# American Institute of Mathematical Sciences

December  2021, 6(4): 391-408. doi: 10.3934/puqr.2021019

## Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficients

 1 Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  August 11, 2021 Accepted  December 06, 2021 Published  December 2021

Fund Project: This research has been supported by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling, and Simulation (Grant No. EP/S023925/1).

This paper is dedicated to solving high-dimensional coupled FBSDEs with non-Lipschitz diffusion coefficients numerically. Under mild conditions, we provided a posterior estimate of the numerical solution that holds for any time duration. This posterior estimate validates the convergence of the recently proposed Deep BSDE method. In addition, we developed a numerical scheme based on the Deep BSDE method and presented numerical examples in financial markets to demonstrate the high performance.

Citation: Yifan Jiang, Jinfeng Li. Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficients. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 391-408. doi: 10.3934/puqr.2021019
##### References:
 [1] Aurelien Alfonsi, Affine Diffusions and Related Processes: Simulation, Theory and Applications, Bocconi University Press, 2015. [2] John C Cox, Jonathan E Ingersoll Jr and Stephen A Ross, A theory of the term structure of interest rates, In: Sudipto Bhattacharya and George M Constantinides(eds.), Theory of Valuation, 2nd ed., World Scientific, 2005, 129–164. [3] Jaksa Cvitanic and Jianfeng Zhang, The steepest descent method for forward-backward SDEs, Electronic Journal of Probability, 2005, 10: 1468−1495. [4] Griselda Deelstra and Freddy Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term, Applied Stochastic Models and Data Analysis, 1998, 14(1): 77−84. doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2. [5] Steffen Dereich, Andreas Neuenkirch and Lukasz Szpruch, An Euler-type method for the strong approximation of the cox-ingersoll-ross process, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012, 468(2140): 1105−1115. [6] Weinan E, Jiequn Han and Arnulf Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Communications in Mathematics and Statistics, 2017, 5(4): 349−380. doi: 10.1007/s40304-017-0117-6. [7] Jiequn Han and Jihao Long, Convergence of the deep BSDE method for coupled FBSDEs, Probability, Uncertainty and Quantitative Risk, 2020, 5: 5, doi: 10.1186/s41546-020-00047-w. [8] Shaolin Ji, Shige Peng, Ying Peng and Xichuan Zhang, Three algorithms for solving high-dimensional fully-coupled FBSDEs through deep learning, IEEE Intelligent Systems, 2020, 35(3): 71−84. doi: 10.1109/MIS.2020.2971597. [9] Jin Ma and Jiongmin Yong, Forward-Backward Stochastic Differential Equations and their Applications, Springer, Berlin, 1999. [10] Jin Ma, Philip Protter and Jiongmin Yong, Solving forward-backward stochastic differential equations explicitly-a four step scheme, Probability Theory and Related Fields, 1994, 98(3): 339−359. doi: 10.1007/BF01192258. [11] Etienne Pardoux and Shanjian Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probability Theory and Related Fields, 1999, 114(2): 123−150. doi: 10.1007/s004409970001. [12] Shige Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastic Reports, 1991, 37(1−2): 61−74. doi: 10.1080/17442509108833727. [13] Shige Peng and Zhen Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM Journal on Control and Optimization, 1999, 37(3): 825−843. doi: 10.1137/S0363012996313549. [14] Toshio Yamada and Shinzo Watanabe, On the uniqueness of solutions of stochastic differential equations, Journal of Mathematics of Kyoto University, 1971, 11(1): 155−167. [15] Shinzo Watanabe and Toshio Yamada, On the uniqueness of solutions of stochastic differential equations Ⅱ, Journal of Mathematics of Kyoto University, 1971, 11(3): 553−563. [16] Jianfeng Zhang, Backward stochastic differential equations, In: Backward Stochastic Differential Equations, Probability Theory and Stochastic Modelling, Springer, New York, 2017, 86: 79–99.

