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A stochastic-statistical residential burglary model with independent Poisson clocks

Part of: Equations of mathematical physics and other areas of application Stochastic processes

Published online by Cambridge University Press:  05 February 2020

CHUNTIAN WANG Open the ORCID record for CHUNTIAN WANG [Opens in a new window] ,
YUAN ZHANG Open the ORCID record for YUAN ZHANG [Opens in a new window] ,
ANDREA L. BERTOZZI Open the ORCID record for ANDREA L. BERTOZZI [Opens in a new window]  and
MARTIN B. SHORT Open the ORCID record for MARTIN B. SHORT [Opens in a new window]
CHUNTIAN WANG
Affiliation:
Department of Mathematics, The University of Alabama, Tuscaloosa, AL35487, USA email: cwang27@ua.edu
YUAN ZHANG
Affiliation:
School of Mathematical Sciences, Peking University, Beijing, China, 100871 email: zhangyuan@math.pku.edu.cn
ANDREA L. BERTOZZI
Affiliation:
Departments of Mathematics and Mechanical and Aerospace Engineering University of California, Los Angeles, Los Angeles, CA90095, USA email: bertozzi@ucla.edu
MARTIN B. SHORT
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA30332, USA email: mbshort@math.gatech.edu
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Abstract

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Residential burglary is a social problem in every major urban area. As such, progress has been to develop quantitative, informative and applicable models for this type of crime: (1) the Deterministic-time-step (DTS) model [Short, D’Orsogna, Pasour, Tita, Brantingham, Bertozzi & Chayes (2008) Math. Models Methods Appl. Sci.18, 1249–1267], a pioneering agent-based statistical model of residential burglary criminal behaviour, with deterministic time steps assumed for arrivals of events in which the residential burglary aggregate pattern formation is quantitatively studied for the first time; (2) the SSRB model (agent-based stochastic-statistical model of residential burglary crime) [Wang, Zhang, Bertozzi & Short (2019) Active Particles, Vol. 2, Springer Nature Switzerland AG, in press], in which the stochastic component of the model is theoretically analysed by introduction of a Poisson clock with time steps turned into exponentially distributed random variables. To incorporate independence of agents, in this work, five types of Poisson clocks are taken into consideration. Poisson clocks (I), (II) and (III) govern independent agent actions of burglary behaviour, and Poisson clocks (IV) and (V) govern interactions of agents with the environment. All the Poisson clocks are independent. The time increments are independently exponentially distributed, which are more suitable to model individual actions of agents. Applying the method of merging and splitting of Poisson processes, the independent Poisson clocks can be treated as one, making the analysis and simulation similar to the SSRB model. A Martingale formula is derived, which consists of a deterministic and a stochastic component. A scaling property of the Martingale formulation with varying burglar population is found, which provides a theory to the finite size effects. The theory is supported by quantitative numerical simulations using the pattern-formation quantifying statistics. Results presented here will be transformative for both elements of application and analysis of agent-based models for residential burglary or in other domains.

Keywords

Applications of continuous-time Markov processes on discrete state spacesapplications of stochastic analysis (to partial differential equations (PDEs), etc.)PDEs in connection with game theoryeconomicssocial and behavioural sciences

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 35Q91: PDEs in connection with game theory, economics, social and behavioral sciences
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 1 , February 2021 , pp. 32 - 58
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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