Correlated Poisson Processes and Their Applications in Financial Modeling

Correlated multivariate Poisson processes and extreme measures

“…We begin with a description of the Common Shock Model (CSM) Lindskog and McNeil (2001) and the motivation of the approach proposed in Duch et al (2014). Afterwards, we discuss the results obtained in Kreinin (2016) for the case of two Poisson processes and describe the computation of the extreme measures in the case J = 2.…”

“…Clearly, the correlation coefficient can only be positive. A more advanced approach to the construction of negatively correlated Poisson processes is based on the idea of the backward simulation of the Poisson processes Kreinin (2016). The conditional distribution of the arrival moments of a Poisson process, conditional on the value of the process at the terminal simulation time, T , is uniform.…”

“…The admissible range of the correlation coefficients can be computed using the Extreme Joint Distributions (EJD) Theorem in Kreinin (2016) (see Theorem 2.2 in this section). The key statement, the characterization of the EJDs, is equivalent to the Frechet-Hoeffding theorem Fréchet (1960) for distributions on the positive quadrant of the two-dimensional lattice, Z…”

“…}. However, taking into account the numerical aspect of the problem, we prefer to use equations, derived in Kreinin (2016), written in terms of the probability density function, not in terms of the cumulative distribution function. Given marginal distributions of the non-negative, integer-valued random variables X1 and X2, with finite first and second moments, there exist two joint distributions, F * (i, j) and F * * (i, j) minimizing and maximizing the correlation, ρ = corr(X1, X2), respectively.…”

“…The EJD theorem in Kreinin (2016) allows one to construct the extreme measures p * and p * * , given marginal distributions of X1 and X2, with the minimal negative correlation ρ * and maximal positive correlation ρ * * , respectively. The extreme correlation coefficient uniquely defines the extreme measure.…”