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...m on the right hand side converges almost surely to sup s1ss . Remark 4.3. When ff = 2, i.e.sis a Brownian motion, the preceding result is well-known (see e.g. Durrett [12], Pitman-Yor [25], RevuzYor =-=[26]-=-, Chapter I, Exercise 1.18, or Bertoin-Werner [3] and [4]), and is a main step in the proof of Spitzer's theorem on the winding number of planar Brownian motion. It is natural to think that the assump...

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...Gamma(b + iu) \Gamma(b) ; so that : OE (b) (u) = log ` \Gamma(b + iu) \Gamma(b) ' : Then, the representation of OE (b) derives from the formulae, valid for Re(z) ? 0, (see Formula (1.3.14) of Lebedev =-=[22]-=-) : log \Gamma(z) = Z z 1 /(x) dx ; /(z) = \Gammafl + Z 1 0 e \Gammat \Gamma e \Gammatz 1 \Gamma e \Gammat dt : (iv) Holder's inequality implies that the function g() = 1 1\Gammaff log E \Theta H ff i...

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... a random L'evy environment. Since the proof of Proposition 2.1 relies heavily on the properties of generalized Ornstein-Uhlenbeck processes (see Hadjiev[18], Novikov [24] and Samorodnitsky and Taqqu =-=[27]-=-, section 3.6), we give in Appendix 1 a summary of the properties of these processes. Eventually, we devote Appendix 2 to the study of a Poisson process with drift : there, we apply the results of Sec...

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...1. The stochastic (difference) equation. The stochastic difference equation Y n = A n Y n\Gamma1 +B n with f(A n ; B n ); n 2 Ng an i.i.d sequence, has been thoroughly studied by Vervaat [29], Kesten =-=[20]-=-, and more recently by Brandt [8]. This equation arises in various disciplines, for example economics, physics, nuclear technology, ... (see Vervaat [29]) If we have the convergence Y n d 7\Gamma! n!1...

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...s ds whensis a Brownian motion with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette [6], Dufresne [11], Geman-Yor [15] and Yor ([31], =-=[30]-=-). c) In her PhD Thesis [23], C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a frozen random potential V . Let us give examples of...

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...s pricing. 6.1. The stochastic (difference) equation. The stochastic difference equation Y n = A n Y n\Gamma1 +B n with f(A n ; B n ); n 2 Ng an i.i.d sequence, has been thoroughly studied by Vervaat =-=[29]-=-, Kesten [20], and more recently by Brandt [8]. This equation arises in various disciplines, for example economics, physics, nuclear technology, ... (see Vervaat [29]) If we have the convergence Y n d...

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... us2t : and : sup t0 1 t E [M 2 t ]s2. Now, remark that the equivalent (4.6) is an immediate consequence of the convergence (4.5) applied to the process \Gamma, the Monotone Density Theorem (see e.g. =-=[5]-=-, Theorem 1.7.2), and the relation : d dt E [log A t (\Gamma)] = E e \Gamma t A t (\Gamma) = E 1 A t () ; (the last equality coming from Lemma 2.3). We now show how to use Girsanov transform to estima...

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...-stopping time T: E [jM1 \Gamma M T \Gamma j=F T ]s2 OE(1) e \Gamma + T \Gamma 2 OE(1) : Hence, the martingale (M t ; ts0) is in BMO, and kMkBMO 1s2 OE(1) . Thanks to John-Nirenberg's inequality (see =-=[10]-=-, Chapter VI, Theorem 109), if 0sff ! 1 2OE(1) , we have: E \Theta e ffA1 ! +1 : In fact, we obtain better results : Proposition 3.3. The law of A1 is determined by its integral moments m n := E [A n ...

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...he distribution of A t () = R t 0 e s ds whensis a Brownian motion with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette [6], Dufresne =-=[11]-=-, Geman-Yor [15] and Yor ([31], [30]). c) In her PhD Thesis [23], C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a frozen random p...

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...t ; ts0 ; P x ; x 2 E), with state space E = R+ ; R or R n , such that : ( 1 a ae X at ; ts0 ; P a ae x ) d = (X t ; ts0 ; P x ) (a ? 0; x 2 E): Such Markov processes have been considered by Lamperti =-=[21]-=- in relation with exponentials of L'evy processes. Let ( t ; ts0 ) be a L'evy process, starting from zero, independent from X, and let A t = A t (\Gamma) = R t 0 e \Gamma s ds. Proposition 5.5. The ge...

