Abstract
We study systems of simple point processes that admit stochastic intensities. We represent these point processes as thinnings of Poisson measures and are interested in a convergence result of such systems. This result states that, if the stochastic intensities of the limit point processes are independent of the underlying Poisson measures, the convergence in distribution in Skorohod topology of the stochastic intensities implies the same convergence for the point processes.
Highlights
In this paper we consider systems of point processes admitting stochastic intensities
We study systems of simple point processes that admit stochastic intensities
We represent these point processes as thinnings of Poisson measures and are interested in a convergence result of such systems. This result states that, if the stochastic intensities of the limit point processes are independent of the underlying Poisson measures, the convergence in distribution in Skorohod topology of the stochastic intensities implies the same convergence for the point processes
Summary
In this paper we consider systems of point processes admitting stochastic intensities. Theorem 1.1, roughly states that if the intensities of point processes converge in distribution and their limits are independent of the underlying Poisson measures, the systems of point processes converge as well. Let us note that, in Theorem 1.1, the processes Y N,k are not assumed to be independent of the Poisson measures πk. Let us note that the condition of independence of the limiting intensities Yk from Poisson measures πk is often satisfied and natural in many examples of application. It holds for example when the limiting intensities are deterministic, or when they are functionals of Brownian motions with respect to the same filtration as πk.
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