Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations

References

[1] L. Abbas-Turki, B. Diallo, and G. Pagès, Conditional Monte Carlo Learning for Diffusions I: main methodology and application to backward stochastic differential equations, hal-02959492, 2020.Suche in Google Scholar

[2] L. Abbas-Turki, B. Diallo, and G. Pagès, Conditional Monte Carlo Learning for Diffusions II: extended methodology and application to risk measures and early stopping problems, hal-02959494, 2020.Suche in Google Scholar

[3] A. Agarwal and J. Claisse, Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method, Stoch. Process. Appl. 130 (2020), No. 8, 5006–5036.10.1016/j.spa.2020.02.009Suche in Google Scholar

[4] V. Bally and G. Pages, Error analysis of the optimal quantization algorithm for obstacle problems, Stoch. Process. Appl. 106 (2003), No. 1, 1–40.10.1016/S0304-4149(03)00026-7Suche in Google Scholar

[5] V. Bally and G. Pagès, A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems, Bernoulli 9 (2003), No. 6, 1003–1049.10.3150/bj/1072215199Suche in Google Scholar

[6] C. Beck, L. Gonon, M. Hutzenthaler, and A. Jentzen, On existence and uniqueness properties for solutions of stochastic fixed point equations, Discrete Contin. Dyn. Syst. Ser. B 26 (2021), No. 9, 4927–4962.10.3934/dcdsb.2020320Suche in Google Scholar

[7] C. Beck, L. Gonon, and A. Jentzen, Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations, arXiv:2003.00596, 2020.Suche in Google Scholar

[8] C. Beck, F. Hornung, M. Hutzenthaler, A. Jentzen, and T. Kruse, Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations, J. Numer. Math. 28 (2020), No. 4, 197–222.10.1515/jnma-2019-0074Suche in Google Scholar

[9] C. Beck, M. Hutzenthaler, and A. Jentzen, On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations, Stochastics and Dynamics 21 (2021), No. 8, 2150048.10.1142/S0219493721500489Suche in Google Scholar

[10] C. Beck, M. Hutzenthaler, A. Jentzen, and B. Kuckuck, An overview on deep learning-based approximation methods for partial differential equations, arXiv:2012.12348, 2020.10.3934/dcdsb.2022238Suche in Google Scholar

[11] C. Beck, A. Jentzen, and T. Kruse, Nonlinear Monte Carlo methods with polynomial runtime for high-dimensional iterated nested expectations, arXiv:2009.13989, 2020.Suche in Google Scholar

[12] S. Becker, R. Braunwarth, M. Hutzenthaler, A. Jentzen, and P. von Wurstemberger, Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations, Commun. Comput. Phys. 28 (2020), No. 5, 2109–2138.10.4208/cicp.OA-2020-0130Suche in Google Scholar

[13] R. Bellman, Dynamic Programming, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2010, Reprint of the 1957 edition, With a new introduction by Stuart Dreyfus.10.1515/9781400835386Suche in Google Scholar

[14] C. Bender and R. Denk, A forward scheme for backward SDEs, Stoch. Process. Appl. 117 (2007), No. 12, 1793–1812.10.1016/j.spa.2007.03.005Suche in Google Scholar

[15] C. Bender, C. Gärtner, and N. Schweizer, Iterative improvement of lower and upper bounds for backward SDEs, SIAM J. Sci. Comput. 39 (2017), No. 2, B442–B466.10.1137/16M1081348Suche in Google Scholar

[16] C. Bender, C. Gärtner, and N. Schweizer, Pathwise dynamic programming, Math. Operations Research 43 (2018), No. 3, 965–995.10.1287/moor.2017.0891Suche in Google Scholar

[17] C. Bender, N. Schweizer, and J. Zhuo, A primal–dual algorithm for BSDEs, Math. Finance 27 (2017), No. 3, 866–901.10.1111/mafi.12100Suche in Google Scholar

