Regression-based Monte Carlo methods for solving nonlinear PDEs

Regression-based Monte Carlo solution of nonlinear parabolic PDEs is based on the theory of backward stochastic differential equations (BSDEs) which gives a stochastic representation for the solution of the PDE in terms of the solution of the BSDE. A discrete-time approximation of the BSDE is used which involves conditional expectations that must be approximated at each time step, much like in the famous Longstaff-Schwartz method for pricing American options. Unfortunately, when applied to nonlinear PDE problems, these methods often become unstable when implemented with small time steps because the variance of the gradient estimates is inversely proportional to the time step.;The main contribution of this thesis is the introduction of simple control variate techniques in order to reduce the variance of the gradient estimates. We propose a first-order technique that keeps the variance bounded for small time steps. We also suggest a higher-order technique that makes the variance proportional to the time step. These techniques are easy to implement and the numerical examples studied in this thesis 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.;In the last part of this thesis, we take a look at several nonlinear PDE problems and demonstrate with numerical examples how the BSDE method and variance reduction techniques studied in this thesis can be used to solve them. The examples include a semilinear reaction-diffusion type PDE with a finite blow-up time, a fully nonlinear PDE arising from the uncertain volatility model, and a system of nonlinear coupled PDEs arising from the relatively new area of PDEs called mean field games. As the main contribution of this part, we propose a new iterative BSDE method for solving the mean field game system. Our method relies on accurate gradient estimates that are made possible by our variance reduction techniques, and as a Monte Carlo method, it is better suitable for higher dimensions than finite-difference methods.