UID:
almafu_9960165853602883
Format:
1 online resource (xx, 463 pages) :
,
digital, PDF file(s).
ISBN:
1-108-57216-2
,
1-108-64831-2
Series Statement:
Institute of Mathematical Statistics monographs
Content:
Stable Lévy processes lie at the intersection of Lévy processes and self-similar Markov processes. Processes in the latter class enjoy a Lamperti-type representation as the space-time path transformation of so-called Markov additive processes (MAPs). This completely new mathematical treatment takes advantage of the fact that the underlying MAP for stable processes can be explicitly described in one dimension and semi-explicitly described in higher dimensions, and uses this approach to catalogue a large number of explicit results describing the path fluctuations of stable Lévy processes in one and higher dimensions. Written for graduate students and researchers in the field, this book systemically establishes many classical results as well as presenting many recent results appearing in the last decade, including previously unpublished material. Topics explored include first hitting laws for a variety of sets, path conditionings, law-preserving path transformations, the distribution of extremal points, growth envelopes and winding behaviour.
Note:
Title from publisher's bibliographic system (viewed on 23 Mar 2022).
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Cover -- Half-title Page -- Series Page -- Title Page -- Copyright Page -- Dedication -- Contents -- Notation -- Preface -- Acknowledgements -- 1 Stable distributions -- 1.1 One-dimensional stable distributions -- 1.2 Characteristic exponent of a one-dimensional stable law -- 1.3 Moments -- 1.4 Normalised one-dimensional stable distributions -- 1.5 Distributional identities -- 1.6 Stable distributions in higher dimensions -- 1.7 Comments -- 2 Lévy processes -- 2.1 Lévy-Itô decomposition -- 2.2 Killing -- 2.3 Path variation and asymmetry -- 2.4 Feller and strong Markov property -- 2.5 Infinitesimal generator -- 2.6 Drifting and oscillating -- 2.7 Moments -- 2.8 Exponential change of measure -- 2.9 Donsker-type convergence -- 2.10 Transience and recurrence -- 2.11 Duality -- 2.12 Hitting points -- 2.13 Regularity of the half-line -- 2.14 Excursions and the Wiener-Hopf factorisation -- 2.15 Reflection -- 2.16 Creeping -- 2.17 First passage problems -- 2.18 Lévy processes in higher dimensions -- 2.19 Comments -- 3 Stable processes -- 3.1 One-dimensional stable processes -- 3.2 Normalised one-dimensional stable processes -- 3.3 Path variation, asymmetry and moments -- 3.4 Path properties in one dimension -- 3.5 Wiener-Hopf factorisation and the first passage problem -- 3.6 Isotropic d-dimensional stable processes -- 3.7 Resolvent density -- 3.8 Comments -- 4 Hypergeometric Lévy processes -- 4.1 β-subordinators -- 4.2 Hypergeometric processes -- 4.3 The subclass of Lamperti-stable processes -- 4.4 The first passage problem -- 4.5 Exponential functionals -- 4.6 Distributional densities of exponential functionals -- 4.7 Distributional tails of exponential functionals -- 4.8 Comments -- 5 Positive self-similar Markov processes -- 5.1 The Lamperti transform -- 5.2 Starting at the origin -- 5.3 Stable processes killed on entering (−∞, 0).
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5.4 Stable processes conditioned to stay positive -- 5.5 Stable processes conditioned to limit to 0 from above -- 5.6 Censored stable processes -- 5.7 The radial part of an isotropic stable process -- 5.8 Comments -- 6 Spatial fluctuations in one dimension -- 6.1 First exit from an interval -- 6.2 Hitting points in an interval -- 6.3 First entrance into a bounded interval -- 6.4 Point of closest and furthest reach -- 6.5 First hitting of a two-point set -- 6.6 First hitting of a point -- 6.7 First exit for the reflected process -- 6.8 Comments -- 7 Doney-Kuznetsov factorisation and the maximum -- 7.1 Kuznetsov's factorisation -- 7.2 Quasi-periodicity -- 7.3 The Law of the maximum at a finite time -- 7.4 Doney's factorisation -- 7.5 Comments -- 8 Asymptotic behaviour for stable processes -- 8.1 Stable subordinators -- 8.2 Upper envelopes for ρ ∈ (0, 1) -- 8.3 Lower envelopes for ρ ∈ (0, 1) -- 8.4 Comments -- 9 Envelopes of positive self-similar Markov processes -- 9.1 Path decompositions for pssMp -- 9.2 Lower envelopes -- 9.3 Upper envelopes -- 9.4 Comments -- 10 Asymptotic behaviour for path transformations -- 10.1 More on hypergeometric Lévy processes -- 10.2 Distributions of pssMp path functionals -- 10.3 Stable processes conditioned to stay positive -- 10.4 Stable processes conditioned to limit to 0 from above -- 10.5 Censored stable processes -- 10.6 Isotropic stable processes -- 10.7 Comments -- 11 Markov additive and self-similar Markov processes -- 11.1 MAPs and the Lamperti-Kiu transform -- 11.2 Distributional and path properties of MAPs -- 11.3 Excursion theory for MAPs -- 11.4 Matrix Wiener-Hopf factorisation -- 11.5 Self-similar Markov processes in R[sup(d)] -- 11.6 Starting at the origin -- 11.7 Comments -- 12 Stable processes as self-similar Markov processes -- 12.1 Stable processes and their h-transforms as ssMp.
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12.2 Stable processes conditioned to avoid or hit 0 -- 12.3 One-dimensional Riesz-Bogdan-Żak transform -- 12.4 First entrance into a bounded interval revisited -- 12.5 First hitting of a point revisited -- 12.6 Riesz-Bogdan-Żak transformation in dimension d ≥ 2 -- 12.7 Radial asymptotics for d ≥ 2 -- 12.8 Comments -- 13 Radial reflection and the deep factorisation -- 13.1 Radially reflected stable processes when α ∈ (0, 1) -- 13.2 Deep inverse factorisation of the stable process -- 13.3 Ladder MAP matrix potentials -- 13.4 Stationary limit of the radially reflected process -- 13.5 Deep factorisation of the stable process -- 13.6 Comments -- 14 Spatial fluctuations and the unit sphere -- 14.1 Sphere inversions -- 14.2 Sphere inversions with reflection -- 14.3 First hitting of a sphere -- 14.4 First entrance and exit of a ball -- 14.5 Walk-on-spheres and first exit of general domains -- 14.6 Comments -- 15 Applications of radial excursion theory -- 15.1 Radial excursions -- 15.2 The Point of closest reach to the origin -- 15.3 Deep factorisation in d-dimensions -- 15.4 Radial reflection -- 15.5 Comments -- 16 Windings and up-crossings of stable processes -- 16.1 Polar decomposition of planar stable processes -- 16.2 Windings at infinity -- 16.3 Windings at the origin -- 16.4 Upcrossings of one-dimensional stable processes -- 16.5 Comments -- Appendix -- A.1 Useful results from complex analysis -- A.2 Mellin and Laplace-Fourier inversion -- A.3 Gamma and beta functions -- A.4 Double gamma function -- A.5 Double sine function -- A.6 Hypergeometric functions -- A.7 Additive and subadditive functions -- A.8 Random difference equations -- A.9 A generalisation of the Borel-Cantelli Lemma -- A.10 Skorokhod space -- A.11 Feller processes -- A.12 Hunt-Nagasawa duality -- A.13 Poisson point processes -- References -- Index.
Additional Edition:
ISBN 1-108-48029-2
Language:
English
URL:
https://doi.org/10.1017/9781108648318
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