## On some exponential functionals of Brownian motion (1992)

Venue: | Adv. Appl. Prob |

Citations: | 129 - 14 self |

### BibTeX

@ARTICLE{Matsumoto92onsome,

author = {Hiroyuki Matsumoto and Marc Yor},

title = {On some exponential functionals of Brownian motion},

journal = {Adv. Appl. Prob},

year = {1992},

pages = {509--531}

}

### Years of Citing Articles

### OpenURL

### Abstract

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.

### Citations

464 |
Special Functions and Their Applications
- Lebedev
- 1972
(Show Context)
Citation Context ...1) Φ(α, γ; z) + z Γ(1 + α − γ) Γ(α) 1−γ Φ(1 + α − γ, 2 − γ; z), Γ(α + k) Γ(α) = α(α + 1) · · · (α + k − 1), k = 1, 2, ... For details about the confluent hypergeometric functions, we refer to Lebedev =-=[38]-=-. Φ(α, γ; z) and Ψ(α, γ; z) are linearly independent solutions for the linear differential equation zu ′′ + (γ − z)u ′ − αu = 0. and We set ν = √ 2λ + µ 2 and define the functions u1 and u2 on (0, ∞) ... |

361 |
Heat kernels and spectral theory
- Davies
- 1989
(Show Context)
Citation Context ...r) for every n ≧ 4: pn+2(t, r) = − e−nt/2 ∂ 2π sinh(r) ∂r pn(t, r). (3.5) For details about the real hyperbolic space Hn and the classical formulae for the heat kernels, we refer the reader to Davies =-=[14]-=-. Gruet [28] has considered the Brownian motion on Hn , which is a diffusion process generated by ∆n/2, and has derived a new integral representation for pn(t, r) by using the explicit expression (1.2... |

237 |
Diffusion Processes and Their Sample Paths
- Itô, Jr
- 1965
(Show Context)
Citation Context ...1λ(ϕt − ϕ0)) | |Zt| = ρ] = ( I|λ| I0 )( |z|ρ t ) . (2.12) Moreover it should also be noted that there exists a one-dimensional Brownian motion {βu} independent of {|Zs|} such that See also Itô-McKean =-=[37]-=-, p.270. ∫ t ϕt − ϕ0 = βαt with αt = 0 ds . (2.13) |Zs| 2H. Matsumoto, M. Yor/Exponential functionals of BM, I 317 Remark 2.3. The function θ(r, t) is, of course, a non-negative function. However, si... |

180 |
A Treatise on the Theory of Bessel Functions, 2nd edition
- Watson
- 1922
(Show Context)
Citation Context ...ost interesting and important, we compute the Laplace transform in t of the right hand sides of (3.3) and (3.6). Then, using the Hankel-Lipschitz formula for the modified Bessel functions (see Watson =-=[57]-=-, p.386), we can check the coincidence of the Laplace transforms or of the expressions for the Green function. For details, see [28], [40], [41]. We give a proof of (3.6) and see how the exponential f... |

128 |
The value of an Asian option
- Rogers, Shi
- 1995
(Show Context)
Citation Context ... {(St − k)+} is also a submartingale. Hence we obtain −rT 1 CA(k, T) ≦ e T ∫ T E Q [(ST − k)+]dt 0 = e −rT E Q [(ST − k)+] = CE(k, T). For more discussions on CA(k, T), see Geman-Yor [24], Rogers-Shi =-=[55]-=- and the references cited therein. By using explicit expressions for the density of A (µ) t discussed in Part I, we obtain several integral representations for CA(k, T). However, they are complicated.... |

103 |
The distribution of a perpetuity, with applications to risk theory and pension funding
- Dufresne
- 1990
(Show Context)
Citation Context ...For details, see Part I and the references cited therein. Another important fact, which has been used in several domains and also discussed in Part I, is the following identity in law due to Dufresne =-=[18]-=-. Let µ > 0. Then one has A (−µ) ∞ ∫ ∞ ≡ 0 exp(2B (−µ) s )ds (law) = 1 , (1.5) 2γµ where γµ is a gamma random variable with parameter µ, that is, P(γµ ∈ dx) = 1 Γ(µ) xµ−1 e −x dx, x ≧ 0. The purpose o... |

