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10
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 (10 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.
A particle migrating randomly on a sphere
 J. Theoretical Prob
, 1997
"... Consider a particle moving on the surface of the unit sphere in R 3 and heading towards a specific destination with a constant average speed, but subject to random deviations. The motion is modeled as a diffusion with drift restricted to the surface of the sphere. Expressions are set down for variou ..."
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Cited by 21 (11 self)
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Consider a particle moving on the surface of the unit sphere in R 3 and heading towards a specific destination with a constant average speed, but subject to random deviations. The motion is modeled as a diffusion with drift restricted to the surface of the sphere. Expressions are set down for various characteristics of the process including expected travel time to a cap, the limiting distribution, the likelihood ratio and some estimates for parameters appearing in the model. KEY WORDS: Drift; great circle path; likelihood ratio; poleseeking; skew product; spherical Brownian motion; stochastic differential equation; travel time. 1.
Directional Mixture Models and Optimal Estimation of the Mixing Density
, 2000
"... The authors develop consistent nonparametric estimation techniques for the directional mixing density. Classical spherical harmonics are used to adapt Euclidean techniques to this directional environment. Minimax rates of convergence are obtained for rotationally invariant densities verifying variou ..."
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Cited by 3 (2 self)
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The authors develop consistent nonparametric estimation techniques for the directional mixing density. Classical spherical harmonics are used to adapt Euclidean techniques to this directional environment. Minimax rates of convergence are obtained for rotationally invariant densities verifying various smoothness conditions. It is found that the di#erence in smoothness between the Laplace, the Gaussian and the von MisesFisher distributions, lead to contrasting inferential conclusions. R ESUM E Les auteurs developpent des techniques d'estimation non parametriques convergentes pour la densitedelavariablemelangeante dans un contexte directionnel. Pour adapter les techniques euclidiennes, ils font appel aux harmoniques spheriques classiques. Ils determinent le taux de convergence minimax sous di#erentes conditions de regularite de la densite invariante par rotation. Les lois de Laplace, de Gauss et de vonMises Fisher ne jouissant pastoutesdumeme degrederegularite, on comprend alors pourq...
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.
Directed polymers and the quantum Toda lattice
, 2009
"... We give a characterization of the law of the partition function of a Brownian directed polymer model in terms of the eigenfunctions of the quantum Toda lattice. This is obtained via a multidimensional generalization of theorem of Matsumoto and Yor concerning exponential functionals of Brownian motio ..."
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Cited by 3 (1 self)
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We give a characterization of the law of the partition function of a Brownian directed polymer model in terms of the eigenfunctions of the quantum Toda lattice. This is obtained via a multidimensional generalization of theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion.
The diffusion of Radon shape
 Adv. App. Prob
, 2006
"... Almost thirty years ago, D.G. Kendall [8] considered diffusions of shape induced by independent Brownian motions in Euclidean space. In this paper, we consider a different class of diffusions of shape, induced by the projections of a randomly rotating labelled ensemble. In particular, we study diffu ..."
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Cited by 2 (1 self)
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Almost thirty years ago, D.G. Kendall [8] considered diffusions of shape induced by independent Brownian motions in Euclidean space. In this paper, we consider a different class of diffusions of shape, induced by the projections of a randomly rotating labelled ensemble. In particular, we study diffusions of shapes induced by projections of planar triangular configurations of labelled points onto a fixed straight line. That is, we consider the process in Σ 3 1 (the shape space of labelled triads in R 1) that results from extracting the “shape information ” from the projection of a given labelled planar triangle, as this evolves under the action of a Brownian motion in SO(2). We term the thus defined diffusions Radon diffusions and derive explicit stochastic differential equations and stationary distributions. The latter belong to the family of angular central Gaussian distributions. In addition, we discuss how these Radon diffusions and their limiting distributions are related to the shape of the initial triangle, and explore whether the relationship is bijective. The triangular case is then used as a pivot for the study of processes in Σ k 1 arising from projections of an arbitrary number k of labelled points on the plane. Finally, we discuss the problem of Radon diffusions in general shapespace Σ k n. Keywords: Single particle Biophysics; circular Brownian motion; D.G. Kendall’s shape theory; angular central Gaussian distribution; integral geometry; stochastic geometry; random processes of geometrical
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
ON TURÁN TYPE INEQUALITIES FOR MODIFIED BESSEL FUNCTIONS
"... Abstract. In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Turán type inequalities for these functions. Moreover, we present some new Turán ..."
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Abstract. In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Turán type inequalities for these functions. Moreover, we present some new Turán type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has an application in the study of stability of radially symmetric solutions in a generalized FitzHughNagumo equation in two spatial dimensions. At the end of this note an open problem is posed, which may be of interest for further research. 1. Some inequalities for modified Bessel functions Let us denote with Iν and Kν the modified Bessel functions of the first and second kind, respectively. For definitions, recurrence formulas and other properties of modified Bessel functions of the first and second kind we refer to the classical book of Watson [35]. In 2007, motivated by a problem which arises in biophysics, Penfold et al. [31]
Dedicated to Albert N. Shiryaev on the occasion of his 70th birthday On the Fundamental Solution of the KolmogorovShiryaev Equation
, 2005
"... 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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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.
Existence and properties of pseudoinverses for Bessel and related processes
, 2008
"... Abstract It is shown that the tail probability of a Bessel process is the distributio function of a random time which is related to first and last passage times of Bessel processes. ..."
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Abstract It is shown that the tail probability of a Bessel process is the distributio function of a random time which is related to first and last passage times of Bessel processes.