Modelling, simulation and analysis of interacting agent systems is a broad field of research, with existing approaches reaching from traditional informal descriptions of interaction dynamics to more formal, mathematical models. We formulate a continuous-time stochastic agent-based model and define the corresponding Markov jump process. Describing its approximation by ordinary and stochastic differential equations we demonstrate the advantages of an SDE (Stochastic Differential Equation) model for different scenarios of interacting agent systems with medium or large population sizes. In comparison to the ODE (Ordinary Differential Equation) limit model, the SDE gives a higher order approximation of the underlying Markov jump process, both on a pathwise level and regarding the process' moments. In particular, the SDE approach is able to retain metastabilty in the dynamics, which gets lost in a deterministic ODE description, and to capture the distribution of rare and unlikely extreme events.