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...of A t () = R t 0 e s ds whensis a Brownian motion with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette [6], Dufresne [11], Geman-Yor =-=[15]-=- and Yor ([31], [30]). c) In her PhD Thesis [23], C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a frozen random potential V . Let...

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...ation. The stochastic difference equation Y n = A n Y n\Gamma1 +B n with f(A n ; B n ); n 2 Ng an i.i.d sequence, has been thoroughly studied by Vervaat [29], Kesten [20], and more recently by Brandt =-=[8]-=-. This equation arises in various disciplines, for example economics, physics, nuclear technology, ... (see Vervaat [29]) If we have the convergence Y n d 7\Gamma! n!1 Y , then Y satisfies the stochas...

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...ammat dt : (iv) Holder's inequality implies that the function g() = 1 1\Gammaff log E \Theta H ff is convex on R+ . Since g(0) = 0 and g( + 1) = g() + log( + 1), we conclude from Theorem 2.1 of Artin =-=[2]-=- (see also Artin [1] or Bourbaki [7], Chapter 7 Proposition 1 and exercises 1 and 2), that fors? 0, g() = log \Gamma(+1). Hence, by analytic continuation : E \Theta H iu ff = \Gamma(1 + iu) 1\Gammaff ...

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...t 0 e s ds whensis a Brownian motion with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette [6], Dufresne [11], Geman-Yor [15] and Yor (=-=[31]-=-, [30]). c) In her PhD Thesis [23], C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a frozen random potential V . Let us give examp...

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...nterested in the distribution of A t () = R t 0 e s ds whensis a Brownian motion with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette =-=[6]-=-, Dufresne [11], Geman-Yor [15] and Yor ([31], [30]). c) In her PhD Thesis [23], C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a ...

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...ntial functional naturally arises in the study of a one-dimensional diffusion in a frozen random potential V . Let us give examples of physical quantities considered in [23] and in Comtet-Monthus-Yor =-=[9]-=- : ffl The thermal-average time spent in the interval [x; x+dx] is T (x) dx, where T (x) = 2 Z +1 x e V (y) dy : ffl The stationary flow JN of particles going through a disordered sample of size N wit...

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...lly, we devote Appendix 2 to the study of a Poisson process with drift : there, we apply the results of Sections 2, 3, 4, and make explicit the connection with the Az'ema-Emery martingales (see Emery =-=[13]-=-). Contents 1. Introduction 1 2. An integro-differential equation satisfied by the density of the exponential functional 4 2.1. Proof of Proposition 2.1. 7 3. The moments formulae 10 3.1. A functional...

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... the second term on the right hand side converges almost surely to sup s1ss . Remark 4.3. When ff = 2, i.e.sis a Brownian motion, the preceding result is well-known (see e.g. Durrett [12], Pitman-Yor =-=[25]-=-, RevuzYor [26], Chapter I, Exercise 1.18, or Bertoin-Werner [3] and [4]), and is a main step in the proof of Spitzer's theorem on the winding number of planar Brownian motion. It is natural to think ...

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...) is a L'evy process such that for each a ? 0, Z a is a gamma variable of shape parameter a. Then, we consider (Z (j) ; j 2 N) a family of independent gamma processes (Z (j) a ; as0).Following Gordon =-=[17]-=-, we fix b ? 0 and define j (b) a := \Gammaafl + 1 X j=0 a j + 1 \Gamma Z (j) a j + b ; where fl = \Gamma\Gamma 0 (1) denotes the Euler constant. This definition makes sense, as we may write alternati...

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...implies that the function g() = 1 1\Gammaff log E \Theta H ff is convex on R+ . Since g(0) = 0 and g( + 1) = g() + log( + 1), we conclude from Theorem 2.1 of Artin [2] (see also Artin [1] or Bourbaki =-=[7]-=-, Chapter 7 Proposition 1 and exercises 1 and 2), that fors? 0, g() = log \Gamma(+1). Hence, by analytic continuation : E \Theta H iu ff = \Gamma(1 + iu) 1\Gammaff (u 2 R) ; which implies the desired ...

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... derivative of the Gamma function. (iv) For each ff 2 (0; 1), we have : Lsff = log H 1\Gammasff d = j (1)sff Remark 3.5. The fact that log Z a is infinitely divisible is well-known: see e.g. Shanbhag =-=[28]-=-, who shows that log Z a is self-decomposable, hence infinitely divisible. Proof. (i) Let a 0 ? 0. Since, for each j 2 N, (Z (j) a+a 0 \Gamma Z (j) a 0 ; as0) is a gamma process independent of ( Z (j)...

