Poisson processes directed by subordinators, stuttering poisson and pseudo-poisson processes, with applications to actuarial mathematics

Weconsider a PSI-process, that is a sequence of random variables (&), i = 0.1,…, which is subordinated by a continuous-time non-decreasing integer-valued process N(t): <K0 = ÇN(ty Our main example is when /V(t) itself is obtained as a subordination of the standard Poisson process 77(s) by a non-decreasing Lévy process S(t): N(t) = 77(S(t)).The values (&)one interprets as random claims, while their accumulated intensity S(t) is itself random. We show that in this case the process 7V(t) is a compound Poisson process of the stuttering type and its rate depends just on the value of theLaplace exponent of S(t) at 1. Under the assumption that the driven sequence (&) consists of i.i.d. random variables with finite variance we calculate a correlation function of the constructed PSI-process. Finally, we show that properly rescaled sums of such processes converge to the Ornstein-Uhlenbeck process in the Skorokhod space. We suppose that the results stated in the paper mightbe interesting for theorists and practitioners in insurance, in particular, for solution of reinsurance tasks.

Convergence of random bounded linear operators in the Skorokhod space

We introduce the space of random bounded linear operators on a separable Banach space such that their range belong to the Skorokhod space of right-continuous with left-hand limits functions. We call these random operators D-valued random variables. Almost sure and weak convergence results for the sequences of such random variables are proved by martingale methods. An application is described for a regime-switching inhomogeneous Lévy dynamics of a risky asset.

Functional limit theorems for the number of busy servers in a G/G/∞ queue

We discuss weak convergence of the number of busy servers in a G/G/∞ queue in the J1-topology on the Skorokhod space. We prove two functional limit theorems with random and nonrandom centering, thereby solving two open problems stated in Mikosch and Resnick (2006). A new integral representation for the limit Gaussian process is given.