Time discretization of FBSDE with polynomial growth drivers and reaction-diffusion PDEs. The Annals of Applied Probability 25, 5 (2015), 2563-2625. · Zbl 1342.65011 |

[76] | Ma, J., Protter, P., San Martín, J., and Torres, S. Numerical method for backward stochastic differential equations. The Annals of Applied Probability 12, 1 (2002), 302-316. · Zbl 1017.60074 |

[77] | Ma, J., Protter, P., and Yong, J. M. Solving forward-backward stochastic differential equations explicitly—a four step scheme. Probability Theory and Related Fields 98, 3 (1994), 339-359. · Zbl 0794.60056 |

[78] | Ma, J., and Yong, J. Forward-backward stochastic differential equations and their applications, vol. 1702 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1999. · Zbl 0927.60004 |

[79] | McKean, H. P. Application of Brownian Motion to the Equation of Kolmogorov-Petrovskii-Piskunov. Communications on pure and applied mathematics 28, 3 (1975), 323-331. · Zbl 0316.35053 |

[80] | Merton, R. C. Theory of rational option pricing. Theory of Valuation (1973), 229-288. · Zbl 1257.91043 |

[81] | Milstein, G. N. Numerical integration of stochastic differential equations, vol. 313. Springer Science & Business Media, 1994. · Zbl 0810.65144 |

[82] | Milstein, G. N., and Tretyakov, M. V. Numerical algorithms for forward-backward stochastic differential equations. SIAM Journal on Scientific Computing 28, 2 (2006), 561-582. · Zbl 1114.60054 |

[83] | Milstein, G. N., and Tretyakov, M. V. Discretization of forward-backward stochastic differential equations and related quasi-linear parabolic equations. IMA Journal of Numerical Analysis 27, 1 (2007), 24-44. · Zbl 1109.65009 |

[84] | Nabian, M. A., and Meidani, H. A deep neural network surrogate for high-dimensional random partial differential equations. arXiv:1806.02957 (2018). |

[85] | Novak, E., and Woźniakowski, H. Tractability of Multivariate Problems: Standard information for functionals, vol. 12. European Mathematical Society, 2008. · Zbl 1241.65025 |

[86] | Pardoux, E., and Peng, S. Adapted solution of a backward stochastic differential equation. Systems & Control Letters 14, 1 (1990), 55-61. · Zbl 0692.93064 |

[87] | Pardoux, E., and Peng, S. Backward stochastic differential equations and quasilinear parabolic partial differential equations. In Stochastic partial differential equations and their applications. Springer, 1992, pp. 200-217. · Zbl 0766.60079 |

[88] | Pardoux, E., and Tang, S. Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probability Theory and Related Fields 114, 2 (1999), 123-150. · Zbl 0943.60057 |

[89] | Pham, H. Feynman-Kac representation of fully nonlinear PDEs and applications. Acta Mathematica Vietnamica 40, 2 (2015), 255-269. · Zbl 1322.60133 |

[90] | Prévôt, C., and Röckner, M. A concise course on stochastic partial differential equations, vol. 1905. Springer, 2007. · Zbl 1123.60001 |

[91] | Raissi, M. Forward-backward stochastic neural networks: Deep learning of high-dimensional partial differential equations. arXiv:1804.07010 (2018). |

[92] | Rasulov, A., Raimova, G., and Mascagni, M. Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation. Mathematics and Computers in Simulation 80, 6 (2010), 1118-1123. · Zbl 1198.65023 |

[93] | Ruijter, M. J., and Oosterlee, C. W. A Fourier cosine method for an efficient computation of solutions to BSDEs. SIAM Journal on Scientific Computing 37, 2 (2015), A859-A889. · Zbl 1314.65011 |

[94] | Ruijter, M. J., and Oosterlee, C. W. Numerical Fourier method and second-order Taylor scheme for backward SDEs in finance. Applied Numerical Mathematics. An IMACS Journal 103 (2016), 1-26. · Zbl 1386.91166 |

[95] | Ruszczynski, A., and Yao, J. A dual method for backward stochastic differential equations with application to risk valuation. arXiv:1701.06234 (2017). |

[96] | Sirignano, J., and Spiliopoulos, K. DGM: A deep learning algorithm for solving partial differential equations. Journal of Computational Physics 375 (2018), 1339-1364. · Zbl 1416.65394 |

[97] | Skorokhod, A. V. Branching diffusion processes. Theory of Probability & Its Applications 9, 3 (1964), 445-449. · Zbl 0264.60058 |

[98] | Tadmor, E. A review of numerical methods for nonlinear partial differential equations. Bulletin of the American Mathematical Society 49, 4 (2012), 507-554. · Zbl 1258.65073 |

[99] | Thomée, V. Galerkin finite element methods for parabolic problems, vol. 1054. Springer, 1984. · Zbl 0528.65052 |

[100] | Turkedjiev, P. Two algorithms for the discrete time approximation of Markovian backward stochastic differential equations under local conditions. Electronic Journal of Probability 20 (2015), no. 50, 49. · Zbl 1322.60139 |

[101] | Von Petersdorff, T., and Schwab, C. Numerical solution of parabolic equations in high dimensions. ESAIM: Mathematical Modelling and Numerical Analysis 38, 1 (2004), 93-127. · Zbl 1083.65095 |

[102] | Warin, X. Variations on branching methods for non linear PDEs. arXiv:1701.07660 (2017). |

[103] | Warin, X. Monte Carlo for high-dimensional degenerated Semi Linear and Full Non Linear PDEs. arXiv:1805.05078 (2018). · Zbl 07008421 |

[104] | Warin, X. Nesting Monte Carlo for high-dimensional non-linear PDEs. Monte Carlo Methods and Applications 24, 4 (2018), 225-247. · Zbl 07008421 |

[105] | Watanabe, S. On the branching process for Brownian particles with an absorbing boundary. Journal of Mathematics of Kyoto University 4, 2 (1965), 385-398. · Zbl 0134.34401 |

[106] | Wu, Z., and Yu, Z. Probabilistic interpretation for a system of quasilinear parabolic partial differential equation combined with algebra equations. Stochastic Processes and their Applications 124, 12 (2014), 3921-3947. · Zbl 1314.60135 |

[107] | Zhang, G., Gunzburger, M., and Zhao, W. A sparse-grid method for multi-dimensional backward stochastic differential equations. Journal of Computational Mathematics 31, 3 (2013), 221-248. · Zbl 1289.65011 |

[108] | Zhang, J. A numerical scheme for BSDEs. The Annals of Applied Probability 14, 1 (2004), 459-488. · Zbl 1056.60067 |

[109] | Zhao, W. |

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.