Gillespie stochastic simulation is used extensively to investigate stochastic phenomena in many fields, ranging from chemistry to biology to ecology. The inverse problem, however, has remained largely unsolved: How to reconstruct the underlying reactions de novo from sparse observations. A key challenge is that often only aggregate concentrations, proportional to the population numbers, are observable intermittently. We discovered that under specific assumptions, the set of relative population updates in phase space forms a convex polytope whose vertices are indicative of the dominant underlying reactions. We demonstrate the validity of this simple principle by reconstructing stochastic models (reaction structure plus propensities) from a variety of simulated and experimental systems, where hundreds and even thousands of reactions may be occurring in between observations. In some cases, the inferred models provide mechanistic insight. This principle can lead to the understanding of a broad range of phenomena, from molecular biology to population ecology.