[18] C. Bender and J. Steiner, Least-squares Monte Carlo for backward SDES, In: Numerical Methods in Finance, Springer, 2012, pp. 257–289.10.1007/978-3-642-25746-9_8Suche in Google Scholar

[19] C. Bender and J. Zhang, Time discretization and Markovian iteration for coupled FBSDEs, The Annals of Applied Probability 18 (2008), No. 1, 143–177.10.1214/07-AAP448Suche in Google Scholar

[20] B. Bouchard and N. Touzi, Discrete-time approximation and Monte Carlo simulation of backward stochastic differential equations, Stoch. Process. Appl. 111 (2004), No. 2, 175–206.10.1016/j.spa.2004.01.001Suche in Google Scholar

[21] P. Briand, B. Delyon, and J. Mémin, Donsker-type theorem for BSDEs, Electronic Commun. Probability 6 (2001), 1–14.10.1214/ECP.v6-1030Suche in Google Scholar

[22] P. Briand, C. Geiss, S. Geiss, and C. Labart, Donsker-type theorem for BSDEs: rate of convergence, Bernoulli 27 (2021), No. 2, 899–929.10.3150/20-BEJ1259Suche in Google Scholar

[23] P. Briand and C. Labart, Simulation of BSDEs by Wiener chaos expansion, Ann. Appl. Probab. 24 (2014), No. 3, 1129–1171.10.1214/13-AAP943Suche in Google Scholar

[24] D. Chang, H. Liu, and J. Xiong, A branching particle system approximation for a class of FBSDEs, Probab. Uncertain. Quant. Risk 1 (2016), No. 1, 1–34.10.1186/s41546-016-0007-ySuche in Google Scholar

[25] J.-F. Chassagneux, Linear multistep schemes for BSDEs, SIAM J. Numer. Anal. 52 (2014), No. 6, 2815–2836.10.1137/120902951Suche in Google Scholar

[26] J.-F. Chassagneux and D. Crisan, Runge–Kutta schemes for backward stochastic differential equations, Ann. Appl. Probab. 24 (2014), No. 2, 679–720.10.1214/13-AAP933Suche in Google Scholar

[27] J.-F. Chassagneux and Adrien Richou, Numerical stability analysis of the Euler scheme for BSDEs, SIAM J. Numer. Anal. 53 (2015), No. 2, 1172–1193.10.1137/140977047Suche in Google Scholar

[28] J.-F. Chassagneux and A. Richou, Numerical simulation of quadratic BSDEs, Ann. Appl. Probab. 26 (2016), No. 1, 262–304.10.1214/14-AAP1090Suche in Google Scholar

[29] J.-F. Chassagneux and C. Garcia Trillos, Cubature method to solve BSDEs: Error expansion and complexity control, Math. Computation 89 (2020), No. 324, 1895–1932.10.1090/mcom/3522Suche in Google Scholar

[30] Y. Chen and J. W. L. Wan, Deep neural network framework based on backward stochastic differential equations for pricing and hedging American options in high dimensions, Quantitative Finance 21 (2021), No. 1, 45–67.10.1080/14697688.2020.1788219Suche in Google Scholar

[31] D. Chevance, Numerical methods for backward stochastic differential equations, In: Numerical Methods Finance (Eds. L. C. G. Rogers and D. Talay), Carbridge Univ. Press, Carbridge, 1997, pp. 232–244.10.1017/CBO9781139173056.013Suche in Google Scholar

[32] S. Cox, M. Hutzenthaler, A. Jentzen, J. van Neerven, and T. Welti, Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions, IMA J. Numer. Anal. 41 (2021), No. 1, 493–548.10.1093/imanum/drz063Suche in Google Scholar

[33] S. Crépey, Financial Modeling, Springer Finance, Springer, Heidelberg–New York–Dordrecht–London, 2013.10.1007/978-3-642-37113-4Suche in Google Scholar

