  We study in this article the stochastic Zakharov-Kuznetsov equation driven by
a multiplicative noise. We establish, in space dimensions two and three the
global existence of martingale solutions, and in space dimension two the global
pathwise uniqueness and the existence of pathwise solutions. New methods are
employed in the passage to the limit on a special type of boundary conditions
and in the verification of the pathwise uniqueness of martingale solutions with
a lack of regularity, where both difficulties arise due to the partly
hyperbolic feature of the model.
