Backward stochastic differential equations (BSDE) are a fundamental tool in the mathematical modelling of financial problems. Through the famous nonlinear extensions to the Feynman-Kac formula, they do not merely provide a stochastic representation of the solution to large classes of partial differential equations such as pricing- or Hamilton- Jacobi-Bellman equations, but also include sensitivities, corresponding to derivatives of the solution, which are crucial in many financial mathematical applications. Henceforth, they simultaneously represent option pricing and hedging problems, and form a natural framework for the numerical treatment of stochastic optimal control.

The main challenge in the numerical approximation of such equations is the computation of conditional expectations over potentially high-dimensional spaces. In classical settings, where the dimensionality of the underlying randomness is moderate, many approaches have been proposed in the literature. However, for high-dimensional problems, one has to resort toMonte Carlo methods. In recent years, a new class of regression Monte Carlo methods has arisen in the literature, so called deep BSDE methods, which practically approximate the solution of BSDEs in a neural network regression Monte Carlo framework, after forming a suitable loss function motivated either by stochastic optimal control or the martingale representation theorem. These classes of methods can roughly be divided into two main categories. Forward methods, where the solution of the associated backward SDE is simultaneously optimized in a global optimization, minimizing a loss function stemming from a stochastic target problem reformulation. Alternatively, backward methods have been investigated, where the numerical resolution of the equation is decomposed into smaller sub-optimizations corresponding to a discrete set of points in a suitable time discretization. These methods enabled the numerical treatment of longstanding open challenges, such as the pricing and deltahedging of multi-asset financial options up to d = 100 risk factors and beyond.

The goal of this thesis is to analyze such modern machine learning based numerical methods, and apply them in the financial mathematical context. We propose numerical extensions of these methods in high-dimensional frameworks, analyze their convergence properties in discrete time, and investigate their robustness and accuracy in practical applications such as hedging and stochastic optimal control. Ourmain contributions in each chapter can be summarized as follows…