Approximation of a generalized continuous-state branching process with interaction
Résumé
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).