Recently, Professor Wang Wansheng's team from the SHNU School of Mathematics and Physics published a research paper titled Deep Learning Numerical Methods for High Dimensional Quasi Linear PIDEs and Coupled FBSDEs with Jumps in SIAM Journal on Scientific Computing, one of the top international applied mathematics journals. This study proposes a novel deep learning algorithm aimed at efficiently solving high-dimensional quasi linear parabolic integral differential equations (PIDEs) and jump coupled forward backward stochastic differential equations (FBSDEJs). This study breaks through the bottleneck of the curse of dimensionality encountered by traditional numerical methods in high-dimensional problems, and provides practical computational solutions for complex problems in fields such as financial mathematics and stochastic optimal control.
The Deep FBSDE method proposed by Professor Wang Wansheng's team cleverly addresses these challenges. The core idea of this method includes three steps: firstly, using the nonlinear Feynman Kac formula to transform the high-dimensional PIDE problem into a coupled FBSDEJ problem; Secondly, this FBSDEJ problem is regarded as a stochastic control problem, and a pair of deep neural networks are innovatively introduced to approximate the gradient and integral kernel of the solution; Finally, the neural network is trained by minimizing a global loss function related to terminal conditions to obtain the solution of the equation.
Another important contribution of this research work is its rigorous theoretical analysis, which has received funding support from major research programs of the National Natural Science Foundation of China, general projects of the National Natural Science Foundation of China, and the Science and Technology Innovation Action Plan of the Shanghai Municipal Commission of Science and Technology.
