Deep PDE Solution to BSDE

Abstract

We numerically solve a high dimensional Backward Stochastic Differential Equation (BSDE) by solving the corresponding Partial Differential Equation (PDE) instead. In order to have a good approximation of the gradient of the solution of the PDE, we numerically solve a coupled PDE, consisting of the original semilinear parabolic PDE and the PDEs for its derivatives. We then prove existence and uniqueness of the classical solution of this coupled PDE, and then show how to truncate the unbounded domain to a bounded one, so that the error between the original solution and that of the same coupled PDE but on the bounded domain, is small. We then solve this coupled PDE using neural nets, and proceed to establish a convergence of the numerical solution to the true solution. Finally, we test this on 100 dimensional Allen-Cahn, a nonlinear Black-Scholes and other examples. We also compare our results to the result of solving the BSDE directly.

Keywords: BSDE, PDE, Deep Learning, Deep Galerkin Method, Convergence

JEL Classification: C69