Results 1  10
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19
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 98 (9 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.
Conditioned Brownian trees
"... We consider a Brownian tree consisting of a collection of onedimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the Brownian snake driven by a normalized Brownian excursion, and thus ..."
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Cited by 14 (6 self)
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We consider a Brownian tree consisting of a collection of onedimensional Brownian paths started from the origin, whose genealogical structure is given by the Continuum Random Tree (CRT). This Brownian tree may be generated from the Brownian snake driven by a normalized Brownian excursion, and thus yields a convenient representation of the socalled Integrated SuperBrownian Excursion (ISE), which can be viewed as the uniform probability measure on the tree of paths. We discuss different approaches that lead to the definition of the Brownian tree conditioned to stay on the positive halfline. We also establish a Verwaatlike theorem showing that this conditioned Brownian tree can be obtained by rerooting the unconditioned one at the vertex corresponding to the minimal spatial position. In terms of ISE, this theorem yields the following fact: Conditioning ISE to put no mass on]−∞, −ε [ and letting ε go to 0 is equivalent to shifting the unconditioned ISE to the right so that the leftmost point of its support becomes the origin. We derive a number of explicit estimates and formulas for our conditioned Brownian trees. In particular, the probability that ISE puts no mass on] − ∞, −ε [ is shown to behave like 2ε 4 /21 when ε goes to 0. Finally, for the conditioned Brownian tree with a fixed height h, we obtain a decomposition involving a spine whose distribution is absolutely continuous with respect to that of a ninedimensional Bessel process on the time interval [0,h], and Poisson processes of subtrees originating from this spine. 1
The Most Visited Sites of Symmetric Stable Processes
, 2000
"... this paper remain unchanged if we replace ..."
The First Exit Time of Planar Brownian Motion from the Interior of a Parabola
 Ann. Probab
, 2001
"... Introduction For x 2 IR n nf0g, we let (x) be the angle between x and the point (1; 0; : : : ; 0). The right circular cone of angle 0 < < is the domain = fx 2 IR n : (x) < g. Let fB t : t 0g be the n{dimensional Brownian motion and denote by E x and P x the expectation and probability ass ..."
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Cited by 8 (4 self)
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Introduction For x 2 IR n nf0g, we let (x) be the angle between x and the point (1; 0; : : : ; 0). The right circular cone of angle 0 < < is the domain = fx 2 IR n : (x) < g. Let fB t : t 0g be the n{dimensional Brownian motion and denote by E x and P x the expectation and probability associated with this motion starting at x and denote by = infft > 0 : B t = 2 g its rst exit time from . The following result was
On the fundamental solution of the KolmogorovShiryaev equation. The Shiryaev Festschrift (Metabief 2005
, 2006
"... We derive an integral representation for the fundamental solution of the Kolmogorov forward equation: ft = −((1+µx)f)x + (ν x 2 f)xx associated with the Shiryaev process X solving: dXt = (1+µXt) dt + σXt dBt where µ ∈ IR, ν = σ 2 /2> 0 and B is a standard Brownian motion. The method of proof is base ..."
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Cited by 3 (2 self)
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We derive an integral representation for the fundamental solution of the Kolmogorov forward equation: ft = −((1+µx)f)x + (ν x 2 f)xx associated with the Shiryaev process X solving: dXt = (1+µXt) dt + σXt dBt where µ ∈ IR, ν = σ 2 /2> 0 and B is a standard Brownian motion. The method of proof is based upon deriving and inverting a Laplace transform. Basic properties of X needed in the proof are reviewed. 1.
INFINITELY DIVISIBILITY OF SOLUTIONS OF SOME SEMISTABLE INTEGRODIFFERENTIAL EQUATIONS AND EXPONENTIAL FUNCTIONALS OF LÉVY PROCESSES
, 2006
"... We provide the increasing qharmonic functions associated to the following family of integrodifferential operators, for any α> 0, γ ≥ 0 and f ∈ D(L (α,ψ,γ)), (0.1) L (α,ψ,γ) f(x) = x −α ..."
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Cited by 1 (1 self)
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We provide the increasing qharmonic functions associated to the following family of integrodifferential operators, for any α> 0, γ ≥ 0 and f ∈ D(L (α,ψ,γ)), (0.1) L (α,ψ,γ) f(x) = x −α
LIMITING LAWS ASSOCIATED WITH BROWNIAN MOTION PERTURBED BY NORMALIZED EXPONENTIAL WEIGHTS, I
, 2008
"... Abstract. Let (Bt; t ≥ 0) be a one dimensional Brownian motion, with local time process (Lx t; t ≥ 0, x ∈ R). We determine the rate of decay of Z V [ { t (x): = Ex exp − 1 L ..."
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Abstract. Let (Bt; t ≥ 0) be a one dimensional Brownian motion, with local time process (Lx t; t ≥ 0, x ∈ R). We determine the rate of decay of Z V [ { t (x): = Ex exp − 1 L
The Laguerre process and generalized
, 708
"... In this paper, we study complex Wishart processes or the socalled Laguerre processes (Xt)t≥0. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semigroup. We also give absolute ..."
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In this paper, we study complex Wishart processes or the socalled Laguerre processes (Xt)t≥0. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semigroup. We also give absolutecontinuity relations between different indices. Finally, we compute the density function of the socalled generalized Hartman–Watson law as well as the law of T0: = inf{t,det(Xt) = 0} when the size of the matrix is 2. Keywords: generalized Hartman–Watson law; Gross–Richards formula; Laguerre process; special functions of matrix argument