Well-posedness for anticipated backward stochastic Schrödinger equations

Abstract

AbstractIn this paper, we consider the well-posedness for the anticipated backward stochastic Schrödinger equation in a bounded domain or the whole space Rd, 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 equationanticipated backward stochastic differential equationswell-posednessGalerkin's approximation2010 Mathematics Subject Classifications: 60H0560H1560G99 AcknowledgmentsThe 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 statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work is supported by NSFC [grant numbers 11471190 and 11971260].