Nonhomogeneous Poisson processes (NHPPs) are often used to model recurrent events, and there is thus a need to check model fit for such models. We study the problem of obtaining exact goodness-of-fit tests for certain parametric NHPPs, using a method based on Monte Carlo simulation conditional on sufficient statistics. A closely related way of obtaining exact confidence intervals in parametri...

Shot-noise processes generalize compound Poisson processes in the following way: a jump (the shot) is followed by a decline (noise). This constitues a useful model for insurance claims in many circumstances: claims due to natural catastrophes or self-exciting processes exhibit similar features. We give a general account of shotnoise processes with time-inhomogeneous drivers and derive a number ...

An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions using Wiener processes. Then jump perturbations are added using simple Poisson processes constructing the theo...

Multiplicative processes and multifractals proved useful in various applications ranging from hydrodynamic turbulence to computer network traffic, to name but two. It was recently shown and explained how and why multifractal analysis could be interestingly and fruitfully placed in the more general framework of infinitely divisible cascades. The aim of this contribution is to design processes, c...

Numerical simulations of hierarchical gravitational clustering in an expanding universe show that, as time evolves, small clusters merge with each other to form larger clusters, whereas fragmentation of clusters is relatively uncommon. Stochastic models of the hierarchical clustering process can provide insight into, as well as useful approximations to, the evolution measured in these simulatio...

Motivated, in part, by the desire to develop an information-theoretic foundation for compound Poisson approximation limit theorems (analogous to the corresponding developments for the central limit theorem and for simple Poisson approximation), this work examines sufficient conditions under which the compound Poisson distribution has maximal entropy within a natural class of probability measure...

In this paper we study approximating the total loss associated with the individual insurance risk model by a compound Poisson random variable. By minimizing the expectation of the absolute deviation of the compound Poisson random variable from the true total loss, we investigate not only the optimal compound Poisson random variable but also the numerical calculation of the approximation error. ...

We study some linear and nonlinear shot noise models where the jumps are drawn from a compound Poisson process with jump sizes following an Erlang-m distribution. We show that the associated Master equation can be written as a spatial mth order partial differential equation without integral term. This differential form is valid for statedependent Poisson rates and we use it to characterize, via...