Backward Simulation of Correlated Negative Binomial L'evy Process Process

Abstract

Recent studies on correlated Poisson processes show that the backward simulation methods are computationally efficient, and incorporate flexible and extremal correlation structures in a multivariate risk system. These methods rely on the fact that the past arrival times of a Poisson process given the number of events over a time interval, [0, T ], are the order statistics of uniform random variables on [0, T ]. In this paper, we discuss an extension of the backward methods to a correlated negative binomial Lévy process which is an appealing model for over-dispersed count data such as operational losses. To obtain the conditional uniformity for the negative binomial Lévy process, we consider a particular setting in which the time interval is partitioned into equally spaced sub-intervals with unit length and the terminal time T is set to be the number of sub-intervals. Under this setting, the resulting joint probability of the increment series, conditional on the number of events over [0, T ], say l, is uniform for any points in the support of a {T, l}-simplex lattice. Based on this result, we establish a backward simulation method similar to that of Poisson process. Both the conditional independence and conditional dependence cases are discussed with illustrations of the corresponding time correlation patterns.

Backward Simulation of Multivariate Mixed Poisson Processes

“…The Backward Simulation approach is applicable to any process that exhibits the order statistic property. For example, Backward Simulation is applicable to the class of Negative Binomial Lévy Processes [3]. In fact, it is applicable to a more general class of processes known as Compound Poisson Processes which also posses a linear correlation structure under BS.…”

“…. 3 For discrete distributions, Fréchet-Hoeffding is equivalent to the EJD theorem in 2-dimensions [20].…”