Well-posedness for anticipated backward stochastic Schrödinger equations

Abstract

In this paper, we consider the well-posedness for the anticipated backward stochastic Schrödinger equation in a bounded domain or the whole space ${\mathbb{R}}^d$, which is associated to a stochastic control problem of nonlinear Schrödinger equations with time delay effect. The approach to establish the existence and uniqueness of adapted solutions are mainly based on the complex Itô's formula, the Galerkin's approximation method and the martingale representation theorem.

Keywords:

• Stochastic Schrödinger equation
• anticipated backward stochastic differential equations
• well-posedness
• Galerkin's approximation

2010 Mathematics Subject Classifications:

• 60H05
• 60H15
• 60G99

Acknowledgments

The authors would like to thank the referees for a careful reading of the manuscript and for a number of useful comments and suggestions for improving the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by NSFC [grant numbers 11471190 and 11971260].