  Here is investigated the bilinear optimal control problem of quantum
mechanical systems with final observation governed by a stochastic nonlinear
Schr\"odinger equation perturbed by a linear multiplicative Wiener process. The
existence of an open loop optimal control and first order Lagrange optimality
conditions are derived, via Skorohod's representation theorem, Ekeland's
variational principle and the existence for the linearized dual backward
stochastic equation. Moreover, our approach in particular applies to the
deterministic case.
