  In this paper, we analyze a semi-discrete finite volume scheme for the
three-dimensional barotropic compressible Euler equations driven by a
multiplicative Brownian noise. We derive necessary a priori estimates for
numerical approximations, and show that the Young measure generated by the
numerical approximations converge to a dissipative measure--valued martingale
solution to the stochastic compressible Euler system. These solutions are
probabilistically weak in the sense that the driving noise and associated
filtration are integral part of the solution. Moreover, we demonstrate strong
convergence of numerical solutions to the regular solution of the limit systems
at least on the lifespan of the latter, thanks to the weak
(measure-valued)--strong uniqueness principle for the underlying system. To the
best of our knowledge, this is the first attempt to prove the convergence of
numerical approximations for the underlying system.
