In the framework of the dynamical evolution of the chemical reactions of the sulphur dioxide with the surface of calcium carbonate stones in the process of the degradation of the cultural heritage, starting from a well known deterministic mathematical model, in order to better describe the high variability of the external sulphur dioxide concentration we introduce a suitable stochastic dynamical boundary condition. As boundary condition we take a Jacobi process, solution to a Brownian motion driven stochastic differential equation. We discuss both the mathematical problems arising from considering a lower regular boundary condition and in particular the global existence and (pathwise) uniqueness of the reaction diffusion system coupled with this stochastic boundary condition. The proof relies on a splitting strategy, which allows to deal with the low regularity of the boundary condition. A discretization scheme based on the same splitting is proposed and some numerical samples are shown.