2016/06/02 by Max-Olivier Hongler, Hongler, Max-Olivier, Roger Filliger +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60G20 #82C31 #Cold Atom Physics and Bose-Einstein Condensates #FOS: Physical sciences #Mathematical Physics (math-ph) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1606.00809
openalex publication_date 2016/06/02 · openalex created_date 2022/09/04 · openalex updated_date 2026/07/28
We study some linear and nonlinear shot noise models where the jumps are\ndrawn from a compound Poisson process with jump sizes following an Erlang-m\ndistribution. We show that the associated Master equation can be written as a\nspatial m^ rm th order partial differential equation without integral\nterm. This differential form is valid for state-dependent Poisson rates and we\nuse it to characterize, via a mean-field approach, the collective dynamics of a\nlarge population of pure jump processes interacting via their Poisson rates. We\nexplicitly show that for an appropriate class of interactions, the speed of a\ntight collective traveling wave behavior can be triggered by the jump size\nparameter m. As a second application we consider an exceptional class of\nstochastic differential equations with nonlinear drift, Poisson shot noise and\nan additional White Gaussian Noise term, for which explicit solutions to the\nassociated Master equation are derived.\n