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On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 119 (12 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
Exponential functionals of Lévy processes
 Probabilty Surveys
, 2005
"... Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0 ..."
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Cited by 41 (4 self)
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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
Poisson kernels of half–spaces in real hyperbolic spaces
, 2005
"... We provide an integral formula for the Poisson kernel of halfspaces for Brownian motion in real hyperbolic space H n. This enables us to find asymptotic properties of the kernel. Our starting point is the formula for its Fourier transform. When n = 3, 4 or 6 we give an explicit formula for the Pois ..."
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Cited by 4 (2 self)
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We provide an integral formula for the Poisson kernel of halfspaces for Brownian motion in real hyperbolic space H n. This enables us to find asymptotic properties of the kernel. Our starting point is the formula for its Fourier transform. When n = 3, 4 or 6 we give an explicit formula for the Poisson kernel itself. In the general case we give various asymptotics and show convergence to the Poisson kernel of H n.
Brownian motion and Harmonic functions on Sol(p, q)
, 2011
"... The Lie group Sol(p, q) is the semidirect product induced by the action of R on R2 which is given by (x, y) ↦ → (epzx, e−qzy), z ∈ R. Viewing Sol(p, q) as a 3dimensional manifold, it carries a natural Riemannian metric and LaplaceBeltrami operator. We add a linear drift term in the zvariable to ..."
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Cited by 1 (1 self)
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The Lie group Sol(p, q) is the semidirect product induced by the action of R on R2 which is given by (x, y) ↦ → (epzx, e−qzy), z ∈ R. Viewing Sol(p, q) as a 3dimensional manifold, it carries a natural Riemannian metric and LaplaceBeltrami operator. We add a linear drift term in the zvariable to the latter, and study the associated Brownian motion with drift. We derive a central limit theorem and compute the rate of escape. Also, we introduce the natural geometric compactification of Sol(p, q) and explain how Brownian motion converges almost surely to the boundary in the resulting topology. We also study all positive harmonic functions for the Laplacian with drift, and determine explicitly all minimal harmonic functions. All this is carried out with a strong emphasis on understanding and using the geometric features of Sol(p, q), and in particular the fact that it can be described as the horocyclic product of two hyperbolic planes with curvatures −p2 and −q2, respectively.
Poisson kernel of half spaces in real hyperbolic spaces
, 2006
"... To cite this version: Tomasz Byczkowski, Piotr Graczyk, Andrzej Stos. Poisson kernel of half spaces in real hyper ..."
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To cite this version: Tomasz Byczkowski, Piotr Graczyk, Andrzej Stos. Poisson kernel of half spaces in real hyper