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.

Bounds on the mean and squared coefficient of variation of phase-type distributions

We consider a class of phase-type distributions (PH-distributions), to be called the MMPP class of PH-distributions, and find bounds of their mean and squared coefficient of variation (SCV). As an application, we have shown that the SCV of the event-stationary inter-event time for Markov modulated Poisson processes (MMPPs) is greater than or equal to unity, which answers an open problem for MMPPs. The results are useful for selecting proper PH-distributions and counting processes in stochastic modeling.

Generalized Counting Processes in a Stochastic Environment

This paper addresses the generalization of counting processes through the age formalism of Lévy Walks. Simple counting processes are introduced and their properties are analyzed: Poisson processes or fractional Poisson processes can be recovered as particular cases. The stationarity assumption in the renewal mechanism characterizing simple counting processes can be modified in different ways, leading to the definition of generalized counting processes. In the case that the transition mechanism of a counting process depends on the environmental conditions—i.e., the parameters describing the occurrence of new events are themselves stochastic processes—the counting processes is said to be influenced by environmental stochasticity. The properties of this class of processes are analyzed, providing several examples and applications and showing the occurrence of new phenomena related to the modulation of the long-term scaling exponent by environmental noise.

Invariant measure of stochastic higher order KdV equation driven by Poisson processes

The current paper is devoted to stochastic damped KdV equations of higher order driven by Poisson process. We establish the well-posedness of stochastic damped higher-order KdV equations, and prove that there exists an unique invariant measure for deterministic initial conditions. Some discussion on the general pure jump noise case are also provided.