[34] D. Crisan and K. Manolarakis, Solving backward stochastic differential equations using the cubature method: application to nonlinear pricing, SIAM J. Financial Math. 3 (2012), No. 1, 534–571.10.1142/9789814313179_0050Suche in Google Scholar

[35] D. Crisan, K. Manolarakis, and N. Touzi, On the Monte Carlo simulation of BSDEs: an improvement on the Malliavin weights, Stoch. Process. Appl. 120 (2010), No. 7, 1133–1158.10.1016/j.spa.2010.03.015Suche in Google Scholar

[36] D. Crisan and K. Manolarakis, Probabilistic methods for semilinear partial differential equations. Applications to finance, M2AN Math. Model. Numer. Anal. 44 (2010), No. 5, 1107–1133.10.1051/m2an/2010054Suche in Google Scholar

[37] D. Crisan and K. Manolarakis, Second order discretization of backward SDEs and simulation with the cubature method, Ann. Appl. Probab. 24 (2014), No. 2, 652–678.10.1214/13-AAP932Suche in Google Scholar

[38] J. Cvitanic and J. Zhang, The steepest descent method for forward-backward SDEs, Electr. J. Probability 10 (2005), 1468–1495.10.1214/EJP.v10-295Suche in Google Scholar

[39] P. E. Chaudru de Raynal and C. A. Garcia Trillos, A cubature based algorithm to solve decoupled McKean–Vlasov forward–backward stochastic differential equations, Stoch. Process. Appl. 125 (2015), No. 6, 2206–2255.10.1016/j.spa.2014.11.018Suche in Google Scholar

[40] F. Delarue and S. Menozzi, A forward–backward stochastic algorithm for quasi-linear PDEs, Ann. Appl. Probab. 16 (2006), No. 1, 140–184.10.1214/105051605000000674Suche in Google Scholar

[41] F. Delarue and S. Menozzi, An interpolated stochastic algorithm for quasi-linear PDEs, Math. Computation 77 (2008), No. 261, 125–158.10.1090/S0025-5718-07-02008-XSuche in Google Scholar

[42] Ł. Delong, Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications, Springer, 2013.10.1007/978-1-4471-5331-3Suche in Google Scholar

[43] J. Douglas, Jr., J. Ma, and P. Protter, Numerical methods for forward–backward stochastic differential equations, Ann. Appl. Probab. 6 (1996), No. 3, 940–968.10.1214/aoap/1034968235Suche in Google Scholar

[44] Weinan E, J. Han, and A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun. Math. Stat. 5 (2017), No. 4, 349–380.10.1007/s40304-017-0117-6Suche in Google Scholar

[45] Weinan E, J. Han, and A. Jentzen, Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning, Nonlinearity 35 (2022), No. 1, 278–310.10.1088/1361-6544/ac337fSuche in Google Scholar

[46] Weinan E, M. Hutzenthaler, A. Jentzen, and T. Kruse, On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations, J. Sci. Comput. 79 (2019), No. 3, 1534–1571.10.1007/s10915-018-00903-0Suche in Google Scholar

[47] Weinan E, M. Hutzenthaler, A. Jentzen, and T. Kruse, Multilevel Picard iterations for solving smooth semilinear parabolic heat equations, Part. Differ. Equ. Appl. 2 (2021), No. 6, 1–31.10.1007/s42985-021-00089-5Suche in Google Scholar

[48] N. El Karoui, S. Peng, and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance 7 (1997), No. 1, 1–71.10.1111/1467-9965.00022Suche in Google Scholar

[49] Y. Fu, W. Zhao, and T. Zhou, Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), No. 9, 3439–3458.10.3934/dcdsb.2017174Suche in Google Scholar