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##### References:
 [1] Aurelien Alfonsi, Affine Diffusions and Related Processes: Simulation, Theory and Applications, Bocconi University Press, 2015. [2] John C Cox, Jonathan E Ingersoll Jr and Stephen A Ross, A theory of the term structure of interest rates, In: Sudipto Bhattacharya and George M Constantinides(eds.), Theory of Valuation, 2nd ed., World Scientific, 2005, 129–164. [3] Jaksa Cvitanic and Jianfeng Zhang, The steepest descent method for forward-backward SDEs, Electronic Journal of Probability, 2005, 10: 1468−1495. [4] Griselda Deelstra and Freddy Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term, Applied Stochastic Models and Data Analysis, 1998, 14(1): 77−84. doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2. [5] Steffen Dereich, Andreas Neuenkirch and Lukasz Szpruch, An Euler-type method for the strong approximation of the cox-ingersoll-ross process, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012, 468(2140): 1105−1115. [6] Weinan E, Jiequn Han and Arnulf Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Communications in Mathematics and Statistics, 2017, 5(4): 349−380. doi: 10.1007/s40304-017-0117-6. [7] Jiequn Han and Jihao Long, Convergence of the deep BSDE method for coupled FBSDEs, Probability, Uncertainty and Quantitative Risk, 2020, 5: 5, doi: 10.1186/s41546-020-00047-w. [8] Shaolin Ji, Shige Peng, Ying Peng and Xichuan Zhang, Three algorithms for solving high-dimensional fully-coupled FBSDEs through deep learning, IEEE Intelligent Systems, 2020, 35(3): 71−84. doi: 10.1109/MIS.2020.2971597. [9] Jin Ma and Jiongmin Yong, Forward-Backward Stochastic Differential Equations and their Applications, Springer, Berlin, 1999. [10] Jin Ma, Philip Protter and Jiongmin Yong, Solving forward-backward stochastic differential equations explicitly-a four step scheme, Probability Theory and Related Fields, 1994, 98(3): 339−359. doi: 10.1007/BF01192258. [11] Etienne Pardoux and Shanjian Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probability Theory and Related Fields, 1999, 114(2): 123−150. doi: 10.1007/s004409970001. [12] Shige Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastic Reports, 1991, 37(1−2): 61−74. doi: 10.1080/17442509108833727. [13] Shige Peng and Zhen Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM Journal on Control and Optimization, 1999, 37(3): 825−843. doi: 10.1137/S0363012996313549. [14] Toshio Yamada and Shinzo Watanabe, On the uniqueness of solutions of stochastic differential equations, Journal of Mathematics of Kyoto University, 1971, 11(1): 155−167. [15] Shinzo Watanabe and Toshio Yamada, On the uniqueness of solutions of stochastic differential equations Ⅱ, Journal of Mathematics of Kyoto University, 1971, 11(3): 553−563. [16] Jianfeng Zhang, Backward stochastic differential equations, In: Backward Stochastic Differential Equations, Probability Theory and Stochastic Modelling, Springer, New York, 2017, 86: 79–99.
Relative error of the bond price (left) and the loss (right) against the number of the iteration steps
Relative error of the bond price under multi-dimensional CIR model (left) and the loss (right) against the number of the iteration steps
Numerical simulation of CIR bond
 Step Mean of $Y_{0}$ Standard deviation of $Y_{0}$ Mean of loss Standard deviation of loss 500 0.4643 9.58E-2 8.46E-2 1.27E-1 1000 0.4136 2.55E-2 7.13E-3 1.23E-2 2000 0.3972 1.21E-3 8.47E-4 6.23E-4 3000 0.3972 3.69E-4 5.80E-4 3.20E-4
 Step Mean of $Y_{0}$ Standard deviation of $Y_{0}$ Mean of loss Standard deviation of loss 500 0.4643 9.58E-2 8.46E-2 1.27E-1 1000 0.4136 2.55E-2 7.13E-3 1.23E-2 2000 0.3972 1.21E-3 8.47E-4 6.23E-4 3000 0.3972 3.69E-4 5.80E-4 3.20E-4
Numerical simulation of multi-dimensional CIR bond
 Step Mean of $Y_{0}$ Standard deviation of $Y_{0}$ Mean of loss Standard deviation of loss 500 0.3773 8.77E-2 1.15E-1 1.66E-1 1000 0.3228 2.03E-2 5.51E-3 8.33E-3 2000 0.3100 1.63E-3 4.50E-4 1.12E-4 3000 0.3095 8.28E-4 3.89E-4 7.20E-5
 Step Mean of $Y_{0}$ Standard deviation of $Y_{0}$ Mean of loss Standard deviation of loss 500 0.3773 8.77E-2 1.15E-1 1.66E-1 1000 0.3228 2.03E-2 5.51E-3 8.33E-3 2000 0.3100 1.63E-3 4.50E-4 1.12E-4 3000 0.3095 8.28E-4 3.89E-4 7.20E-5
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