Numerical inverse Lévy measure method for infinite shot noise series representation

Infinitely divisible random vectors without Gaussian component admit representations with shot noise series. We analyze four known methods of deriving kernels of the series and reveal the superiority of the inverse Lévy measure method over the other three methods for simulation use. We propose a numerical approach to the inverse Lévy measure method, which in most cases, provides no explicit kernel. We also propose to apply the quasi-Monte Carlo procedure to the inverse Lévy measure method to enhance the numerical efficiency. It is known that the efficiency of the quasi-Monte Carlo could be enhanced by sensible alignment of low discrepancy sequence. In this paper we apply this idea to exponential interarrival times in the shot noise series representation. The proposed method paves the way for simulation use of shot noise series representation for any infinite Lévy measure and enables one to simulate entire approximate trajectory of stochastic differential equations with jumps based on infinite shot noise series representation. Although implementation of the proposed method requires a small amount of initial work, it is applicable to general Lévy measures and has the potential to yield substantial improvements in simulation time and estimator efficiency. Numerical results are provided to support our theoretical analysis and confirm the effectiveness of the proposed method for practical use.