# 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
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##### References:
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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