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E-grāmata: Stable Levy Processes via Lamperti-Type Representations

, (University of Bath)
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Stable Levy Processes via Lamperti-Type RepresentationsPalielināt
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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many classical results as well as presenting many recent results appearing in the last decade, including previously unpublished material.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.

Recenzijas

'This treatise takes readers on a superb journey through the fascinating worlds of stable Lévy processes and of a rich variety of further naturally related random processes. Andreas Kyprianou and Juan Carlos Pardo masterfully deploy an arsenal of techniques, which are already interesting on their own right, to reveal many classical or more recent high level results on the distributions of functionals and on the path behaviours stable processes. It is indeed remarkable that their methods lead to so many explicit formulas, some amazingly simple, some more complex. The authors should be praised for making accessible as a coherent whole a vast literature that has been developed over several decades, including the latest developments.' Jean Bertoin, University of Zurich

Papildus informācija

A systematic treatment of stable Lévy processes and self-similar Markov processes, for graduate students and researchers in the field.
Notation xii
Preface xix
Acknowledgements xx
1 Stable distributions
1(26)
1.1 One-dimensional stable distributions
1(3)
1.2 Characteristic exponent of a one-dimensional stable law
4(5)
1.3 Moments
9(2)
1.4 Normalised one-dimensional stable distributions
11(2)
1.5 Distributional identities
13(7)
1.6 Stable distributions in higher dimensions
20(5)
1.7 Comments
25(2)
2 Levy processes
27(31)
2.1 Levy-Ito decomposition
27(3)
2.2 Killing
30(1)
2.3 Path variation and asymmetry
31(3)
2.4 Feller and strong Markov property
34(1)
2.5 Infinitesimal generator
35(1)
2.6 Drifting and oscillating
35(1)
2.7 Moments
36(1)
2.8 Exponential change of measure
37(2)
2.9 Donsker-type convergence
39(1)
2.10 Transience and recurrence
39(2)
2.11 Duality
41(1)
2.12 Hitting points
42(2)
2.13 Regularity of the half-line
44(1)
2.14 Excursions and the Wiener-Hopf factorisation
44(4)
2.15 Reflection
48(1)
2.16 Creeping
49(1)
2.17 First passage problems
49(5)
2.18 Levy processes in higher dimensions
54(2)
2.19 Comments
56(2)
3 Stable processes
58(20)
3.1 One-dimensional stable processes
58(2)
3.2 Normalised one-dimensional stable processes
60(2)
3.3 Path variation, asymmetry and moments
62(2)
3.4 Path properties in one dimension
64(2)
3.5 Wiener-Hopf factorisation and the first passage problem
66(4)
3.6 Isotropic d-dimensional stable processes
70(2)
3.7 Resolvent density
72(4)
3.8 Comments
76(2)
4 Hypergeometric Levy processes
78(37)
4.1 y3-subordinators
78(3)
4.2 Hypergeometric processes
81(6)
4.3 The subclass of Lamperti-stable processes
87(2)
4.4 The first passage problem
89(4)
4.5 Exponential functionals
93(10)
4.6 Distributional densities of exponential functionals
103(8)
4.7 Distributional tails of exponential functionals
111(2)
4.8 Comments
113(2)
5 Positive self-similar Markov processes
115(38)
5.1 The Lamperti transform
115(2)
5.2 Starting at the origin
117(5)
5.3 Stable processes killed on entering (--∞, 0)
122(6)
5.4 Stable processes conditioned to stay positive
128(5)
5.5 Stable processes conditioned to limit to 0 from above
133(3)
5.6 Censored stable processes
136(8)
5.7 The radial part of an isotropic stable process
144(7)
5.8 Comments
151(2)
6 Spatial fluctuations in one dimension
153(30)
6.1 First exit from an interval
153(7)
6.2 Hitting points in an interval
160(1)
6.3 First entrance into a bounded interval
161(5)
6.4 Point of closest and furthest reach
166(2)
6.5 First hitting of a two-point set
168(4)
6.6 First hitting of a point
172(4)
6.7 First exit for the reflected process
176(5)
6.8 Comments
181(2)
7 Doney-Kuznetsov factorisation and the maximum
183(23)
7.1 Kuznetsov's factorisation
183(2)
7.2 Quasi-periodicity
185(5)
7.3 The Law of the maximum at a finite time
190(6)
7.4 Doney's factorisation
196(8)
7.5 Comments
204(2)
8 Asymptotic behaviour for stable processes
206(21)
8.1 Stable subordinators
206(8)
