Hyperbolic and fractional hyperbolic Brownian motion

Lao, Lan-Jun; Orsingher, Enzo

doi:10.1080/17442500701433509

Cited by 18 publications

(22 citation statements)

References 12 publications

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“…If ν = 1 2 it was shown in [46] that p (·) is the density of the composition of an hyperbolic Brownian motion with a reflecting standard Brownian motion. In this framework other cases can be investigated in much detail.…”

Section: Perspectives In Fractional Calculus: Fdes In Non-euclidean Wmentioning

confidence: 99%

Fractional Calculus: Quo Vadimus? (Where are we Going?)

Machado

Mainardi

Kiryakova

2015

FCAA

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Discussions under this title were held during a special session in frames of the International Conference "Fractional Differentiation and Applications" (ICFDA '14) held in Catania (Italy), 23-25 June 2014, see details at http://www.icfda14.dieei.unict.it/. Along with the presentations made during this session, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend.The intention of this special session was to continue the useful traditions from the first conferences on the Fractional Calculus (FC) topics, to pose open problems, challenging hypotheses and questions "where to go", to discuss them and try to find ways to resolve.MSC 2010 : Primary 26A33, 01A67; Secondary 34A08, 35R11, 60G22

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Section: Perspectives In Fractional Calculus: Fdes In Non-euclidean Wmentioning

confidence: 99%

Fractional Calculus: Quo Vadimus? (Where are we Going?)

Machado

Mainardi

Kiryakova

2015

FCAA

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“…Our result and formula (16) in [5] are related. Byczkowski and Ma lecki [5, Theorem 3.2] obtain, with a different method, the Poisson kernel for the Poincaré model D n .…”

Section: Two-dimensional Casementioning

confidence: 50%

“…In [5] the Poisson kernel of a ball in the n-dimensional Poincaré disc D n , where n > 2, is obtained in formula (16) by combining the results of Theorems 2.2, 3.1, and 3.2 therein. We obtain the same result, up to a conformal mapping, for the half-plane model H n by means of a simple alternative proof based on the method of separation of variables.…”

Section: Introductionmentioning

confidence: 99%

Hitting Spheres on Hyperbolic Spaces

Cammarota¹,

Orsingher²

2013

Theory Probab. Appl.

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For a hyperbolic Brownian motion on the Poincaré half-plane H 2 , starting from a point z = (η, α) inside a hyperbolic disc U of radiusη, we obtain the probability of hitting the boundary ∂U at the point (η,ᾱ). Forη → ∞ we derive the asymptotic Cauchy hitting distribution on ∂H 2 . In particular, it follows that the hyperbolic Brownian motion starting at (x, y) ∈ H 2 "hits" the boundary of H 2 at a point which is Cauchy distributed with scale parameter y = y/(x 2 +y 2 ) and position parameter x = x/(x 2 + y 2 ). For small values of η andη we obtain the classical Euclidean Poisson kernel. The exit probabilities from a hyperbolic annulus in H 2 of radii η 1 and η 2 are derived and the transient behavior of hyperbolic Brownian motion is considered. Similar probabilities are calculated also for a Brownian motion on the surface of the three-dimensional sphere. For the hyperbolic half-space H n we obtain, with a proof based on the method of separation of variables, the Poisson kernel of a ball. For small domains in H n we obtain the n-dimensional Euclidean Poisson kernel. The exit probabilities from an annulus are derived also in the n-dimensional case.

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“…Similar probabilities (yet not this one in particular), of hitting some boundary, or a ball around it, have been studied for the hyperbolic Brownian motion in [6,7,14,21,22] In Section 2, we analyse the dynamics of (1.1) under different parameter configurations, and propose several space transformations to translate properties of one model configuration to the other. In Section 3, we use these maps to derive an exact formula for P for general parameter values.…”

Section: Introductionmentioning

confidence: 93%

On the probability of hitting the boundary for Brownian motions on the SABR plane

Gulisashvili¹,

Horvath

Jacquier

2016

Electron. Commun. Probab.

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Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models???related to the SABR model in mathematical finance???which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change methods

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Hyperbolic and fractional hyperbolic Brownian motion

Cited by 18 publications

References 12 publications

Fractional Calculus: Quo Vadimus? (Where are we Going?)

Fractional Calculus: Quo Vadimus? (Where are we Going?)

Hitting Spheres on Hyperbolic Spaces

On the probability of hitting the boundary for Brownian motions on the SABR plane

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