A stability result for solutions of stochastic equations driven by point processes
We consider a sequence \((X^n)_n\) of processes which satisfy stochastic differential equations of the following form: \[ dX^n_t = \int_{\{|u |\leq 1\}} f^n \bigl( X^n_\bullet (\omega), s,u \bigr) q^n (ds,du) + \int_{\{|u |> 1\}} f^n \bigl( X^n_\bullet (\omega), s,u \bigr) p^n (ds,du), \] where \(p^n\) is a stochastic point process, i.e., an integer-valued random measure \(p^n (\omega, ds,du)\) (defined on a given stochastic basis) and \(q^n\) stands for the martingale measure \(p^n - \lambda^n\), \(\lambda^n\) denoting the compensator of \(p^n\); the coefficients \(f^n (X^n_\bullet (\omega), t,u)\), for every couple \((t,u)\), may depend on the whole past of the solution-process before time \(t\). For the processes \((X^n)_n\) we prove a stability theorem in the sense of weak convergence of their laws \((\Lambda^n)_n\) on the Skorokhod space. In fact, we state sufficient conditions under which, as \(n\) tends to infinity, \(\Lambda^n\) converges narrowly (or weakly) to the law of the solution \(X\) of the following equation: \[ dX_t = \int_{\{|u |\leq 1\}} f \bigl( X_\bullet (\omega), s,u \bigr) q(ds, du) + \int_{\{|u |> 1\}} f \bigl( X_\bullet (\omega), s,u \bigr) p(ds,du), \] where \(p\) is a Poisson random measure with a preassigned Lévy measure \(\alpha\) and \(q = p - ds \otimes \alpha (du)\).
scientific article; zbMATH DE number 794495
scientific article