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abstract: 'We prove a maximum principle for local solutions of quasilinear stochastic PDEs with obstacle (in short OSPDE). The proofs are based on a version of Itô’s formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.'
address:
- 'Université d’Evry-Val-d’Essonne-FRANCE\'
- ' LUNAM Université, Université du Maine - FRANCE\'
- 'Université d’Evry-Val-d’Essonne -FRANCE\'
author:
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title: Maximum Principle for Quasilinear Stochastic PDEs with Obstacle 
---

Introduction
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In this paper, we consider an obstacle problem for the following parabolic Stochastic PDE (SPDE in short) $$\label{SPDEO}\left\{ \begin{split}&du_t(x)=\partial_i  \left(a_{i,j}(x)\partial_ju_t(x)+g_i(t,x,u_t(x),\nabla
u_t(x))\right)dt+f(t,x,u_t(x),\nabla u_t(x))dt \\&\quad \quad\ \  \
\ \ \ +\sum_{j=1}^{+\infty}h_j(t,x,u_t(x),\nabla
u_t(x))dB^j_t +\nu (t,dx), \\
&u_t\geq S_t \, , \ \ \\ &u_0=\xi\, .\
\end{split}\right.$$ Here, $S$ is the given obstacle, $a$ is a matrix defining a symmetric operator on an open bounded domain ${\ensuremath{\mathcal O}}$, $f,g,h$ are random coefficients.\
In a recent work [@DMZ12] we have proved existence and uniqueness of the solution of equation (\[SPDEO\]) under standard Lipschitz hypotheses and $L^2$-type integrability conditions on the coefficients. Let us recall that the solution is a couple $(u,\nu )$, where $u$ is a process with values in the first order Sobolev space 