Figure 1.
Relative error of the bond price (left) and the loss (right) against the number of the iteration steps
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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 |
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.
Mathematics Subject Classification:
60H10, 60H30, 60H35.
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
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References:
| [1] |
Aurelien Alfonsi, Affine Diffusions and Related Processes: Simulation, Theory and Applications, Bocconi University Press, 2015. Google Scholar |
| [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. Google Scholar |
| [3] |
Jaksa Cvitanic and Jianfeng Zhang, The steepest descent method for forward-backward SDEs, Electronic Journal of Probability, 2005, 10: 1468−1495. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [16] |
Jianfeng Zhang, Backward stochastic differential equations, In: Backward Stochastic Differential Equations, Probability Theory and Stochastic Modelling, Springer, New York, 2017, 86: 79–99. Google Scholar |
show all references
References:
| [1] |
Aurelien Alfonsi, Affine Diffusions and Related Processes: Simulation, Theory and Applications, Bocconi University Press, 2015. Google Scholar |
| [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. Google Scholar |
| [3] |
Jaksa Cvitanic and Jianfeng Zhang, The steepest descent method for forward-backward SDEs, Electronic Journal of Probability, 2005, 10: 1468−1495. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [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. Google Scholar |
| [16] |
Jianfeng Zhang, Backward stochastic differential equations, In: Backward Stochastic Differential Equations, Probability Theory and Stochastic Modelling, Springer, New York, 2017, 86: 79–99. Google Scholar |
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
| Step | Mean of | Standard deviation of | 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 | Standard deviation of | 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 |
Table 2.
Numerical simulation of multi-dimensional CIR bond
| Step | Mean of | Standard deviation of | 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 | Standard deviation of | 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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