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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.
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 ..."
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Cited by 65 (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...
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
Maximum Of Fractional Brownian Motion: Probabilities Of Small Values
, 2000
"... Let b fl (t), b fl (0) = 0 be fractional Brownian motion, i.e., a Gaussian process with the structure function Ejb fl (t)\Gammab fl (s)j 2 = jt\Gammasj fl , 0 ! fl ! 2. We study the logarithmic asymptotics of P T = Pfb fl (t) ! 1; t 2 T \Deltag as T !1, where \Delta is either the interval (0; ..."
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Cited by 8 (2 self)
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Let b fl (t), b fl (0) = 0 be fractional Brownian motion, i.e., a Gaussian process with the structure function Ejb fl (t)\Gammab fl (s)j 2 = jt\Gammasj fl , 0 ! fl ! 2. We study the logarithmic asymptotics of P T = Pfb fl (t) ! 1; t 2 T \Deltag as T !1, where \Delta is either the interval (0; 1) or a bounded region that contains a vicinity of 0 for the case of multidimensional time. It is shown that log P T = \GammaD log T (1 + o(1)), where D is the dimension of zeroes of b fl (t) in the former case and the dimension of time in the latter. AMS 1991 subject classifications. Primary 60G15, 60G18, 60G70 Key words: Extreme values, Gaussian processes, fractional Brownian motion, selfsimilar processes. Running title  Maximum FBM : Probabilities of Small Values 1 1. INTRODUCTION AND RESULTS In connection with an analysis of the fractal nature of solutions of the Burgers equation with random initial data [13,14], Ya.Sinai posed the problem of finding the asymptotics for the probabi...
The Mean Velocity Of A Brownian Motion In A Random Lévy Potential
"... . A Brownian motion in a random L'evy potential V , is the informal solution of the stochastic differential equation dX t = dB t \Gamma 1 2 V 0 (X t ) dt ; where B is a Brownian motion independent of V . We generalize some results of KawazuTanaka [8], who considered for V a Brownian motion wi ..."
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Cited by 7 (0 self)
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. A Brownian motion in a random L'evy potential V , is the informal solution of the stochastic differential equation dX t = dB t \Gamma 1 2 V 0 (X t ) dt ; where B is a Brownian motion independent of V . We generalize some results of KawazuTanaka [8], who considered for V a Brownian motion with drift, by proving that X t t converges almost surely to a constant, the mean velocity, which we compute in terms of the L'evy exponent OE of V , defined by : E \Theta e mV (t) = e \GammatOE(m) . 1. Introduction Example 1. Given a Poisson cloud (oe i ; i 2 Z) on the real line, we consider a process X such that: ffl X behaves like a Brownian motion between adjacent barriers oe i and oe i+1 ; ffl when X hits a barrier, he flips a coin and goes to the right with probability p, to the left with probability q = 1 \Gamma p. Observe that it is natural to require stationarity of the random media, that is invariance in law under translations, so that the intervals Date: November 2...
Annealed tail estimates for a Brownian motion in a drifted Brownian potential
, 2006
"... Talet We study Brownian motion in a drifted Brownian potential. Kawazu and Tanaka [23] exhibited two speed regimes for this process, depending on the drift. They supplemented these laws of large numbers by central limit theorems, which were recently completed by Hu, Shi and Yor [19] using stochastic ..."
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Talet We study Brownian motion in a drifted Brownian potential. Kawazu and Tanaka [23] exhibited two speed regimes for this process, depending on the drift. They supplemented these laws of large numbers by central limit theorems, which were recently completed by Hu, Shi and Yor [19] using stochastic calculus. We studied large deviations [34], showing among other results that the rate function in the annealed setting, that is after averaging over the potential, has a flat piece in the ballistic regime. In this paper, we focus on this subexponential regime, proving that the probability of deviating below the almost sure speed has a polynomial rate of decay, and computing the exponent in this power law. This provides the continuoustime analogue of what Dembo, Peres and Zeitouni proved for the transient random walk in random environment [13]. Our method takes a completely different route, making use of Lamperti’s representation together with an iteration scheme. 1
Electronic addresses:
, 1995
"... The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts such as continuous time finance models and onedimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a ..."
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The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts such as continuous time finance models and onedimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.
LOCAL TIME OF A DIFFUSION IN A STABLE LÉVY ENVIRONMENT
, 909
"... Abstract. We consider a onedimensional diffusion in a stable Lévy environment. We show that the normalized local time process refocused at the bottom of the standard valley with height log t, (LX(t,mlog t + x)/t,x ∈ R), converges in law to a functional of two independent Lévy processes conditioned ..."
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Abstract. We consider a onedimensional diffusion in a stable Lévy environment. We show that the normalized local time process refocused at the bottom of the standard valley with height log t, (LX(t,mlog t + x)/t,x ∈ R), converges in law to a functional of two independent Lévy processes conditioned to stay positive. To prove this result, we show that the law of the standard valley is close to a twosided Lévy process conditioned to stay positive. We also obtain the limit law of the supremum of the normalized local time. This result has been obtained by Andreoletti and Diel [1] in the case of a Brownian environment.
unknown title
, 2009
"... Estimates on the speedup and slowdown for a diffusion in a drifted brownian potential. ..."
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Estimates on the speedup and slowdown for a diffusion in a drifted brownian potential.