[50] M. Fujii, A. Takahashi, and M. Takahashi, Asymptotic expansion as prior knowledge in deep learning method for high dimensional BSDEs, Asia-Pacific Financial Markets 26 (2017), No. 3, 391–408.10.1007/s10690-019-09271-7Suche in Google Scholar

[51] C. Geiss and C. Labart, Simulation of BSDEs with jumps by Wiener chaos expansion, Stoch. Process. Appl. 126 (2016), No. 7, 2123–2162.10.1016/j.spa.2016.01.006Suche in Google Scholar

[52] C. Geiss, C. Labart, A. Luoto, Random walk approximation of BSDEs with Hölder continuous terminal condition, Bernoulli 26 (2020), No. 1, 159–190.10.3150/19-BEJ1120Suche in Google Scholar

[53] C. Geiss, C. Labart, and A. Luoto, Mean square rate of convergence for random walk approximation of forward–backward SDEs, Adv. Appl. Probability 52 (2020), No. 3, 735–771.10.1017/apr.2020.17Suche in Google Scholar

[54] M. B. Giles, A. Jentzen, and T. Welti, Generalised multilevel Picard approximations, arXiv:1911.03188, 2019.Suche in Google Scholar

[55] E. Gobet, J. G. López-Salas, P. Turkedjiev, and C. Vázquez, Stratified regression Monte–Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs, SIAM J. Sci. Comput. 38 (2016), No. 6, C652–C677.10.1137/16M106371XSuche in Google Scholar

[56] E. Gobet and C. Labart, Error expansion for the discretization of backward stochastic differential equations, Stoch. Process. Appl. 117 (2007), No. 7, 803–829.10.1016/j.spa.2006.10.007Suche in Google Scholar

[57] E. Gobet and C. Labart, Solving BSDE with adaptive control variate, SIAM J. Numer. Anal. 48 (2010), No. 1, 257–277.10.1137/090755060Suche in Google Scholar

[58] E. Gobet and J.-P. Lemor, Numerical simulation of BSDEs using empirical regression methods: theory and practice, arXiv:0806.4447, 2008.10.1214/105051605000000412Suche in Google Scholar

[59] E. Gobet, J.-P. Lemor, and X. Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, Ann. Appl. Probab. 15 (2005), No. 3, 2172–2202.10.1214/105051605000000412Suche in Google Scholar

[60] E. Gobet and P. Turkedjiev, Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression, Bernoulli 22 (2016), No. 1, 530–562.10.3150/14-BEJ667Suche in Google Scholar

[61] E. Gobet and P. Turkedjiev, Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions, Math. Comp. 85 (2016), No. 299, 1359–1391.10.1090/mcom/3013Suche in Google Scholar

[62] M. Hairer and H. Shen, The dynamical sine-Gordon model, Commun. Math. Phys. 341 (2016), No. 3, 933–989.10.1007/s00220-015-2525-3Suche in Google Scholar

[63] S. Heinrich, Monte Carlo complexity of global solution of integral equations, J. Complexity 14 (1998), No. 2, 151–175.10.1006/jcom.1998.0471Suche in Google Scholar

[64] S. Heinrich, Multilevel Monte Carlo methods, In: Large-Scale Scientific Computing, Lecture Notes in Computer Science, Vol. 2179, Springer, 2001, pp. 58–67.10.1007/3-540-45346-6_5Suche in Google Scholar

[65] S. Heinrich and E. Sindambiwe, Monte Carlo complexity of parametric integration, J. Complexity 15 (1999), No. 3, 317–341.10.1006/jcom.1999.0508Suche in Google Scholar

[66] P. Henry-Labordère, Counterparty risk valuation: a marked branching diffusion approach, arXiv:1203.2369, 2012.10.2139/ssrn.1995503Suche in Google Scholar

[67] P. Henry-Labordère, Deep Primal-Dual Algorithm for BSDEs: Applications of Machine Learning to CVA and IM, Available at SSRN: http://dx.doi.org/10.2139/ssrn.3071506 2017.10.2139/ssrn.3071506Suche in Google Scholar

