A multiscale stochastic criminal behavior model and the convergence to a piecewise-deterministic-Markov-process limit

Ever since the pioneering human-environment interaction model of criminal behavior [M. B. Short, M. R. DOrsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249-1267] was published, many mathematical agent-based residential burglary models have been proposed. In order to reach an improved balance among model accuracy, analysis simplicity and real-world data fitting tractability, we introduce in this paper a multi-scale hybrid interacting-particle-system model of criminal behavior in a discrete setting. We assume that agents' actions are governed by independent Poisson clocks, while the environment variable evolves on a separate finer discrete spatial-temporal scale. Furthermore, as we refine the second scale to its scaling limit, the hybrid system converges to a piecewise deterministic Markov process (PDMP). Through a martingale approach and infinitesimal generator analysis, we provide a formal derivation for the convergence. Computer simulations of coupled hybrid and PDMP systems both exhibit spatio-temporal aggregates of crime and show excellent agreement between the two, which supports our theoretical derivation of the scaling limit. The methodology and results we develop here indicate a way to establish connections for the proposed crime models.