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Deep learning numerical methods for high-dimensional fully nonlinear PIDEs and coupled FBSDEs with jumps. arXiv:2301.12895

Preprint, arXiv:2301.12895 [math.NA] (2023).
Summary: 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.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C20 Probabilistic models, generic numerical methods in probability and statistics
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
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