Finite volume schemes for hyperbolic balance laws with multiplicative noise

We consider finite volume schemes for a scalar stochastic balance law with multiplicative noise. For a class of monotone numerical fluxes we establish the pathwise convergence of a semi-discrete finite volume solution towards a stochastic entropy solution. Main tool is a stochastic version of the compensated compactness approach. The approach relies solely on L p -estimates. It avoids the use of a maximum principle and total-variation estimates. These are typical tools in the deterministic case but are not available for the non-deterministic model. cal results illustrate the analytical findings.