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.

show all references

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.

Figure 1.  Relative error of the bond price (left) and the loss (right) against the number of the iteration steps
Figure 2.  Relative error of the bond price under multi-dimensional CIR model (left) and the loss (right) against the number of the iteration steps
Table 1.  Numerical simulation of CIR bond
StepMean of $Y_{0}$Standard deviation of $Y_{0}$Mean of lossStandard 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
StepMean of $Y_{0}$Standard deviation of $Y_{0}$Mean of lossStandard 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
Table 2.  Numerical simulation of multi-dimensional CIR bond
StepMean of $Y_{0}$Standard deviation of $Y_{0}$Mean of lossStandard 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
StepMean of $Y_{0}$Standard deviation of $Y_{0}$Mean of lossStandard 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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