76 |
Bessel processes, Asian options and perpetuities
- Geman, Yor
- 1993
(Show Context)
Citation Context ...gain, we see that {(St − k)+} is also a submartingale. Hence we obtain −rT 1 CA(k, T) ≦ e T ∫ T E Q [(ST − k)+]dt 0 = e −rT E Q [(ST − k)+] = CE(k, T). For more discussions on CA(k, T), see Geman-Yor =-=[24]-=-, Rogers-Shi [55] and the references cited therein. By using explicit expressions for the density of A (µ) t discussed in Part I, we obtain several integral representations for CA(k, T). However, they... |

74 |
Semi-stable Markov processes
- LAMPERTI
- 1972
(Show Context)
Citation Context ...nt of view. The starting important fact is the Lamperti relation, which says that there exists a Bessel process R (µ) = {R (µ) u } with R (µ) 0 exp(B (µ) t ) = R (µ) A (µ) t For details, see Lamperti =-=[40]-=- and also p.452 of [57]. = 1 and index µ satisfying , t ≧ 0. (2.1)that H. Matsumoto, M. Yor/Exponential functionals of BM, I 314 When µ < 0, we easily see that limt→∞ A (µ) t A (µ) ∞ = inf{s; R (µ) s... |

70 |
Some aspects of Brownian motion
- Yor
- 1992
(Show Context)
Citation Context ... 0 has been obtained by Dufresne [19] and the extension (5.2) at the process level has been given in [47]. Proof. We sketch a proof based on the theory of initial enlargements of filtrations (cf. Yor =-=[59]-=-). Another proof based on some properties of Bessel processes has been given in [47]. Let B (−µ) t = σ{B (−µ) s show that there exists a { ̂ B (−µ) t such that B (−µ) t , s ≦ t} and ̂ B (−µ) t = B (−µ... |

63 |
Recherches sur les fractions continues, Annales de la Faculté des Sciences de Toulouse
- Stieltjes
- 1993
(Show Context)
Citation Context ... not easy to carry out numerical computations, especially when t is small. See [5], [36] and [44] for some results and discussions on the numerical computations. Remark 2.4. In his fundamental memoir =-=[63]-=-, Stieltjes considered a similar integral to that on the right hand side of (2.9). He has shown that ∫ ∞ e −ξ2 ( ) /2t πξ sinh(nξ)sin dξ = 0 (2.14) t 0 holds for any n ∈ N and for any t > 0. Setting µ... |

43 |
One-dimensional Brownian motion and the three-dimensional Bessel process
- Pitman
- 1997
(Show Context)
Citation Context ... (µ) t , t ≧ 0. The study of the stochastic process Z(B (µ)) is in fact the main object of the } is a diffusion = σ{Z(µ) , s ≦ t}, and that this result gives rise to an extension of Pitman’s theorem (=-=[51]-=-,[54]). A key fact in the proof of the above mentioned result is the following Proposition 5.2, which also plays an important role in the rest of this section. Before mentioning the proposition, we no... |

34 |
A One-Dimensional Diffusion Process in a Wiener medium
- Brox
(Show Context)
Citation Context ...c) (Xt(W))} may be represented as a random time change of another Brownian motion and, based on this representation, several interesting results have been obtained. For these results, see, e.g., Brox =-=[8]-=-, Hu-Shi-Yor [30], Kawazu-Tanaka [36]. In the rest of this section, we assume c > 0. Then we have S (c) (∞) = ∞ and S (c) (−∞) > −∞, and therefore, max t≧0 X(t) < ∞, P-a.s. The question we discuss in ... |

28 |
Fourier coefficients of the resolvent for a Fuchsian group
- Fay
- 1977
(Show Context)
Citation Context ...gnetic field. It is essentially the same as the Maass Laplacian which plays an important role in several domains of mathematics, e.g., number theory, representation theory and so on. For details, see =-=[22]-=-, [31] and the references cited therein. In [31], the authors have started their arguments from the Brownian motion on H 2 given in the above proof of Theorem 3.1 and have discussed about explicit and... |

28 |
The inverse Gaussian distribution
- Seshadri
- 1998
(Show Context)
Citation Context ...hat I (−δ) z and γz are independent, we have I (δ) (−δ) z = I z +2zγδ. In fact, more general identities in law for generalized inverse Gaussian and gamma random variables are well known. See Seshadri =-=[56]-=-, [46] and the references therein.H. Matsumoto, M. Yor/Exponential functionals of BM, II 371 Hence, from the identity (5.6) considered for µ = 0, we obtain E[f(e Bt )e δBt |Zt] = Kδ(1/z) K0(1/z) E[f(... |

