Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local discontinuous Galerkin method for nonlinear BSPDEs of Neumann boundary conditions with deep backward dynamic programming time-marching

Yixiang Dai, Yunzhang Li, Jing Zhang

TL;DR

The results show the effectiveness and accuracy of the LDG method in tackling BSPDEs with Neumann boundary conditions.

Abstract

This paper aims to present a local discontinuous Galerkin (LDG) method for solving backward stochastic partial differential equations (BSPDEs) with Neumann boundary conditions. We establish the $L^2$-stability and optimal error estimates of the proposed numerical scheme. Two numerical examples are provided to demonstrate the performance of the LDG method, where we incorporate a deep learning algorithm to address the challenge of the curse of dimensionality in backward stochastic differential equations (BSDEs). The results show the effectiveness and accuracy of the LDG method in tackling BSPDEs with Neumann boundary conditions.

Local discontinuous Galerkin method for nonlinear BSPDEs of Neumann boundary conditions with deep backward dynamic programming time-marching

TL;DR

The results show the effectiveness and accuracy of the LDG method in tackling BSPDEs with Neumann boundary conditions.

Abstract

This paper aims to present a local discontinuous Galerkin (LDG) method for solving backward stochastic partial differential equations (BSPDEs) with Neumann boundary conditions. We establish the -stability and optimal error estimates of the proposed numerical scheme. Two numerical examples are provided to demonstrate the performance of the LDG method, where we incorporate a deep learning algorithm to address the challenge of the curse of dimensionality in backward stochastic differential equations (BSDEs). The results show the effectiveness and accuracy of the LDG method in tackling BSPDEs with Neumann boundary conditions.
Paper Structure (14 sections, 6 theorems, 130 equations, 2 figures, 2 tables)

This paper contains 14 sections, 6 theorems, 130 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and definition of solutions to BSPDEs
  4. The LDG method
  5. The LDG method and the main results
  6. Proofs
  7. The well-posedness of the approximating equations
  8. The stability of the LDG method
  9. The optimal estimate error of the LDG method
  10. Numerical Experiments
  11. Deep backward dynamic programming algorithm
  12. Numerical Examples
  13. BSPDEs with globally Lipschitz continuous coefficients
  14. BSPDEs with polynomial growing coefficients

Key Result

Theorem 2.1

Let assumptions $({\mathcal{A}}_{1})-({\mathcal{A}}_{4})$ hold. Then BSPDE BSPDE_in_Qiu_paper with zero Neumann boundary condition admits a unique strong solution $(u,\psi)$ satisfying where $\Gamma^0 := \Gamma(\cdot, 0,0,0)$, and the constant $C$ depends on $L,\ \kappa,\ K$ and $T$.

Figures (2)

  • Figure 1: Accuracy on \ref{['bspde_ex_1']}: $T=0.5$, $k=2$ (left), $k=3$ (right).
  • Figure 2: Accuracy on \ref{['bspde_ex_2']}: $T=0.5$, $k=2$ (left), $k=3$ (right).

Theorems & Definitions (14)

  • Definition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Remark 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.1
  • Theorem 3.3
  • Lemma 4.1
  • proof
  • ...and 4 more