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2018 Approximation of a generalized continuous-state branching process with interaction
Ibrahima Dramé, Étienne Pardoux
Electron. Commun. Probab. 23: 1-14 (2018). DOI: 10.1214/18-ECP176

Abstract

In this work, we consider a continuous–time branching process with interaction where the birth and death rates are non linear functions of the population size. We prove that after a proper renormalization our model converges to a generalized continuous state branching process solution of the SDE \[\begin{aligned} Z_t^x=& x + \int _{0}^{t} f(Z_r^x) dr + \sqrt{2c} \int _{0}^{t} \int _{0}^{Z_{r}^x }W(dr,du) + \int _{0}^{t}\int _{0}^{1}\int _{0}^{Z_{r^-}^x}z \ \overline{M} (ds, dz, du)\\ &+ \int _{0}^{t}\int _{1}^{\infty }\int _{0}^{Z_{r^-}^x}z \ M(ds, dz, du), \end{aligned} \] where $W$ is a space-time white noise on $(0,\infty )^2$ and $\overline{M} (ds, dz, du)= M(ds, dz, du)- ds \mu (dz) du$, with $M$ being a Poisson random measure on $(0,\infty )^3$ independent of $W,$ with mean measure $ds\mu (dz)du$, where $(1\wedge z^2)\mu (dz)$ is a finite measure on $(0, \infty )$.

Citation

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Ibrahima Dramé. Étienne Pardoux. "Approximation of a generalized continuous-state branching process with interaction." Electron. Commun. Probab. 23 1 - 14, 2018. https://doi.org/10.1214/18-ECP176

Information

Received: 25 May 2018; Accepted: 2 October 2018; Published: 2018
First available in Project Euclid: 17 October 2018

zbMATH: 1401.60159
MathSciNet: MR3866046
Digital Object Identifier: 10.1214/18-ECP176

Subjects:
Primary: 60F17, 60J80, 92D25

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