[68] P. Henry-Labordère, N. Oudjane, X. Tan, N. Touzi, and X. Warin, Branching diffusion representation of semilinear PDEs and Monte Carlo approximation, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 55 (2019), No. 1, 184–210.10.1214/17-AIHP880Suche in Google Scholar

[69] P. Henry-Labordère, X. Tan, and N. Touzi, A numerical algorithm for a class of BSDEs via the branching process, Stoch. Process. Appl. 124 (2014), No. 2, 1112–1140.10.1016/j.spa.2013.10.005Suche in Google Scholar

[70] Y. Hu, D. Nualart, and X. Song, Malliavin calculus for backward stochastic differential equations and application to numerical solutions, The Annals of Applied Probability 21 (2011), No. 6, 2379–2423.10.1214/11-AAP762Suche in Google Scholar

[71] T. P. Huijskens, M. J. Ruijter, and C. W. Oosterlee, Efficient numerical Fourier methods for coupled forward–backward SDEs, J. Comput. Appl. Math. 296 (2016), 593–612.10.1016/j.cam.2015.10.019Suche in Google Scholar

[72] M. Hutzenthaler, A. Jentzen, and T. Kruse, Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities, Found. Comput. Math. (2021), 1–62.10.1007/s10208-021-09514-ySuche in Google Scholar

[73] M. Hutzenthaler, A. Jentzen, T. Kruse, and T. A. Nguyen, Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities, arXiv:2009.02484, 2020.Suche in Google Scholar

[74] M. Hutzenthaler, A. Jentzen, T. Kruse, T. A. Nguyen, and P. von Wurstemberger, Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations, Proc. of the Royal Society A 476 (2020), No. 2244, 20190630.10.1098/rspa.2019.0630Suche in Google Scholar PubMed PubMed Central

[75] M. Hutzenthaler, A. Jentzen, and P. von Wurstemberger, Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks, Electron. J. Probab. 25 (2020), 1–73.10.1214/20-EJP423Suche in Google Scholar

[76] M. Hutzenthaler and T. Kruse, Multilevel Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities, SIAM J. Numer. Anal. 58 (2020), No. 2, 929–961.10.1137/17M1157015Suche in Google Scholar

[77] P. Imkeller, G. dos Reis, and J. Zhang, Results on numerics for FBSDE with drivers of quadratic growth, In: Contemporary Quantitative Finance, Springer, 2010, pp. 159–182.10.1007/978-3-642-03479-4_9Suche in Google Scholar

[78] C. Labart and J. Lelong, A parallel algorithm for solving BSDEs, Monte Carlo Methods Appl. 19 (2013), No. 1, 11–39.10.1515/mcma-2013-0001Suche in Google Scholar

[79] A. Le Cavil, N. Oudjane, and F. Russo, Particle system algorithm and chaos propagation related to non-conservative McKean type stochastic differential equations, Stochastics and Partial Differential Equations: Analysis and Computations 5 (2017), No. 1, 1–37.10.1007/s40072-016-0079-9Suche in Google Scholar

[80] A. Le Cavil, N. Oudjane, and F. Russo, Monte Carlo algorithms for a forward Feynman–Kac-type representation for semilinear nonconservative partial differential equations, Monte Carlo Methods and Applications 24 (2018), No. 1, 55–70.10.1515/mcma-2018-0005Suche in Google Scholar

[81] A. Le Cavil, N. Oudjane, and F. Russo, Forward Feynman–Kac type representation for semilinear non-conservative partial differential equations, Stochastics 91 (2019), No. 8, 1206–1248.10.1080/17442508.2019.1594809Suche in Google Scholar

[82] J.-P. Lemor, E. Gobet, and X. Warin, Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations, Bernoulli 12 (2006), No. 5, 889–916.10.3150/bj/1161614951Suche in Google Scholar

