On simulation from infinitely divisible distributions

Bondesson, Lennart

doi:10.2307/1427027

Cited by 122 publications

(63 citation statements)

References 12 publications

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“…By applying the generalized shot noise method [6,33], we get the desired series representation. To prove the claim for {X 2,n } n∈N , consider the probability measure on…”

Section: Resultsmentioning

confidence: 99%

On finite truncation of infinite shot noise series representation of tempered stable laws

Imai

Kawai

2011

Physica A: Statistical Mechanics and its Applications

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Tempered stable processes are widely used in various fields of application as alternatives with finite second moment and long-range Gaussian behaviors to stable processes. Infinite shot noise series representation is the only exact simulation method for the tempered stable process and has recently attracted attention for simulation use with ever improved computational speed. In this paper, we derive series representations for the tempered stable laws of increasing practical interest through the thinning, rejection, and inverse Lévy measure methods. We make a rigorous comparison among those representations, including the existing one due to [18, 34], in terms of the tail mass of Lévy measures which can be simulated under a common finite truncation scheme. The tail mass are derived in closed form for some representations thanks to various structural properties of the tempered stable laws. We prove that the representation via the inverse Lévy measure method achieves a much faster convergence in truncation to the infinite sum than all the other representations. Numerical results are presented to support our theoretical analysis.

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“…By applying the generalized shot noise method [6,33], we get the desired series representation. To prove the claim for {X 2,n } n∈N , consider the probability measure on…”

Section: Resultsmentioning

confidence: 99%

On finite truncation of infinite shot noise series representation of tempered stable laws

Imai

Kawai

2011

Physica A: Statistical Mechanics and its Applications

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“…Both [14] and [3] describe how to simulate shot-noise Cox processes using the techniques developed in [2]. Just as with the simple cluster process model it is necessary to simulate ζ on a larger region than W , but furthermore it is only possible to simulate a finite number of atoms in any bounded area (see [3] for a further discussion of this).…”

Section: The General Methodsmentioning

confidence: 99%

Simulation of cluster point processes without edge effects

Brix

Kendall

2002

Adv. Appl. Probab.

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The usual direct method of simulation for cluster processes requires the generation of the parent point process over a region larger than the actual observation window, since we have to allow for all possible parents giving rise to observed daughter points, and some of these parents may fall outwith the observation window. When there is no a priori bound on the distance between parent and child then we have to take care to control approximations arising from edge effects. In this paper, we present a simulation method which requires simulation only of those parent points actually giving rise to observed daughter points, thus avoiding edge effect approximation. The idea is to replace the cluster distribution by one which is conditioned to plant at least one daughter point in the observation window, and to modify the parent process to have an inhomogeneous intensity exactly balancing the effect of the conditioning. We furthermore show how the method extends to cases involving infinitely many potential parents, for example gamma-Poisson processes and shot-noise G-Cox processes, allowing us to avoid approximation due to truncation of the parent process.

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“…We solved the WM propagator by a second-order implicit-explicit (IMEX) time splitting scheme. 7 For spatial discretization we used the Fourier collocation method. The reference solution was established by running the Burgers equation with ξ taking all the possible values.…”

Section: The Burgers Equation With One Rvmentioning

confidence: 99%

“…. , ξ d ) is meaningful to be considered because many stochastic processes have series representations, e.g., Karhunen-Loeve expansion for the Gaussian process[24,32], and shot noise expansion for Levy pure jump processes[7,43,44].Downloaded 10/07/15 to 18.111.37.34. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php…”

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confidence: 99%

Adaptive Wick--Malliavin Approximation to Nonlinear SPDEs with Discrete Random Variables

Zheng¹,

Rozovsky²,

Karniadakis³

2015

SIAM J. Sci. Comput.

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We propose an adaptive Wick-Malliavin (WM) expansion in terms of the Malliavin derivative of order Q to simplify the propagator of general polynomial chaos (gPC) of order P (a system of deterministic equations for the coefficients of gPC) and to control the error growth with respect to time. Specifically, we demonstrate the effectiveness of the WM method by solving a stochastic reaction equation and a Burgers equation with several discrete random variables. Exponential convergence is shown numerically with respect to Q when Q ≥ P − 1. We also analyze the computational complexity of the WM method and identify a significant speedup with respect to gPC, especially in high dimensions.

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On simulation from infinitely divisible distributions

Abstract: A general method based on a limit theorem for generation of random numbers from infinitely divisible distributions with essentially given Lévy measure is studied. Some classes of infinitely divisible distributions that appear in a natural way in this context are paid particular attention.

Cited by 122 publications

References 12 publications

On finite truncation of infinite shot noise series representation of tempered stable laws

On finite truncation of infinite shot noise series representation of tempered stable laws

Simulation of cluster point processes without edge effects

Adaptive Wick--Malliavin Approximation to Nonlinear SPDEs with Discrete Random Variables

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