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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 ..."
Abstract

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.
On The Distribution And Asymptotic Results For Exponential Functionals Of Lévy Processes
, 1997
"... . The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional A t := R t 0 e s ds, where ( s ; s 0) denotes a L'evy process. When A1 ! 1, we show that in most cases, the law of A1 is a solution of an integrodifferential equation ; moreover, this ..."
Abstract

Cited by 66 (8 self)
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. The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional A t := R t 0 e s ds, where ( s ; s 0) denotes a L'evy process. When A1 ! 1, we show that in most cases, the law of A1 is a solution of an integrodifferential equation ; moreover, this law is characterized by its integral moments. When the process is asymptotically ffstable, we prove that t \Gamma1=ff log A t converges in law, as t !1, to the supremum of an ffstable L'evy process ; in particular, if E [ 1 ] ? 0, then ff = 1 and (1=t) log A t converges almost surely to E [ 1 ]. Eventually, we use Girsanov's transform to give the explicit behavior of E \Theta (a +A t ()) \Gamma1 as t ! 1, where a is a constant, and deduce from this the rate of decay of the tail of the distribution of the maximum of a diffusion process in a random L'evy environment. 1. Introduction We first describe three different sources of interest for exponential functionals of Brownian mot...
Breaking supersymmetry in a onedimensional random Hamiltonian
, 805
"... The onedimensional supersymmetric random Hamiltonian Hsusy = − d2 dx 2 +φ 2 +φ ′ , where φ(x) is a Gaussian white noise of zero mean and variance g, presents particular spectral and localization properties at low energy: a Dyson singularity in the integrated density of states (IDoS) N(E) ∼ 1 / ln ..."
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The onedimensional supersymmetric random Hamiltonian Hsusy = − d2 dx 2 +φ 2 +φ ′ , where φ(x) is a Gaussian white noise of zero mean and variance g, presents particular spectral and localization properties at low energy: a Dyson singularity in the integrated density of states (IDoS) N(E) ∼ 1 / ln 2 E and a delocalization transition related to the behaviour of the Lyapunov exponent (inverse localization length) vanishing like γ(E) ∼ 1/  lnE  as E → 0. We study how this picture is affected by breaking supersymmetry with a scalar random potential: H = Hsusy + V (x) where V (x) is a Gaussian white noise of variance σ. In the limit σ ≪ g 3, a fraction of states N(0) ∼ g / ln 2 (g 3 /σ) migrate to the negative spectrum and the Lyapunov exponent reaches a finite value γ(0) ∼ g / ln(g 3 /σ) at E = 0. Exponential (Lifshits) tail of the IDoS for E → − ∞ is studied in detail and is shown to involve a competition between the two noises φ(x) and V (x) whatever the larger is. This analysis relies on analytic results for N(E) and γ(E) obtained by two different methods: a stochastic method and the replica method. The problem of extreme value statistics of eigenvalues is also considered (distribution of the n−th excited state energy). The results are analyzed in the context of classical diffusion in a random force field in the presence of random annihilation/creation local rates. 1