28 |
On the entire moments of self-similar Markov process and exponential functionals of Lévy processes
- Bertoin, Yor
(Show Context)
Citation Context ...the reciprocal ) −1 is determinate. By using this fact and formula (4.1), he has also shown another representation for the density of A (µ) t after giving another expression for the moments. See also =-=[6]-=- for a similar result for exponential functionals of Lévy processes. (A (µ) t We go back to formula (5.4). Let us consider an exponential random variable e whose density is e−x , x ≧ 0. Then we have E... |

25 |
Risk theory in a stochastic economic environment
- Paulsen
- 1993
(Show Context)
Citation Context ...ngs to the type IV family of Pearson distributions (cf. [34]). It should be mentioned that the functional ∫ ∞ 0 exp(B(−µ) s )dW (−ν) s has been much studied in the context of risk theory. See Paulsen =-=[50]-=- and the references cited therein about this. In [50] the density is derived when ν > 1. See also [2] and [3] about some results in special cases. For further related discussions, see [47] and [64]. W... |

23 |
Loi de l’indice du lacet Brownien et Distribution de HartmanWatson
- Yor
- 1980
(Show Context)
Citation Context ...family of probability measures ηr(dt), r > 0, on R+, from the identity ∫ ∞ 0 e −µ2 ( t/2 I|µ| ηr(dt) = I0 ) (r), µ ∈ R. For more probabilistic discussions of the Hartman-Watson distributions, see Yor =-=[67]-=-. In [67] an integral representation for the density of ηr(dt) has been obtained. Let us define the function θ(r, t), r > 0, t > 0, by θ(r, t) = r (2π 3 t) 1/2 eπ2 /2t ∫ ∞ 0 e −ξ2 /2t e −r cosh(ξ) sin... |

20 |
Laguerre Series for Asian and Other Options
- Dufresne
- 2000
(Show Context)
Citation Context ...ty of the Stieltjes moment problem does not hold for the (conditional) distribution of At. Recently Hörfelt [34] announced that he has proven the indeterminacy for the moment problem for At. Dufresne =-=[22]-=- has shown that the Stieltjes moment problem for the reciprocal ) −1 is determinate. By using this fact and formula (4.1), he has also shown another representation for the density of A (µ) t after giv... |

20 |
Sur certaines fonctionnelles exponentielles du mouvement Brownien reel
- Yor
- 1992
(Show Context)
Citation Context ...that the distributions of the corresponding perpetual functionals are determined by their negative moments. Moreover, they have given another proof of the following theorem. See also [57], p.452, and =-=[70]-=-. Theorem 6.2. Let µ > 0. Then, A (−µ) ∞ is distributed as (2γµ) −1 , where γµ denotes a gamma random variable with parameter µ. Proof. Differently from the proofs given in the above cited references ... |

19 |
The nonlinear transformation of Gaussian measure on Banach space and its absolute continuity
- KUSUOKA
- 1982
(Show Context)
Citation Context ...y. To show the proposition, we prove an identity for an anticipative transform on path space, which may be regarded as an example of the Ramer-Kusuoka formula. For the Ramer-Kusuoka formula, see [9], =-=[37]-=-, [52] and [58].H. Matsumoto, M. Yor/Exponential functionals of BM, II 376 Another proof for the proposition which uses several properties of Bessel processes has been given in [45] and a proof based... |

18 |
A diffusion process in a brownian environment with drift
- Tanaka, Kawazu
- 1997
(Show Context)
Citation Context ...random time change of another Brownian motion and, based on this representation, several interesting results have been obtained. For these results, see, e.g., Brox [8], Hu-Shi-Yor [30], Kawazu-Tanaka =-=[36]-=-. In the rest of this section, we assume c > 0. Then we have S (c) (∞) = ∞ and S (c) (−∞) > −∞, and therefore, max t≧0 X(t) < ∞, P-a.s. The question we discuss in the present section is how the tail p... |

18 |
Present value distributions with applications to ruin theory and stochastic equations. Stochastic Process
- Gjessing, Paulsen
- 1997
(Show Context)
Citation Context ...rocesses {Yt} and {Zt}, which are given by the sums of constant drifts, Brownian motions and compound Poisson processes, the perpetual functional ∫ ∞ 0 exp(Yt)dZt has been studied in Gjessing-Paulsen =-=[27]-=-, Nilsen-Paulsen [54], Paulsen [56] and so on. For details, see the original papers. 7. Some limiting distributions In this section, following Hariya-Yor [32], we present the results on the asymptotic... |

