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

• URL: http://arxiv.org/abs/2301.12895v1
• Date: Mon, 30 Jan 2023 13:55:42 GMT
• Title: Deep learning numerical methods for high-dimensional fully nonlinear PIDEs and coupled FBSDEs with jumps
• Authors: Wansheng Wang, Jie Wang, Jinping Li, Feifei Gao, Yi Fu
• Abstract summary: We propose a deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) The jump-diffusion process are derived by a Brownian motion and an independent compensated Poisson random measure. To derive the error estimates for this deep learning algorithm, the convergence of Markovian, the error bound of Euler time discretization, and the simulation error of deep learning algorithm are investigated.
• Score: 26.28912742740653
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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