Hostname: page-component-6bf8c574d5-w79xw Total loading time: 0 Render date: 2025-02-19T21:45:50.426Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximations of small jumps of Lévy processes with a view towards simulation

Part of: Probabilistic methods, simulation and stochastic differential equations Limit theorems Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Søren Asmussen  and
Jan Rosiński
Søren Asmussen*
Affiliation:
Lund University
Jan Rosiński*
Affiliation:
University of Tennessee
*
∗ Postal address: Mathematical Statistics, Centre of Mathematical Sciences, Lund University, Box 118, S-221 00 Lund, Sweden. Email address: [email protected]
∗∗ Postal address: Mathematics Department, University of Tennessee, Knoxville, TN 37996-1300, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X = (X(t):t ≥ 0) be a Lévy process and X∊ the compensated sum of jumps not exceeding ∊ in absolute value, σ2(∊) = var(X∊(1)). In simulation, X - X∊ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X∊/σ(∊) can be approximated by another Brownian term. A necessary and sufficient condition in terms of σ(∊) is given, and it is shown that when the condition fails, the behaviour of X∊/σ(∊) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.

Keywords

Berry-Esseen theoremBrownian motioncumulantsEdgeworth expansionfunctional central limit theoremGamma processinfinitely divisible distributionnormal inverse Gaussian processstable process

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60G52: Stable processes 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60E07: Infinitely divisible distributions; stable distributions 65C99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 482 - 493
Copyright
Copyright © Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (1999). Stochastic Simulation With a View Towards Stochastic Processes (MaPhySto Lecture Notes 2). Centre for Mathematical Physics and Stochastics, Aarhus.Google Scholar
Barndorff-Nielsen, O. (1997). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bhattacharya, R. N., and Ranga Rao, R. (1976). Normal Approximation and Asymptotic Expansions. John Wiley, Chichester.Google Scholar
Bondesson, L. (1982). On simulation from infinitely divisible distributions. Adv. Appl. Prob. 14, 855–869.CrossRefGoogle Scholar
Chambers, J. M., Mallows, C. L., and Stuck, B. W. (1976). A method for simulating stable random variables. J. Amer. Statist. Assoc. 71, 340–344.CrossRefGoogle Scholar
De la Peüa, V. H. and Giné, E. (1999). Decoupling. From Dependence to Independence. Springer, Berlin.Google Scholar
Götze, F. (1985). Asymptotic expansions in functional limit theorems. J. Multivar. Analysis 16, 1–20.Google Scholar
Kallenberg, O. (1998). Foundations of Modern Probability. Springer, Berlin.Google Scholar
Lorz, U., and Heinrich, L. (1991). Normal and Poisson approximation of infinitely divisible distributions. Statist. 22, 627–649.Google Scholar
Marcus, M. B. and Rosiński, J. (2001). L1-norm of infinitely divisible random vectors and certain stochastic integrals. Elect. Commun. Prob. 6, 15–29.Google Scholar
Petrov, V. V. (1995). Limit Theorems of Probability Theory. Oxford University Press.Google Scholar
Pollard, D. (1984). Convergence of Stochastic Processes. Springer, Berlin.Google Scholar
Rosiński, J. (2001). Series representations of Lévy processes from the perspective of point processes. In Lévy Processes—Theory and Applications, eds Barndorff-Nielsen, O. E., Mikosch, T. and Resnick, S. Birkhäuser, Boston.Google Scholar
Rydberg, T. (1997). The normal inverse Gaussian Lévy processes: simulation and approximation. Stoch. Models 13, 887–910.CrossRefGoogle Scholar
Samorodnitsky, G., and Taqqu, M. L. (1994). Non-Gaussian Stable Processes. Chapman and Hall, New York.Google Scholar
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Skovgaard, I. (1986). On multivariate Edgeworth expansions. Int. Statist. Rev. 54, 169–186 (correction (1989) 57, 183).CrossRefGoogle Scholar