Principal Values of the Integral Functionals of Brownian Motion: Existence, Continuity and an Extension of Itô’s Formula

Cherny, Alexander S.

doi:10.1007/978-3-540-44671-2_24

Cited by 17 publications

(6 citation statements)

References 10 publications

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“…At the start of this study, there is an expression of the winding process as an Itô stochastic integral; • Azéma-Yor [1] obtained their solution of the Skorokhod problem with the help of martingale -stochastic calculus arguments; L.C.G. Rogers [30] subsequently derived the excursion arguments essentially presented in Section 5; • about Section 6, the second author also developed a purely stochastic calculus approach, based on extensions of Itô's formula to functions f with f ∈ L 2 loc (see [40], as well as Yamada's survey [38] and Cherny [8]). Very recently, Kasahara-Watanabe [19] kept developing close links between extended Itô's formulae, excursion theory and Krein theory.…”

Section: Itô Calculus Versus Itô Excursion Theorymentioning

confidence: 99%

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Itô’s excursion theory and its applications

Pitman

Yor

2007

Jpn. J. Math.

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Section: Itô Calculus Versus Itô Excursion Theorymentioning

confidence: 99%

“…Other principal values, defined from dx |x| β sgn(x), for β < 3 2 allow to recover all symmetric stable Lévy processes, with indexes α ∈ (0, 2), α and β being again related by (8).…”

Section: Principal Values Of Brownian Local Timesmentioning

confidence: 99%

Itô’s excursion theory and its applications

Pitman

Yor

2007

Jpn. J. Math.

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“…Within the probabilistic context, our results are closely related to (singular) integrals of the local time of the Brownian motion. For further discussion, we refer to [27]; here we mention the fundamental work of Biane and Yor [6], as well as more recent [9]. We also stress that the elliptic operator L generates a diffusion the half-plane, and the corresponding Dirichlet-to-Neumann operator is the generator of the trace of this diffusion on the boundary.…”

Section: Introductionmentioning

confidence: 99%

Harmonic extension technique for non-symmetric operators with completely monotone kernels

Kwaśnicki¹

2019

Preprint

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We identify a class of non-local integro-differential operators K in R with Dirichlet-to-Neumann maps in the half-plane R × (0, ∞) for appropriate elliptic operators L. More precisely, we prove a bijective correspondence between Lévy operators K with non-local kernels of the form ν(y − x), where ν(x) and ν(−x) are completely monotone functions on (0, ∞), and elliptic operators L = a(y)∂ xx + 2b(y)∂ xy + ∂ yy . This extends a number of previous results in the area, where symmetric operators have been studied: the classical identification of the Dirichlet-to-Neumann operator for the Laplace operator in R × (0, ∞) with − √ −∂ xx , the square root of one-dimensional Laplace operator; the Caffarelli-Silvestre identification of the Dirichlet-to-Neumann operator for ∇ • (y 1−α ∇) with (−∂ xx ) α/2 for α ∈ (0, 2); and the identification of Dirichlet-to-Neumann maps for operators a(y)∂ xx + ∂ yy with complete Bernstein functions of −∂ xx due to Mucha and the author. Our results rely on recent extension of Krein's spectral theory of strings by Eckhardt and Kostenko.

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“…In particular, their work provides a representation of stable Lévy processes in terms of principal values of the local time for the Brownian motion on R -rather than usual integrals on [0, ∞), as in Corollary 1.3. Almost sure convergence and convergence in probability of such principal value integrals is studied in [12]; Remark (iv) under Theorem 3.2 therein essentially cover a special case of Corollary 1.3 for subordinators (non-negative Lévy processes) Z(t).…”

Section: Introductionmentioning

confidence: 99%

Boundary traces of shift-invariant diffusions in half-plane

Kwaśnicki¹

2019

Preprint

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We study boundary traces of shift-invariant diffusions: two-dimensional diffusions in the upper half-plane R × [0, ∞) (or in R × [0, R)) invariant under horizontal translations. We prove that the corresponding trace processes are Lévy processes with completely monotone jumps, and, conversely, every Lévy process with completely monotone jumps is a boundary trace of some shift-invariant diffusion. Up to some natural transformations of space and time, this correspondence is bijective. We also reformulate this result in the language of additive functionals of the Brownian motion in [0, ∞) (or in [0, R)), and Brownian excursions. Our main tool is the recent extension of Krein's spectral theory of strings, due to Eckhardt and Kostenko.

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Principal Values of the Integral Functionals of Brownian Motion: Existence, Continuity and an Extension of Itô’s Formula

Cited by 17 publications

References 10 publications

Itô’s excursion theory and its applications

Itô’s excursion theory and its applications

Harmonic extension technique for non-symmetric operators with completely monotone kernels

Boundary traces of shift-invariant diffusions in half-plane

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