Approximation of a generalized continuous-state branching process with interaction

Ibrahima Dramé (1) , Etienne Pardoux (1)

1 I2M - Institut de Mathématiques de Marseille

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 Z(t)(x) = x + integral(t)(0) f(Z(r)(x))dr + root 2c integral(t)(0)integral(Zrx)(0) W(dr, du) + integral(t)(0)integral(1)(0)integral(zr-x)(0) z (M) over bar (ds, dz, du) + integral(t)(0)integral(infinity)(0)integral(zr-x)(0) z M(ds; dz; du), where W is a space-time white noise on (0;infinity)(2) and (M) over bar (ds; dz; du) = M(ds; dz; du) - ds mu(dz)du, with M being a Poisson random measure on (0;infinity) 3 independent of W; with mean measure ds mu(dz)du, where (1 boolean AND z(2))mu(dz) is a finite measure on (0;infinity).

Mots clés: continuous-state branching processes; interaction; Galton-Watson processes; tightness

Domaines: Probabilités [math.PR]

Ibrahima Dramé, Etienne Pardoux. Approximation of a generalized continuous-state branching process with interaction. Electronic Communications in Probability, 2018, 23, pp.73. ⟨10.1214/18-ECP176⟩. ⟨hal-01974468⟩