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

Yifan Jiang; Jinfeng Li

doi:10.3934/puqr.2021019

Article Contents

2021, Volume 6, Issue 4: 391-408. Doi: 10.3934/puqr.2021019

This issue Previous Article Extended conditional G-expectations and related stopping times Next Article On the laws of the iterated logarithm under sub-linear expectations

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

Yifan Jiang^1, and
Jinfeng Li^2, ,

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

Author Bio: E-mail: yifan.jiang@maths.ox.ac.uk; E-mail: lijinfeng@fudan.edu.cn

Received: August 11, 2021

Accepted: December 06, 2021

Published: December 2021

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).

Abstract Full Text(HTML) Figure(2) / Table(2) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 60H10, 60H30, 60H35.

Citation:

FullText(HTML)

Figure 1. Relative error of the bond price (left) and the loss (right) against the number of the iteration steps

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

Table 1. 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

| Show Table

DownLoad: CSV

Table 2. 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

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.