### Journals Information

Mathematics and Statistics Vol. 7(5), pp. 191 - 196
DOI: 10.13189/ms.2019.070505
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## Backward Simulation of Correlated Negative Binomial L'evy Process Process

Taehan Bae *, Maral Mazjini

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 L´evy process which is an appealing model for over-dispersed count data such as operational losses. To obtain the conditional uniformity for the negative binomial L´evy 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.

KEYWORDS
Backward Simulation, Negative Binomial L´evy Process, Correlation Structure

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Taehan Bae , Maral Mazjini , "Backward Simulation of Correlated Negative Binomial L'evy Process Process," Mathematics and Statistics, Vol. 7, No. 5, pp. 191 - 196, 2019. DOI: 10.13189/ms.2019.070505.

(b). APA Format:
Taehan Bae , Maral Mazjini (2019). Backward Simulation of Correlated Negative Binomial L'evy Process Process. Mathematics and Statistics, 7(5), 191 - 196. DOI: 10.13189/ms.2019.070505.