Numerical Simulations Of Reflected BSDEs And BSDEs With Mean Reflection Using Deep BSDE Method

Deep BSDE Method proposed by[Weinan et al.,2017]and[Han et al.,2018]utilized deep neural networks as function approximators,which alleviated the issue of'curse of dimensionality' of simulating high-dimensional models and so improved the efficiency of the numerical simulations of high-dimensional Marko-vian BSDEs and corresponding parabolic PDEs.In this paper,we extend the Deep BSDE Method to Reflected BSDEs and the newly-proposed BSDEs with mean reflection[Briand et al.,2018].For Reflected BSDEs,two new loss function terms,which become larger when the reflected condition and the Skorohod condition are not satisfied,are added to the origi-nal loss.Similarly,for the BSDEs with mean reflections,the similar loss terms are added to constrain the mean reflected condition and the similar Skorohod condition.Because the Markovian property of BSDEs with mean reflection has not been proved yet,we also proposed two variants of the extended deep BSDE methods that don't rely on the markovian property.The first one let the control term Ztn be a function of both the state of Xtn and the state of the previous time step Xtn-1.The second one induced an architecture similar to recurrent neural networks,making the control term Ztn a function of all previous state.Then,we implemented these extended methods with Pytorch Toolbox.A 100-dimensional reflected BSDE for American option pricing and a 100-dimensional BSDE with mean reflection named "super hedging model with risk constraint" is simulated.Control experiments were implemented with respect to the different number of layers in the neural networks,different terminal conditions,and the different number of time steps in Euler discretization.Especially,for BSDEs with mean reflections,two methods that don't rely on the Markovian assumption mentioned above are implemented and compared.Finally,for the one-dimensional reflected BSDEs and BSDEs with mean reflec-tion,the comparison between the traditional numerical method based on binomial trees and the extended Deep BSDE Method is given in order to test the accuracy of the proposed methods.