金融工程研究中心学术报告：A Willow-Tree-Based Numerical Framework for Forward-Backward Stochastic Differential Equations

Forward–backward stochastic differential equations (FBSDEs) provide a powerful probabilistic framework for nonlinear partial differential equations, stochastic control, and mathematical finance. However, their numerical solution is challenging because backward discretization schemes require repeated evaluations of conditional expectations. This paper develops a willow-tree-based numerical method for decoupled Markovian FBSDEs. The forward diffusion is approximated by a finite-state willow tree constructed through Johnson transformations and moment matching, allowing non-Gaussian distributional features to be captured with a relatively small number of nodes. Conditional expectations in the backward recursion are then evaluated directly by transition probability matrices, avoiding nested Monte Carlo simulation. In addition, we establish a convergence framework for the proposed method by separating the total error into time discretization error, Johnson transformation error, and willow spatial approximation error. Although the analysis and numerical experiments focus on decoupled FBSDEs, the proposed willow-tree construction is based on finite-state conditional expectation approximation and is therefore naturally applicable to second-order backward stochastic differential equations (2BSDEs) and other stochastic dynamic systems requiring repeated conditional expectations. Numerical experiments illustrate that the proposed method achieves accurate and stable approximations while retaining computational efficiency. The results demonstrate that willow trees offer a flexible and effective deterministic framework for conditional expectation approximation in FBSDEs, 2BSDEs, and related stochastic dynamic models. Dr. Xu is currently an associate professor in the Department of Mathematics at Toronto Metropolitan University. Before joining TMU, he worked as an associate professor at Tongji University, Shanghai, China, and as a visiting professor in Math Faculty at University of Waterloo in 2014 and 2015. Through these teaching and researching experiences, he obtained excellent knowledge and understanding in financial theories and models, which results in over 30 publications in reviewed journals and two co-authored books. In addition, Dr. Xu also has a good understanding of industry best practice and regulatory requirements, which he obtained through being a co-founder and director of research and development division in a start-up company, and collaborating with many financial institutions in China.