Deep learning numerical methods for high-dimensional fully nonlinear PIDEs and coupled FBSDEs with jumps
This work addresses computational challenges in high-dimensional jump-diffusion models for researchers in mathematical finance and computational mathematics, representing an incremental improvement over existing deep FBSDE methods.
The authors tackled the problem of solving high-dimensional parabolic integro-differential equations (PIDEs) and forward-backward stochastic differential equations with jumps (FBSDEJs) by proposing a deep learning algorithm that uses neural networks to approximate the gradient and integral kernel. The result is demonstrated through two numerical examples showing the algorithm's efficiency, though no specific performance metrics or comparisons are provided.
We propose a deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) and high-dimensional forward-backward stochastic differential equations with jumps (FBSDEJs), where the jump-diffusion process are derived by a Brownian motion and an independent compensated Poisson random measure. In this novel algorithm, a pair of deep neural networks for the approximations of the gradient and the integral kernel is introduced in a crucial way based on deep FBSDE method. To derive the error estimates for this deep learning algorithm, the convergence of Markovian iteration, the error bound of Euler time discretization, and the simulation error of deep learning algorithm are investigated. Two numerical examples are provided to show the efficiency of this proposed algorithm.