  The flow of nematic liquid crystals can be described by a highly nonlinear
stochastic hydrodynamical model, thus is often influenced by random
fluctuations, such as uncertainty in specifying initial conditions and boundary
conditions. In this article, we consider the $2$-D stochastic nematic liquid
crystals with the velocity field perturbed by affine-linear multiplicative
white noise, with random initial data and random boundary conditions. Our main
objective is to establish the global well-posedness of the stochastic equations
under certain sufficient Malliavin regularity of the initial conditions and the
boundary conditions. The Malliavin calculus techniques play important roles in
proving the global existence of the solutions to the stochastic nematic liquid
crystal models with random initial and random boundary conditions. It should be
pointed out that the global well-posedness is also true when the stochastic
system is perturbed by the noise on the boundary.