8.2 Upper envelopes for ρ ε (0,1)
214(5)
8.3 Lower envelopes for ρ ε (0,1)
219(7)
8.4 Comments
226(1)
9 Envelopes of positive self-similar Markov processes
227(25)
9.1 Path decompositions for pssMp
227(7)
9.2 Lower envelopes
234(6)
9.3 Upper envelopes
240(10)
9.4 Comments
250(2)
10 Asymptotic behaviour for path transformations
252(34)
10.1 More on hypergeometric Levy processes
252(8)
10.2 Distributions of pssMp path functionals
260(5)
10.3 Stable processes conditioned to stay positive
265(7)
10.4 Stable processes conditioned to limit to 0 from above
272(3)
10.5 Censored stable processes
275(4)
10.6 Isotropic stable processes
279(6)
10.7 Comments
285(1)
11 Markov additive and self-similar Markov processes
286(20)
11.1 MAPs and the Lamperti-Kiu transform
286(2)
11.2 Distributional and path properties of MAPs
288(4)
11.3 Excursion theory for MAPs
292(3)
11.4 Matrix Wiener-Hopf factorisation
295(5)
11.5 Self-similar Markov processes in Rd
300(3)
11.6 Starting at the origin
303(1)
11.7 Comments
304(2)
12 Stable processes as self-similar Markov processes
306(24)
12.1 Stable processes and their A-transforms as ssMp
306(8)
12.2 Stable processes conditioned to avoid or hit 0
314(2)
12.3 One-dimensional Riesz-Bogdan-Zak transform
316(2)
12.4 First entrance into a bounded interval revisited
318(4)
12.5 First hitting of a point revisited
322(2)
12.6 Riesz-Bogdan-Zak transformation in dimension d ≥ 2
324(4)
12.7 Radial asymptotics for d ≥ 2
328(1)
12.8 Comments
329(1)
13 Radial reflection and the deep factorisation
330(21)
13.1 Radially reflected stable processes when α ε (0,1)
330(2)
13.2 Deep inverse factorisation of the stable process
332(3)
13.3 Ladder MAP matrix potentials
335(9)
13.4 Stationary limit of the radially reflected process
344(4)
13.5 Deep factorisation of the stable process
348(1)
13.6 Comments
349(2)
14 Spatial fluctuations and the unit sphere
351(32)
14.1 Sphere inversions
351(3)
14.2 Sphere inversions with reflection
354(1)
14.3 First hitting of a sphere
355(10)
14.4 First entrance and exit of a ball
365(9)
14.5 Walk-on-spheres and first exit of general domains
374(7)
14.6 Comments
381(2)
15 Applications of radial excursion theory
383(29)
15.1 Radial excursions
383(6)
15.2 The Point of closest reach to the origin
389(14)
15.3 Deep factorisation in d-dimensions
403(3)
15.4 Radial reflection
406(5)
15.5 Comments
411(1)
16 Windings and up-crossings of stable processes
412(17)
16.1 Polar decomposition of planar stable processes
412(2)
16.2 Windings at infinity
414(3)
16.3 Windings at the origin
417(4)
16.4 Upcrossings of one-dimensional stable processes
421(7)
16.5 Comments
428(1)
Appendix
429(17)
A.1 Useful results from complex analysis
429(1)
A.2 Mellin and Laplace-Fourier inversion
430(1)
A.3 Gamma and beta functions
431(1)
A.4 Double gamma function
432(2)
A.5 Double sine function
434(1)
A.6 Hypergeometric functions
435(1)
A.7 Additive and subadditive functions
436(1)
A.8 Random difference equations
437(1)
A.9 A generalisation of the Borel-Cantelli Lemma
438(1)
A.10 Skorokhod space
439(1)
A.11 Feller processes
440(1)
A.12 Hunt-Nagasawa duality
441(2)
A.13 Poisson point processes
443(3)
References 446(12)
Index 458
Andreas E. Kyprianou was educated at the University of Oxford and University of Sheffield and is currently a professor of mathematics at the University of Bath. He has spent over 25 years working on the theory and application of path-discontinuous stochastic processes and has over 130 publications, including a celebrated graduate textbook on Lévy processes. During his time in Bath, he co-founded and directed the Prob-L@B (Probability Laboratory at Bath), was PI for a multi-million-pound EPSRC Centre for Doctoral Training, and is currently the Director of the Bath Institute for Mathematical Innovation. Juan Carlos Pardo is a full professor at the department of Probability and Statistics at Centro de Investigación en Matemįticas (CIMAT). He was educated at the Universidad Nacional Autónoma de México (UNAM) and Université de Paris VI (Sorbonne Université). He has spent over 13 years working on the theory and application of path-discontinuous stochastic processes and has more than 50 publications in these areas. During the academic year 2018-2019, he held the David Parkin visiting professorship at the University of Bath.
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