Weak approximations of solutions of a first order hyperbolic stochastic partial differential equation

The present article focuses on the use of difference methods in order to approximate the solutions of stochastic partial differential equations of Itô type, in particular hyperbolic equations. We develop the main notions of deterministic difference methods, i.e. convergence, consistency and stability for the stochastic case. We prove a stochastic version of Lax-Richtmyer theorem giving the existence of a weak convergent subsequence of the approximating scheme if the scheme is both consistent and stable.