A backward construction and simulation of correlated Poisson processes

Backward Simulation of Correlated Negative Binomial L'evy Process Process

“…As in Bae and Kreinin [2], the goal is to simulate correlated increment processes for a bivariate NBLP {N 1 (t), N 2 (t)} t≤T , given the correlation, ρ(T ), at the terminal time T . Based on (4), the backward construction relies on the conditional dependence structure between the increment processes given a realization (N 1 (T ) = l 1 , N 2 (T ) = l 2 ) at the terminal time T .…”

“…Bae and Kreinin [2] construct bivariate Poisson processes with flexible time correlation structures using the Marshall-Olkin bivariate binomial distribution for the conditional law and some parametric families of bivariate copulas. A similar method can be implemented for the backward simulation of correlated NBLPs.…”

“…Note that, with various combinations of the parameters in (10) or by using other parametric copula families such as Clayton, Farlie-Gumbel-Morgenstern, Ali-Mikhail-Haq or Gumbel's survival copula (Bae and Kreinin [2]), a variety of time correlation structures may be realized. For instance, Figure 3 gives the correlation patterns realized under the Calyton copula, c(u, v) = max{u −θ + v −θ − 1, 0} −1/θ , θ ∈ [−1, ∞)/{0}, with a few prescribed values of parameter θ (and T = 12, p 1 = p 2 = 0.4, ρ = 0.5 as in Figure 1).…”

“…Due to the randomness in the number of exponential inter-arrival times, this method can be computationally demanding especially when the time horizon is long and the mean Poisson rate is large. As discussed in Kreinin [10] and Bae and Kreinin [2], this method also lacks ability to realize the extremal (maximal or minimal) correlation structure which makes this forward method less appealing for quantitative risk applications. On the other hand, the backward methods which have recently been studied by Duch et al [7] and Kreinin [10], use the fact that the past arrival times of Poisson jumps, given the total number of Poisson jumps over the time interval, have the distribution equivalent to the order statistics of uniform random variates on the time interval.…”

“…More importantly, under the conditional independence framework, Kreinin [10] shows that the backward method allows for extremal correlations, and thus it is appealing for scenario analysis of portfolio risks. In Bae and Kreinin [2], the conditional independence assumption has been alleviated by using several parametric copulas, which results in quite flexible time correlation patterns.…”