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

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

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: asmus@maths.lth.se
∗∗ Postal address: Mathematics Department, University of Tennessee, Knoxville, TN 37996-1300, USA.
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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 

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