Stability of Regression-Based Monte Carlo Methods for Solving Nonlinear PDEs

The regression-based Monte Carlo methods for backward stochastic differential equations (BSDEs) have been the object of considerable research, particularly for solving nonlinear partial differential equations (PDEs). Unfortunately, such methods often become unstable when implemented with small time steps because the variance of gradient estimates is inversely proportional to the time step (sigma(2)similar to 1/ t). Recently new variance reduction techniques were introduced to address this problem in a paper by the author and Avellaneda. The purpose of this paper is to provide a rigorous justification for these techniques in the context of the discrete-time BSDE scheme of Bouchard and Touzi. We also suggest a new higher-order scheme that makes the variance proportional to the time step (sigma(2)similar to t). These techniques are easy to implement. Numerical examples strongly indicate that they render the regression-based Monte Carlo methods stable for small time steps and thus viable for numerical solution of nonlinear PDEs.(c) 2016 Wiley Periodicals, Inc.