We develop a primal-dual convex minimization framework to solve a class of stochastic convex three-composite problem with a linear operator. We consider the cases where the problem is both convex and strongly convex and analyze the convergence of the proposed algorithm in both cases. In addition, we extend the proposed framework to deal with additional constraint sets and multiple non-smooth terms. We provide numerical evidence on graph-guided sparse logistic regression, fused lasso and overlapped group lasso, to demonstrate the superiority of our approach to the state-of-the-art.