16 | Rates of convergence of diffusions with drifted brownian potentials
- Hu, Shi, et al.
- 1999
(Show Context)
Citation Context ...be represented as a random time change of another Brownian motion and, based on this representation, several interesting results have been obtained. For these results, see, e.g., Brox [8], Hu-Shi-Yor =-=[30]-=-, Kawazu-Tanaka [36]. In the rest of this section, we assume c > 0. Then we have S (c) (∞) = ∞ and S (c) (−∞) > −∞, and therefore, max t≧0 X(t) < ∞, P-a.s. The question we discuss in the present secti... |

15 |
Exponential functionals of Brownian motion and disordered systems
- Comtet, Monthus, et al.
- 1998
(Show Context)
Citation Context ...entioned that, although A (µ) t is of a simple form, only integral representations for the density (of somewhat complicated forms!) are known by the results due to Alili-Gruet [3], Comtet-Monthus-Yor =-=[15]-=-, Dufresne [23], Yor [69]. See also [8],[14], [44], [50], [52] and [61]. We mention the results obtained in [3], [15], [23] and [69] in some detail and we show some related identities, which are of in... |

15 |
The Integral of geometric Brownian motion
- Dufresne
- 2001
(Show Context)
Citation Context ... t This identity is equivalent to (5.8) because of (5.4) or (5.5). A (µ) t ) m] . Let f (µ)(a, t) be the density of (2A (µ) t ) −1 . Then the following “recursion” formula, originally due to Dufresne =-=[20]-=-, is deduced from (5.8). Theorem 5.5. Let ν < µ. Then, for any t > 0, one has, with δ = (µ − ν)/2, e ν2 t/2 f (ν) (a, t) = e µ 2 t/2 a−m e −a Γ(δ) ∫ a (a − b) 0 δ−1 b m e b f (µ) (b, t)db, a > 0. 6. S... |

15 |
The path integral on the Poincaré upper half-plane with magnetic field and for the Morse potential
- Grosche
- 1988
(Show Context)
Citation Context ...equivalent, whichH. Matsumoto, M. Yor/Exponential functionals of BM, II 358 may be directly verified by Fourier analysis and has been already pointed out in Comtet [11], Debiard-Gaveau [15], Grosche =-=[26]-=- and so on. See also [31]. In the rest of this section, we restrict ourselves to the case n = 2 and consider two questions related to the results and formulae presented above. For other related topics... |

15 |
Some properties of Wishart process and a matrix extension of the Hartman-Watson
- Donati-Martin, Matsumoto
(Show Context)
Citation Context ...nuity relationships and an extension of formula (2.8) below also hold for the Wishart processes, which are diffusion processes with values in the space of positive definite matrices. For details, see =-=[17]-=-. The important relationships (2.2) and (2.3) are shown in the following way. Since formula (2.2) has been presented on p. 450, [57] and is proven by an easy modification of the arguments below, we on... |

14 |
Exemples de théorèmes locaux sur les groupes résolubles
- Bougerol
- 1983
(Show Context)
Citation Context ...= e n2t/2 , n = 1, 2, ..., and that the moment problem for the log-normal distribution is indeterminate. 3. Bougerol’s identity In his discussion of stochastic analysis on hyperbolic spaces, Bougerol =-=[11]-=- has obtained an interesting and important identity in law. It plays an important role in several domains and in the following discussions. We show it in the simplest form. Some extensions and discuss... |

14 |
On certain Markov processes attached to exponential functionals of Brownian motion; application to Asian options
- Donati-Martin, Ghomrasni, et al.
- 2001
(Show Context)
Citation Context ... − 2kt) a+1 dt. The same formula has been proven in [24] with the help of some properties of Bessel processes. We next present another proof of Theorem 2.2, following Donati-Martin, Ghomrasni and Yor =-=[16]-=-, who have used, as an auxiliary tool, the stochastic process Y (µ) (x) = {Y (µ) t (x)} given by ( x + Y (µ) t (x) = exp(2B (µ) t ) ∫ t 0 exp(−2B (µ) ) s )ds . Y (µ)(x) is a diffusion process with gen... |

13 |
Brownian Motion on the Hyperbolic plane and Selberg Trace Formula
- Ikeda
- 1999
(Show Context)
Citation Context ...umoto, M. Yor/Exponential functionals of BM, II 358 may be directly verified by Fourier analysis and has been already pointed out in Comtet [11], Debiard-Gaveau [15], Grosche [26] and so on. See also =-=[31]-=-. In the rest of this section, we restrict ourselves to the case n = 2 and consider two questions related to the results and formulae presented above. For other related topics, see, e.g., [2] and [29]... |

