We study stochastic optimization problems with decisions truncated by random variables. The technical difficulty is that the optimization problems may not be convex programs due to the truncation. We develop a transformation technique to convert the original non-convex problems to equivalent convex ones. Our transformation allows us to prove the preservation of some desired structural properties, such as convexity, submodularity, and L-natural-convexity, under optimization operations, that are critical for identifying the structures of optimal policies and developing efficient algorithms. We demonstrate the applications of our approach to several important models in inventory control and revenue management.