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

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.

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

 Citation: • • 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

 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

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
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Tables(2)

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