[83] A. Lionnet, G. dos Reis, and L. Szpruch, Time discretization of FBSDE with polynomial growth drivers and reaction–diffusion PDEs, Ann. Appl. Probab. 25 (2015), No. 5, 2563–2625.10.1214/14-AAP1056Suche in Google Scholar

[84] J. Ma, P. Protter, J. San Martín, and S. Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab. 12 (2002), No. 1, 302–316.10.1214/aoap/1037125868Suche in Google Scholar

[85] J. Ma, P. Protter, and J. M. Yong, Solving forward–backward stochastic differential equations explicitly—a four step scheme, Probab. Theory Related Fields 98 (1994), No. 3, 339–359.10.1007/BF01192258Suche in Google Scholar

[86] J. Ma and J. Yong, Forward–Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics, Vol. 1702, Springer-Verlag, Berlin, 1999.Suche in Google Scholar

[87] H. P. McKean, Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov, Comm. Pure Appl. Math. 28 (1975), No. 3, 323–331.10.1002/cpa.3160280302Suche in Google Scholar

[88] G. N. Milstein and M. V. Tretyakov, Numerical algorithms for forward–backward stochastic differential equations, SIAM J. Sci. Comput. 28 (2006), No. 2, 561–582.10.1137/040614426Suche in Google Scholar

[89] G. N. Milstein and M. V. Tretyakov, Discretization of forward–backward stochastic differential equations and related quasi-linear parabolic equations, IMA J. Numer. Anal. 27 (2007), No. 1, 24–44.10.1093/imanum/drl019Suche in Google Scholar

[90] E. Novak and K. Ritter, The curse of dimension and a universal method for numerical integration, In: Multivariate Approximation and Splines (Mannheim, 1996), Internat. Ser. Numer. Math., Vol. 125, Birkhäuser, Basel, 1997, pp. 177–187.10.1007/978-3-0348-8871-4_15Suche in Google Scholar

[91] E. Novak and H. Woźniakowski, Tractability of Multivariate Problems. Vol. 1: Linear Information, EMS Tracts in Mathematics, Vol. 6, European Mathematical Society, Zürich, 2008.10.4171/026Suche in Google Scholar

[92] B. Øksendal, Stochastic Differential Equations. An Introduction with Applications. Universitext, Springer-Verlag, Berlin, 1985.10.1007/978-3-662-13050-6Suche in Google Scholar

[93] É. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), No. 1, 55–61.10.1016/0167-6911(90)90082-6Suche in Google Scholar

[94] É. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, In: Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991), Lect. Notes Control Inf. Sci., Vol. 176, Springer, Berlin, 1992, pp. 200–217.10.1007/BFb0007334Suche in Google Scholar

[95] E. Pardoux and A. Raşcanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Stochastic Modelling and Applied Probability, Springer International Publishing, 2014.10.1007/978-3-319-05714-9Suche in Google Scholar

[96] É. Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, In: Nonlinear Analysis, Differential Equations and Control, Springer, 1999, pp. 503–549.10.1007/978-94-011-4560-2_9Suche in Google Scholar

[97] S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and stochastics reports 37 (1991), No. 1-2, 61–74.10.1080/17442509108833727Suche in Google Scholar

[98] H. Pham, Continuous-Time Stochastic Control and Optimization with Financial Applications, Vol. 61, Springer Science & Business Media, 2009.10.1007/978-3-540-89500-8Suche in Google Scholar

[99] H. Pham, Feynman–Kac representation of fully nonlinear PDEs and applications, Acta Math. Vietnam. 40 (2015), No. 2, 255–269.10.1007/s40306-015-0128-xSuche in Google Scholar

[100] A. Rasulov, G. Raimova, and M. Mascagni, Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation, Math. Comput. Simulation 80 (2010), No. 6, 1118–1123.10.1016/j.matcom.2009.12.009Suche in Google Scholar

