  In this paper, we present a backward deep BSDE method applied to Forward
Backward Stochastic Differential Equations (FBSDE) with given terminal
condition at maturity that time-steps the BSDE backwards. We present an
application of this method to a nonlinear pricing problem - the differential
rates problem. To time-step the BSDE backward, one needs to solve a nonlinear
problem. For the differential rates problem, we derive an exact solution of
this time-step problem and a Taylor-based approximation. Previously backward
deep BSDE methods only treated zero or linear generators. While a Taylor
approach for nonlinear generators was previously mentioned, it had not been
implemented or applied, while we apply our method to nonlinear generators and
derive details and present results. Likewise, previously backward deep BSDE
methods were presented for fixed initial risk factor values $X_0$ only, while
we present a version with random $X_0$ and a version that learns portfolio
values at intermediate times as well. The method is able to solve nonlinear
FBSDE problems in high dimensions.
