Backward simulation of multivariate mixed Poisson processes

Abstract

The simulation of correlated multivariate Poisson processes with negative correlation between their components has many important applications in Finance, Insurance, Geophysics, and many other areas of applied probability. Introduced in our earlier work, the Backward Simulation (BS) approach to the simulation of correlated multivariate Poisson processes is able to capture a wide range of correlation, including extreme positive and extreme negative correlation, that is not possible with other approaches such as the forward simulation approach. Moreover, the BS approach enables simple and efficient generation of sample paths of correlated multivariate Poisson processes. In this work, we extend the BS approach to multivariate mixed Poisson processes.

Keywords:

• Backward simulation
• extreme joint distribution
• multivariate mixed Poisson processes
• order statistic property

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This is also known as the order statistic property [35].

2 The d that we use here for the dimension of a generic vector should not be confused with the dimension of a multivariate mixed Poisson process in Section 4.

3 For discrete distributions, Fréchet–Hoeffding is equivalent to the EJD theorem in 2 dimensions [17].

4 In practice, probability distributions are truncated to some desired accuracy; we are really dealing with linear programs.

5 This is also the subject of future work.

Additional information

Funding

This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada [Grant RGPIN-2016-05637].