  We prove the existence of nonnegative martingale solutions to a class of
stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension
driven thin-film flow influenced by thermal noise. The construction applies to
a range of mobilites including the cubic one which occurs under the assumption
of a no-slip condition at the liquid-solid interface. Since their introduction
more than 15 years ago, by Davidovitch, Moro, and Stone and by Gr\"un, Mecke,
and Rauscher, the existence of solutions to stochastic thin-film equations for
cubic mobilities has been an open problem, even in the case of sufficiently
regular noise. Our proof of global-in-time solutions relies on a careful
combination of entropy and energy estimates in conjunction with a tailor-made
approximation procedure to control the formation of shocks caused by the
nonlinear stochastic scalar conservation law structure of the noise.
