Correlated multivariate Poisson processes and extreme measures

Abstract

Multivariate Poisson processes have many important applications in Insurance, Finance, and many other areas of Applied Probability. In this paper we study the backward simulation approach to modelling multivariate Poisson processes and analyze the connection to the extreme measures describing the joint distribution of the processes at the terminal simulation time.

Backward Simulation of Multivariate Mixed Poisson Processes

“…This allows us to compare the correlations obtained from the FS case to the correlations obtained in the BS case. A comparison of the Forward vs the Backward approaches can be found in [8,20]. We only note here that the BS allows for a greater range of correlations at the terminal time than FS.…”

“…Backward Simulation for the class of Mixed Poisson processes relies on the knowledge of the joint MPD at terminal time T , but how do we construct a multivariate MPD with some desired dependency structure in the first place? In this section, we briefly review the work in [9] and [20] that addresses the general problem of constructing multivariate joint distributions from given marginal distributions such that the linear correlation coefficient between the marginals are equal to some desired correlations. We also discuss how to sample from such multivariate joint distributions.…”

“…The solution to ( 23) is an Extreme Joint Distribution that determines the Extreme Measures P (1) and P (2) which have a one-to-one relationship to the extreme correlations Ĉ (1) and Ĉ (2) [8]. The Extreme Measures (23) lead to extreme correlations since extremizing the bivariate expectation extremizes the linear correlation coefficient as can be seen in (24).…”

“…The BS approach was extended in [20] to the class of bivariate processes containing both Poisson and Wiener components. It also led to the introduction of the Forward Continuation (FC) of Backward Simulation, a method for extending the process simulated by BS to subsequent intervals [nT, (n + 1)T ], where n is some integer, that preserves the joint distribution at various grid points nT [9].…”

“…Extreme joint distributions are used to generate extreme admissible correlations, from which all correlations within the admissible range can be obtained. It was extended to the multivariate setting in [9].…”