Quasi-Monte Carlo Method for Infinitely Divisible Random Vectors via Series Representations

Abstract

Abstract. An infinitely divisible random vector without Gaussian component admits representations of shot noise series. Due to possible slow convergence of the series, they have not been investigated as a device for Monte Carlo simulation. In this paper, we investigate the structure of shot noise series representations from a simulation point of view and discuss the effectiveness of quasi-Monte Carlo methods applied to series representations. The structure of series representations in nature tends to decrease their effective dimension and thus increase the efficiency of quasi-Monte Carlo methods, thanks to the greater uniformity of low-discrepancy sequence in the lower dimension. We illustrate the effectiveness of our approach through numerical results of moment and tail probability estimations for stable and gamma random variables.

Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes

“…In particular, except for the second-order case with a finite Lévy measure, the simulation schemes are all based on the Poisson random variable Z κ,m , the Poisson arrival times { k } k∈N , and a sequence of iid uniform random variables {U k } k∈N , for which variance reduction techniques have been developed in (Imai and Kawai 2010;Kawai and Imai 2012) in a systematic manner.…”

“…However, focusing on univariate background driving processes, even when the background driving process is not a stable process, the law of the stochastic integral t 2 t 1 e a(t 2 −s) dL s (a is now a constant) may be identifiable and exactly (or almost exactly) simulatable for particular characteristics of the background driving process (see Barndorff-Nielsen and Shephard 2001;Imai and Kawai 2010;Imai and Kawai 2011;Imai and Kawai 2013;Kawai and Masuda 2011;Kawai and Masuda 2012;Samorodnitsky and Taqqu 1994;Zhang and Zhang 2008 and references therein). The approach for CARMA(1,0) however cannot be applied to higher order CARMA processes, with multivariate background driving processes.…”

“…Another special case is the class of Lévy-driven CARMA(1,0) processes, that is, Ornstein-Uhlenbeck processes (Barndorff-Nielsen and Shephard 2001;Sato 1999;Samorodnitsky and Taqqu 1994), in which all random elements are univariate. The univariate law of the stochastic integral may be fully characterized and exactly (or almost exactly) simulatable for some background driving Lévy processes, such as gamma, stable, tempered stable, and inverse Gaussian processes (Imai and Kawai 2010;Imai and Kawai 2011, Imai and Kawai 2013, Kawai and Masuda 2011, Kawai and Masuda 2012, Zhang and Zhang 2008. Apart from those special cases, however, it is difficult to construct exact simulation schemes for general higher order CARMA processes with multivariate background driving Lévy processes.…”

On the likelihood function of small time variance Gamma Lévy processes

“…where we have used the gamma density function (10) and the parameter correspondence (12). (See, for example, Lemma 2.3.1 of Kotz et al [2]) The formulation (11) can be used directly for sample path simulation of the variance gamma process, not only for simulation of the marginal distribution.…”

“…(See, for example, Lemma 2.3.1 of Kotz et al [2]) The formulation (11) can be used directly for sample path simulation of the variance gamma process, not only for simulation of the marginal distribution. In fact, the gamma Lévy process can be generated through the so-called infinite shot noise series representation (Bondesson [11] and Imai and Kawai [12] for theory and numerical improvements of series representations of the gamma process).…”