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In this paper, we consider black-box problems where the analytic forms of the objective functions are not available, and the values can only be estimated by output responses from computationally expensive simulations. We apply the sample average approximation method to multi-objective stochastic optimization problems and prove the convergence properties of the method under a set of fairly general regularity conditions. We develop a new algorithm, based on the trust-region method, for approximating the Pareto front of a bi-objective stochastic optimization problem. At each iteration of the proposed algorithm, a trust region is identified and quadratic approximate functions for the expected objective functions are built using sample average values. To determine non-dominated solutions in the trust region, a single-objective optimization problem is constructed based on the approximate objective functions. After updating the set of non-dominated solutions, a new trust region around the most isolated point is determined to explore areas that have not been visited. The numerical results show that our proposed method is feasible, and the performance can be significantly improved with an appropriate sample size.