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In this paper, we study the stochastic version of lot-sizing problems with inventory bounds and order capacities. Customer demands, inventory bounds, and costs are subject to uncertainty and dependent with each other throughout the finite planning horizon. Two models in stochastic programming are developed: the first one has inventory-bound constraints, and the second one has both inventory-bound and order-capacity constraints. We explore structural properties of the two models and develop O(n²) and O(n²Tlogn) dynamic programming algorithms for them, respectively. Our model also generalizes the deterministic lot-sizing problem with inventory bounds. For some cases, when applied to the deterministic versions, our algorithms outperform existing deterministic algorithms.