Results 1  10
of
18
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
Limit laws for transient random walks in random environment on Z
, 2007
"... We consider transient random walks in random environment on Z with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level n converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description o ..."
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Cited by 15 (6 self)
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We consider transient random walks in random environment on Z with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level n converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.
Rates of convergence of diffusions with drifted Brownian potentials
 TRANS. AMER. MATH. SOC
, 1999
"... We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion ..."
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Cited by 15 (5 self)
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We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion process which was previously established by Kawazu and Tanaka, but also obtain all the possible convergence rates. An interesting feature of our approach is that it shows a clear relationship between drifted Brownian potentials and Bessel processes.
The Problem of the Most Visited Site in Random Environment
, 2000
"... this paper is to study the favourite points. For n 0 and x 2 Z, ..."
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Cited by 9 (1 self)
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this paper is to study the favourite points. For n 0 and x 2 Z,
On the concentration of Sinai’s walk
, 2008
"... Abstract: We consider Sinai’s random walk in random environment. We prove that for an interval of time [1,n] Sinai’s walk sojourns in a small neighborhood of the point of localization for the quasi totality of this amount of time. Moreover the local time at the point of localization normalized by n ..."
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Cited by 7 (7 self)
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Abstract: We consider Sinai’s random walk in random environment. We prove that for an interval of time [1,n] Sinai’s walk sojourns in a small neighborhood of the point of localization for the quasi totality of this amount of time. Moreover the local time at the point of localization normalized by n converges in probability to a well defined random variable of the environment. From these results we get applications to the favorite sites of the walk and to the maximum of the local time. 1
RATES OF CONVERGENCE OF A TRANSIENT DIFFUSION IN A SPECTRALLY NEGATIVE LÉVY POTENTIAL
, 2008
"... We consider a diffusion process X in a random Lévy potential V which is a solution of the informal stochastic differential equation dXt = dβt − ..."
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Cited by 4 (1 self)
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We consider a diffusion process X in a random Lévy potential V which is a solution of the informal stochastic differential equation dXt = dβt −
(1.1)
, 906
"... Abstract. It is wellknown that the excursions of a onedimensional diffusion process can be studied by considering a certain Riccati equation associated with the process. We show that, in many cases of interest, the Riccati equation can be solved in terms of an infinite continued fraction. We exami ..."
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Abstract. It is wellknown that the excursions of a onedimensional diffusion process can be studied by considering a certain Riccati equation associated with the process. We show that, in many cases of interest, the Riccati equation can be solved in terms of an infinite continued fraction. We examine the probabilistic significance of the expansion. To illustrate our results, we discuss