Numerical Computation of First-Passage Times of Increasing Lévy Processes

Veillette, Mark S.; Taqqu, Murad S.

doi:10.1007/s11009-009-9158-y

Cited by 32 publications

(35 citation statements)

References 32 publications

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“…It turns out that both Gaver's and Post-Widder formulas have a very slow convergence rate, therefore one has to apply some acceleration algorithm, such as Salzer transformation for the Gaver's formula, which was proposed by Stehfest [98], or Richardson extrapolation for Post-Widder formula which was used by Veillette and Taqqu [105]. We refer to [4] for all the details and background on the Gaver-Stehfest algorithm, here we just present the final expression.…”

Section: The Gaver-stehfest Algorithmmentioning

confidence: 99%

“…Surya [100] presents an algoritm for evaluating the scale function using exponential dampening followed by Laplace inversion; the latter performed in a similar way as Rogers [90]. In a recent paper Veillette and Taqqu [105] compute the distribution of the first passage time for subordinators using two techniques. These are the discretization of the Bromwich integral and Post-Widder formula coupled with Richardson extrapolation.…”

Section: Introductionmentioning

confidence: 99%

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The Theory of Scale Functions for Spectrally Negative Lévy Processes

Kuznetsov

Kyprianou

Rivero

2012

Lecture Notes in Mathematics

249

363

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The purpose of this review article is to give an up to date account of the theory and application of scale functions for spectrally negative Lévy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Lévy processes, in particular a reasonable understanding of the Lévy-Khintchine formula and its relationship to the Lévy-Itô decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Lévy processes;

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Section: The Gaver-stehfest Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Theory of Scale Functions for Spectrally Negative Lévy Processes

Kuznetsov

Kyprianou

Rivero

2012

Lecture Notes in Mathematics

249

363

View full text Add to dashboard Cite

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“…and it is known [39,41,49,50] that for any p > 0, EY p (t) < ∞. Let U (t) = EY (t) be the renewal function.…”

Section: This Means Thatmentioning

confidence: 99%

“…Finally, we study Mixed-Fractional Poisson processes.The paper is organized as follows. In the next section, we collect some known results from the theory of subordinators and inverse subordinators, see [8,36,49,50] among others. In Section 2, we prove a martingale characterization of the FPP, which is a generalization of the Watanabe Theorem.…”

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confidence: 99%

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Fractional Poisson Fields and Martingales

2018

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We present new properties for the Fractional Poisson process and the Fractional Poisson field on the plane. A martingale characterization for Fractional Poisson processes is given. We extend this result to Fractional Poisson fields, obtaining some other characterizations. The fractional differential equations are studied. We consider a more general Mixed-Fractional Poisson process and show that this process is the stochastic solution of a system of fractional differential-difference equations. Finally, we give some simulations of the Fractional Poisson field on the plane.There are several different approaches to the fundamental concept of Fractional Poisson process (FPP) on the real line. The "renewal" definition extends the characterization of the Poisson process as a sum of independent non-negative exponential random variables. If one changes the law of interarrival times to the Mittag-Leffler distribution (see [32,33,44]), the FPP arises. A second approach is given in [6], where the renewal approach to the Fractional Poisson process is developed and it is proved that its one-dimensional distributions coincide with the solution to fractionalized state probabilities. In [34] it is shown that a kind of Fractional Poisson process can be constructed by using an "inverse subordinator", which leads to a further approach.In [26], following this last method, the FPP is generalized and defined afresh, obtaining a Fractional Poisson random field (FPRF) parametrized by points of the Euclidean space R 2 + , in the same spirit it has been done before for Fractional Brownian fields, see, e.g., [17,20,22,30].The starting point of our extension will be the set-indexed Poisson process which is a well-known concept, see, e.g., [17,22,37,38,47].In this paper, we first present a martingale characterization of the Fractional Poisson process. We extend this characterization to FPRF using the concept of increasing path and strong martingales. This characterization permits us to give a definition of a set-indexed Fractional Poisson process. We study the fractional differential equation for FPRF. Finally, we study Mixed-Fractional Poisson processes.The paper is organized as follows. In the next section, we collect some known results from the theory of subordinators and inverse subordinators, see [8,36,49,50] among others. In Section 2, we prove a martingale characterization of the FPP, which is a generalization of the Watanabe Theorem. In Section 3, another generalization called "Mixed-Fractional Poisson process" is introduced and some distributional properties are studied as well as Watanabe characterization is given. Section 4 is devoted to FPRF. We begin by computing covariance for this process, then we give some characterizations using increasing paths and intensities. We present a Gergely-Yeshow characterization § N.

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Nonparametric evaluation of the first passage time of degradation processes

Qin

2018

Appl Stoch Models Bus & Ind

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This paper discusses a nonparametric method to approximate the first passage time (FPT) distribution of the degradation processes incorporating random effects if the process type is unknown. The FPT of a degradation process is unnecessarily observed since its density function can be approximated by inverting the empirical Laplace transform using the empirical saddlepoint method.The empirical Laplace transform is composed of the measured increments of the degradation processes. To evaluate the performance of the proposed method, the approximated FPT is compared with the theoretical FPT assuming a true underlying process. The nonparametric method discussed in this paper is shown to possess the comparatively small relative errors in the simulation study and performs well to capture the heterogeneity in the practical data analysis. To justify the fitting results, the goodness-of-fit tests including Kolmogorov-Smirnov test and Cramér-von Mises test are conducted, and subsequently, a bootstrap confidence interval is constructed in terms of the 90th percentile of the FPT distribution.

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Numerical Computation of First-Passage Times of Increasing Lévy Processes

Cited by 32 publications

References 32 publications

The Theory of Scale Functions for Spectrally Negative Lévy Processes

The Theory of Scale Functions for Spectrally Negative Lévy Processes

Fractional Poisson Fields and Martingales

Nonparametric evaluation of the first passage time of degradation processes

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