Global solutions of stochastic nonlinear Schrödinger system with quadratic interaction

The time-global existence of solutions to a system of stochastic Schrödinger equations with multiplicative noise and the quadratic nonlinear terms are discussed in this paper. The same system in the deterministic treatment was studied in [18] where the mass and energy are conserved. In our stochastic situation, those are not conserved and which causes several difficulties in the arguments of composing a-priori estimate.