Global solution for the stochastic nonlinear Schrödinger system with quadratic interaction in four dimensions

We discuss the global existence of solutions to a system of stochastic Schrödinger equations with multiplicative noise. Our setting of the quadratic nonlinear terms in dimension 4 is $L^2$-critical. We treat the solutions under the ground state. We estimate the time derivative of the quantity of energy by using the cancellation of the cubic terms in the spatial derivative of the solution.