Deep learning-based schemes for solving high-dimensional nonlinear BSDEs and their uncertainty quantification

Backward stochastic differential equations (BSDEs) are widely used in fields such as finance, economics, and physics due to their connection to partial differential equations via the nonlinear Feynman-Kac formula. In option pricing, the solution pair of a BSDE represents both the price and the hedging strategy. While the BSDE under the Black-Scholes framework is linear and admits a closed-form solution, most practical problems involve nonlinear, high-dimensional BSDEs that require advanced numerical methods. Classical schemes often suffer from the curse of dimensionality, prompting the development of deep learning-based approaches. This thesis presents efficient deep learning schemes for solving high-dimensional nonlinear BSDEs and addresses the uncertainty associated with such models. The first part introduces a novel forward scheme that improves stability and accuracy across time steps, demonstrated through various financial examples. The second part develops a differential deep learning framework, training neural networks on both values and derivatives to enhance gradient estimation—crucial for hedging applications. We demonstrate, both theoretically and numerically, that this approach is more efficient than other contemporary deep learning-based schemes. The third part focuses on uncertainty quantification (UQ), proposing the first UQ model tailored to deep BSDE solvers. This model estimates uncertainty from a single run, offering a practical and computationally efficient tool for reliability assessment. Overall, the thesis advances the robustness, accuracy, and interpretability of deep learning methods for solving high-dimensional nonlinear BSDEs.