Strong convergence rate of finite difference approximations for stochastic cubic Schrödinger equations

Jianbo Cui, Jialin Hong, Zhihui Liu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

18 Citations (Scopus)

Abstract

In this paper, we derive a strong convergence rate of spatial finite difference approximations for both focusing and defocusing stochastic cubic Schrödinger equations driven by a multiplicative Q-Wiener process. Beyond the uniform boundedness of moments for high order derivatives of the exact solution, the key requirement of our approach is the exponential integrability of both the exact and numerical solutions. By constructing and analyzing a Lyapunov functional and its discrete correspondence, we derive the uniform boundedness of moments for high order derivatives of the exact solution and the first order derivative of the numerical solution, which immediately yields the well-posedness of both the continuous and discrete problems. The latter exponential integrability is obtained through a variant of a criterion given by Cox, Hutzenthaler and Jentzen [arXiv:1309.5595]. As a by-product of this exponential integrability, we prove that the exact and numerical solutions depend continuously on the initial data and obtain a large deviation-type result on the dependence of the noise with first order strong convergence rate.

Original languageEnglish
Pages (from-to)3687-3713
Number of pages27
JournalJournal of Differential Equations
Volume263
Issue number7
DOIs
Publication statusPublished - 5 Oct 2017
Externally publishedYes

Keywords

  • Central difference scheme
  • Continuous dependence
  • Exponential integrability
  • Stochastic cubic Schrödinger equation
  • Strong convergence rate

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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