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...he 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 motion with drift. a) In =-=[19]-=-, Kawazu and Tanaka investigate the asymptotic behavior of the tail of the distribution of the maximum of a diffusion process in a drifted Brownian environment, precisely : given a Brownian motion (W ...

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...y to the study of the maximum of a diffusion in a random L'evy environment. Since the proof of Proposition 2.1 relies heavily on the properties of generalized Ornstein-Uhlenbeck processes (see Hadjiev=-=[18]-=-, Novikov [24] and Samorodnitsky and Taqqu [27], section 3.6), we give in Appendix 1 a summary of the properties of these processes. Eventually, we devote Appendix 2 to the study of a Poisson process ...

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...o sup s1ss . Remark 4.3. When ff = 2, i.e.sis a Brownian motion, the preceding result is well-known (see e.g. Durrett [12], Pitman-Yor [25], RevuzYor [26], Chapter I, Exercise 1.18, or Bertoin-Werner =-=[3]-=- and [4]), and is a main step in the proof of Spitzer's theorem on the winding number of planar Brownian motion. It is natural to think that the assumptions of the preceding Proposition were too restr...

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...er's inequality implies that the function g() = 1 1\Gammaff log E \Theta H ff is convex on R+ . Since g(0) = 0 and g( + 1) = g() + log( + 1), we conclude from Theorem 2.1 of Artin [2] (see also Artin =-=[1]-=- or Bourbaki [7], Chapter 7 Proposition 1 and exercises 1 and 2), that fors? 0, g() = log \Gamma(+1). Hence, by analytic continuation : E \Theta H iu ff = \Gamma(1 + iu) 1\Gammaff (u 2 R) ; which impl...

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... of the maximum of a diffusion in a random L'evy environment. Since the proof of Proposition 2.1 relies heavily on the properties of generalized Ornstein-Uhlenbeck processes (see Hadjiev[18], Novikov =-=[24]-=- and Samorodnitsky and Taqqu [27], section 3.6), we give in Appendix 1 a summary of the properties of these processes. Eventually, we devote Appendix 2 to the study of a Poisson process with drift : t...

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...on with drift, and t may be finite or infinite. This has been studied by a number of authors, including Bouchaud-Sornette [6], Dufresne [11], Geman-Yor [15] and Yor ([31], [30]). c) In her PhD Thesis =-=[23]-=-, C'ecile Monthus shows how the exponential functional naturally arises in the study of a one-dimensional diffusion in a frozen random potential V . Let us give examples of physical quantities conside...

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... the roots of : 0 = (x + a) 2 (x + b) 2 +s2 (x + b) 2 +s2 (x + a) 2 . (2) If X a := R 1 0 R s (4a) ds, where (R s (ffi); ss0) is the square of a ffidimensional Bessel process starting from 0 (see Yor =-=[32]-=-), then : E exp ` \Gamma 2 2 A1 (h a ) ' = c a E X \Gamma 1 2 a exp(\Gamma 2 2 X a ) : This equation provides another way of obtaining the density of X a . 6.4. The relationship with Az'ema-Emery mart...

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...+s2 , such thats1 (resp.s2 ) has no positive (resp. negative) jumps. Since M ()sM ( 1 ) + M ( 2 ) we may suppose that the jumpssare of a fixed sign, and apply Theorem 8, Chapter IV of Gihman-Skorohod =-=[16]-=-, to conclude. Remark 3.2. For every positive measurable function f , we have : E [f(A T )] = ` Z 1 0 e \Gamma`t E [f(A t )] dt : Hence, the moments formula (1.2) characterizes the law of A t at fixed...

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...ss . Remark 4.3. When ff = 2, i.e.sis a Brownian motion, the preceding result is well-known (see e.g. Durrett [12], Pitman-Yor [25], RevuzYor [26], Chapter I, Exercise 1.18, or Bertoin-Werner [3] and =-=[4]-=-), and is a main step in the proof of Spitzer's theorem on the winding number of planar Brownian motion. It is natural to think that the assumptions of the preceding Proposition were too restrictive, ...

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...method shows that the second term on the right hand side converges almost surely to sup s1ss . Remark 4.3. When ff = 2, i.e.sis a Brownian motion, the preceding result is well-known (see e.g. Durrett =-=[12]-=-, Pitman-Yor [25], RevuzYor [26], Chapter I, Exercise 1.18, or Bertoin-Werner [3] and [4]), and is a main step in the proof of Spitzer's theorem on the winding number of planar Brownian motion. It is ...