This paper is about optimization under uncertainty, when the uncertain parameters are modeled through random variables. Contrary to traditionnal robust approaches which deal with a deterministic problem through a worst-case scenario formulation, stochastic algorithms presented introduce the distribution of the random variables modeling the uncertainty. For mono objective problem such methods are today classical, based on the Robbins-Monro algorithm. When several objectives are involved the optimization problem becomes much harder and the few available methods in the literature are based on genetic approach coupled with Monte-Carlo approaches which are numerically very expensive. We present a new algorithm for solving the expectation formulation of stochastic smooth or nonsmooth multiobjective optimization problems. The proposed method is an extension of the classical stochastic gradient algorithm to multi-objective optimization using the properties of a common descent vector. The mean square and the almost sure convergence of the algorithm are proven. The algorithm effciency is illustrated and assessed on an academic example.