High‐Dimensional PDE and BSDE Solvers

This chapter explores the application of deep learning methods to solve high-dimensional partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in finance. Traditional PDE solvers often struggle with the curse of dimensionality, which limits their scalability beyond a few dimensions. Deep learning methods, such as the deep Galerkin method (DGM), deep BSDE solvers and physics-informed neural networks, have emerged as powerful alternatives that can handle high-dimensional problems efficiently. These methods have been applied to various financial contexts, including multi-asset options, American and Bermudan options, and nonlinear PDEs arising from complex model dynamics. The chapter reviews foundational papers from 2016 to 2024 and provides illustrative Python examples. It is structured to discuss DGM, deep BSDEs, projection and martingale solvers, deep path-dependent PDE approaches, physics-informed neural networks, deep backward dynamic programming and deep splitting. The chapter highlights the scalability and flexibility of deep learning frameworks in addressing high-dimensional financial problems.