In the theory of Partial Differential Equations, the maximum principle plays an important role and there is a huge literature on this subject. It permits one to study the local behavior of solutions of PDE since it gives a relation between the bound of the

solution on the boundary and a bound on the whole domain. The maximum principle for deterministic quasilinear parabolic equations was proved by

Aronson -Serrin.

In two recent papers, we have proved a maximum principle for quasilinear stochastic PDE's. The main step is to get an estimate of the uniform norm of the solution with null boundary Dirichlet condition. In these papers, we have considered a very general case and used sophisticated spaces and tools, so that the links between the deterministic proof given by Aronson-Serrin and our proof in the stochastic case is not clear. In this talk, we explain how one can estimate  the uniform norm of the solution in a simpler case in order to point out the interest of Moser's iteration method.