Numerical inverse Lévy measure method for infinite shot noise series representation

Junichi Imai, Reiichiro Kawai

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)264-283
Number of pages20
JournalJournal of Computational and Applied Mathematics
Volume253
DOIs
Publication statusPublished - 2013

Fingerprint

Shot Noise
Shot noise
Series Representation
Quasi-Monte Carlo
Differential equations
Trajectories
Low-discrepancy Sequences
kernel
Infinitely Divisible
Simulation
Series
Exponential time
Random Vector
Stochastic Equations
Theoretical Analysis
Jump
Alignment
Entire
Trajectory
Differential equation

Keywords

  • Carlo method
  • Infinitely divisible random vector
  • Lévy process
  • Numerical inversion
  • Quasi-Monte
  • Sample paths simulation
  • Stochastic differential equation with jumps

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

Numerical inverse Lévy measure method for infinite shot noise series representation. / Imai, Junichi; Kawai, Reiichiro.

In: Journal of Computational and Applied Mathematics, Vol. 253, 2013, p. 264-283.

Research output: Contribution to journalArticle

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