The well-posedness of the stochastic nonlinear Schrödinger equations in $H^2(R^d)$

The Cauchy problem for the stochastic nonlinear Schrödinger equation with multiplicative noise is considered where the nonlinear term is of power type and the noise coefficients are purely imaginary numbers. The main purpose of this paper is to construct classical solutions in $H^2(\mathbb{R}^d)$ for the problem. The techniques of Kato [21, 22] work well in overcoming smoothness problems even for the stochastic equations.