Results 1  10
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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 (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.
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 34 (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
Pricing equity derivatives subject to bankruptcy
 Mathematical Finance
, 2006
"... We solve in closed form a parsimonious extension of the Black–Scholes–Merton model with bankruptcy where the hazard rate of bankruptcy is a negative power of the stock price. Combining a scale change and a measure change, the model dynamics is reduced to a linear stochastic differential equation who ..."
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Cited by 27 (4 self)
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We solve in closed form a parsimonious extension of the Black–Scholes–Merton model with bankruptcy where the hazard rate of bankruptcy is a negative power of the stock price. Combining a scale change and a measure change, the model dynamics is reduced to a linear stochastic differential equation whose solution is a diffusion process that plays a central role in the pricing of Asian options. The solution is in the form of a spectral expansion associated with the diffusion infinitesimal generator. The latter is closely related to the Schrödinger operator with Morse potential. Pricing formulas for both corporate bonds and stock options are obtained in closed form. Term credit spreads on corporate bonds and implied volatility skews of stock options are closely linked in this model, with parameters of the hazard rate specification controlling both the shape of the term structure of credit spreads and the slope of the implied volatility skew. Our analytical formulas are easy to implement and should prove useful to researchers and practitioners in corporate debt and equity derivatives markets.
Magnetic bottles on the Poincaré halfplane: spectral asymptotics
"... wwwfourier.ujfgrenoble.fr/prepublications.html We consider a magnetic Laplacian −∆A = (id + A) ⋆ (id + A) on the Poincaré upperhalf plane H, when the magnetic field dA is infinite at infinity and such that −∆A has pure discret spectrum. We obtain the asymptotic behavior of the counting function ..."
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Cited by 4 (4 self)
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wwwfourier.ujfgrenoble.fr/prepublications.html We consider a magnetic Laplacian −∆A = (id + A) ⋆ (id + A) on the Poincaré upperhalf plane H, when the magnetic field dA is infinite at infinity and such that −∆A has pure discret spectrum. We obtain the asymptotic behavior of the counting function of the eigenvalues.
The Schrödinger Operator with Morse Potential on the Right Half Line
, 712
"... This paper studies the Schrödinger operator with Morse potential Vk(u) = 1 4 e2u + ke u on a right halfline [u0, ∞), and determines the Weyl asymptotics of eigenvalues for constant boundary conditions at the endpoint u0. In consequence it obtains information on the location of zeros of the Whittak ..."
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Cited by 2 (0 self)
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This paper studies the Schrödinger operator with Morse potential Vk(u) = 1 4 e2u + ke u on a right halfline [u0, ∞), and determines the Weyl asymptotics of eigenvalues for constant boundary conditions at the endpoint u0. In consequence it obtains information on the location of zeros of the Whittaker function Wκ,µ(x), for fixed real parameters κ,x with x> 0, viewed as an entire function in the complex variable µ. In this case all zeros lie on the imaginary axis, with the exception, if k < 0 of a finite number of real zeros which lie in the interval κ  < k. We obtain an asymptotic formula for the number of zeros N(T) = {ρ  Wκ,ρ(x) = 0, Im(ρ)  < T} of the form N(T) = 2 2 πT log T + π (2log 2−1−log x)T +O(1). Parallels are observed with zeros of the Riemann zeta function. 1.
Limiting behaviors of the Brownian motions on
, 901
"... brownian motions on hyperbolic spaces 1 ..."
Limiting behaviors of the Brownian
, 901
"... Abstract: By adopting the upper half space realizations of the real, complex and quaternionic hyperbolic spaces and solving the corresponding stochastic differential equations, we can represent the Brownian motions on these classical families of the hyperbolic spaces as explicit Wiener functionals. ..."
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Abstract: By adopting the upper half space realizations of the real, complex and quaternionic hyperbolic spaces and solving the corresponding stochastic differential equations, we can represent the Brownian motions on these classical families of the hyperbolic spaces as explicit Wiener functionals. Using the representations, we show that the almost sure convergence of the Brownian motions and the central limit theorems for the radial components as time tends to infinity are easily obtained. We also give a straightforward strategy to obtain the explicit expressions for the Poisson kernels by combining the representations with some results on the distributions of the random variables which are defined by the perpetual (infinite) integrals of the usual geometric Brownian motions with negative drifts.
Exponential functionals of Brownian motion and classone Whittaker functions
, 2010
"... Abstract. We consider exponential functionals of a Brownian motion with drift in R n, defined via a collection of linear functionals. We give a characterisation of the Laplace transform of their joint law as the unique bounded solution, up to a constant factor, to a Schrödingertype partial differen ..."
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Abstract. We consider exponential functionals of a Brownian motion with drift in R n, defined via a collection of linear functionals. We give a characterisation of the Laplace transform of their joint law as the unique bounded solution, up to a constant factor, to a Schrödingertype partial differential equation. We derive a similar equation for the probability density. We then characterise all diffusions which can be interpreted as having the law of the Brownian motion with drift conditioned on the law of its exponential functionals. In the case where the family of linear functionals is a set of simple roots, the Laplace transform of the joint law of the corresponding exponential functionals can be expressed in terms of a (classone) Whittaker function associated with the corresponding root system. In this setting, we establish some basic properties of the corresponding diffusion processes. Résumé. Nous étudions certaines fonctionelles d’un mouvement Brownien avec dérive dans R n qui sont définies par une collection de fonctionnelles linéaires. Nous donnons une caractérisation de la transformée de Laplace de leur loi jointe comme l’unique solution bornée, à une constante près d’une équation aux dérivées partielles de type Schrödinger. Nous déduisons une équation similaire pour la densité. Nous caractérisons ensuite toutes les diffusions qui peuvent être interprétées comme ayant la loi d’un mouvement Brownien avec dérive conditionné par la loi de ses fonctionelles exponentielles. Dans le cas où la famille des fonctionelles est un ensemble de racines simples, la transformée de Laplace de la densité jointe des fonctionnelles exponentielles correspondantes peut être exprimée en termes d’une fonction de Whittaker de classe 1 associée au système. Dans ce cadre, nous établissons quelques