[101] A. Richou, Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition, Stoch. Process. Appl. 122 (2012), No. 9, 3173–3208.10.1016/j.spa.2012.05.015Suche in Google Scholar

[102] A. Richou, Numerical simulation of BSDEs with drivers of quadratic growth, The Annals of Applied Probability 21 (2011), No. 5, 1933–1964.10.1214/10-AAP744Suche in Google Scholar

[103] M. J. Ruijter and C. W. Oosterlee, A Fourier cosine method for an efficient computation of solutions to BSDEs, SIAM J. Sci. Comput. 37 (2015), No. 2, A859–A889.10.1137/130913183Suche in Google Scholar

[104] M. J. Ruijter and C. W. Oosterlee, Numerical Fourier method and second-order Taylor scheme for backward SDEs in finance, Appl. Numer. Math. 103 (2016), 1–26.10.1016/j.apnum.2015.12.003Suche in Google Scholar

[105] A. V. Skorohod, Branching diffusion processes, Teor. Verojatnost. i Primenen. 9 (1964), 492–497.Suche in Google Scholar

[106] L. Teng, A. Lapitckii, and M. Günther, A multi-step scheme based on cubic spline for solving backward stochastic differential equations, Appl. Numer. Math. 150 (2020), 117–138.10.1016/j.apnum.2019.09.016Suche in Google Scholar

[107] N. Touzi, Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE, Vol. 29, Springer Science & Business Media, 2012.10.1007/978-1-4614-4286-8Suche in Google Scholar

[108] P. Turkedjiev, Two algorithms for the discrete time approximation of Markovian backward stochastic differential equations under local conditions, Electron. J. Probab. 20 (2015), no. 50, 49.10.1214/EJP.v20-3022Suche in Google Scholar

[109] X. Warin, Variations on branching methods for non linear PDEs, arXiv:1701.07660, 2017.Suche in Google Scholar

[110] S. Watanabe, On the branching process for Brownian particles with an absorbing boundary, J. Math. Kyoto Univ. 4 (1965), 385–398.10.1215/kjm/1250524667Suche in Google Scholar

[111] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Vol. 43, Springer Science & Business Media, 1999.10.1007/978-1-4612-1466-3Suche in Google Scholar

[112] G. Zhang, M. Gunzburger, and W. Zhao, A sparse-grid method for multi-dimensional backward stochastic differential equations, J. Comput. Math. 31 (2013), No. 3, 221–248.10.4208/jcm.1212-m4014Suche in Google Scholar

[113] J. Zhang, Backward Stochastic Differential Equations. From Linear to Fully Nonlinear Theory, Springer, 2017.10.1007/978-1-4939-7256-2Suche in Google Scholar

[114] J. Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab. 14 (2004), No. 1, 459–488.10.1214/aoap/1075828058Suche in Google Scholar

[115] W. Zhao, L. Chen, and S. Peng, A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comp. 28 (2006), No. 4, 1563–1581.10.1137/05063341XSuche in Google Scholar

[116] W. Zhao, Y. Fu, and T. Zhou, New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comp. 36 (2014), No. 4, A1731–A1751.10.1137/130941274Suche in Google Scholar

[117] W. Zhao, Y. Li, and G. Zhang, A generalized theta-scheme for solving backward stochastic differential equations, Discrete & Continuous Dynamical Systems-B 17 (2012), No. 5, 1585.10.3934/dcdsb.2012.17.1585Suche in Google Scholar

[118] W. Zhao, J. Wang, and S. Peng, Error estimates of the theta-scheme for backward stochastic differential equations, Discrete & Continuous Dynamical Systems-B 12 (2009), No. 4, 905.10.3934/dcdsb.2009.12.905Suche in Google Scholar

[119] W. Zhao, G. Zhang, and L. Ju, A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal. 48 (2010), No. 4, 1369–1394.10.1137/09076979XSuche in Google Scholar