Stochastic parabolic equations in Musielak-Orlicz spaces with discontinuous in time N-function

We consider a stochastic parabolic partial differential equation with Dirichlet boundary conditions, multiplicative stochastic noise, and a monotone parabolic operator A. The growth and coercivity of A is controlled by a general N-function M, which depends on time, and spatial variable, but we do not assume any regularity with respect to the former. We show the existence of weak solutions to such system. As auxiliary result, we also provide the proof for the It\^{o}'s formula in Orlicz spaces. This general result applies to the ones studied in the literature, such as p(t, x)-Laplacian and double phase problems.