12 |
On the maximun of a diffusion process in a drifted brownian environment
- Kawazu, Tanaka
- 1993
(Show Context)
Citation Context ... hitting time of a by {Y (ν,µ) t }. For details, see the original paper [4]. 4. Maximum of a diffusion process in random environment The purpose of this section is to survey the work by Kawazu-Tanaka =-=[35]-=- on the maximum of a diffusion process in a drifted random environment. In [35], several equalities and inequalities for the exponential functionals of Brownian motion are used. Let W = {W(y), y ∈ R} ... |

11 |
On the Landau levels on the hyperbolic plane
- Comtet
- 1987
(Show Context)
Citation Context ... 2 2 dx2 + 1 2 |λ|2e2x are unitary equivalent, whichH. Matsumoto, M. Yor/Exponential functionals of BM, II 358 may be directly verified by Fourier analysis and has been already pointed out in Comtet =-=[11]-=-, Debiard-Gaveau [15], Grosche [26] and so on. See also [31]. In the rest of this section, we restrict ourselves to the case n = 2 and consider two questions related to the results and formulae presen... |

11 | Normal” distribution functions on spheres and the modified Bessel functions - Hartman, Watson - 1974 |

11 |
Properties of perpetual integral functionals of Brownian motion with drift
- Salminen, Yor
(Show Context)
Citation Context ...t ∫ ∞ A (µ) ∞ (f) = 0 f(B (µ) s )ds and call it a perpetual functional. At first we show a necessary and sufficient condition in order that A (µ) ∞ (f) is finite almost surely. We follow Salminen-Yor =-=[59]-=-. See also Engelbert-Senf [24] for another approach. Theorem 6.1. Let µ be positive and f be a non-negative and locally integrable Borel function on R. Then, in order for A (µ) ∞ (f) to be finite almo... |

10 |
A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian
- Matsumoto, Yor
- 2001
(Show Context)
Citation Context ...See Paulsen [50] and the references cited therein about this. In [50] the density is derived when ν > 1. See also [2] and [3] about some results in special cases. For further related discussions, see =-=[47]-=- and [64]. We present a probabilistic proof taken from [4], where we also find an analytic proof. Proof. The limiting distribution coincides with that for the stochastic process } given by { ¯ X (ν,µ)... |

8 |
Semi{groupe du mouvement brownien hyperbolique, Stochastics and Stochastic Reports 56
- Gruet
- 1996
(Show Context)
Citation Context ... n ≧ 4: pn+2(t, r) = − e−nt/2 ∂ 2π sinh(r) ∂r pn(t, r). (3.5) For details about the real hyperbolic space Hn and the classical formulae for the heat kernels, we refer the reader to Davies [14]. Gruet =-=[28]-=- has considered the Brownian motion on Hn , which is a diffusion process generated by ∆n/2, and has derived a new integral representation for pn(t, r) by using the explicit expression (1.2) for the jo... |

8 |
An analogue of Pitman’s 2M − X theorem for exponential Wiener functionals. I. A time-inversion approach
- Yor
(Show Context)
Citation Context ...formula, see [9], [37], [52] and [58].H. Matsumoto, M. Yor/Exponential functionals of BM, II 376 Another proof for the proposition which uses several properties of Bessel processes has been given in =-=[45]-=- and a proof based on Theorem 5.1, featuring the generalized Gaussian inverse distributions, has been given in [46]. For our purpose we consider one more transform on path space. For an Rvalued contin... |

8 |
On nonlinear transformations of Gaussian measures
- Ramer
- 1974
(Show Context)
Citation Context ...show the proposition, we prove an identity for an anticipative transform on path space, which may be regarded as an example of the Ramer-Kusuoka formula. For the Ramer-Kusuoka formula, see [9], [37], =-=[52]-=- and [58].H. Matsumoto, M. Yor/Exponential functionals of BM, II 376 Another proof for the proposition which uses several properties of Bessel processes has been given in [45] and a proof based on Th... |

8 |
Exponential functionals of Lévy processes. Probability Surveys 2
- Bertoin, Yor
- 2005
(Show Context)
Citation Context ... important and typical case, f(x) = e−2x . We set ∫ ∞ ( ∫ ∞ ) (law) = . A (−µ) ∞ = 0 exp(2B (−µ) s )ds 0 exp(−2B (µ) s )ds The following Theorem 6.2 is due to Dufresne [21]. Bertoin-Yor [6] (see also =-=[7]-=-) have considered Lévy processes which drift to ∞ as t → ∞ and have shown that the distributions of the corresponding perpetual functionals are determined by their negative moments. Moreover, they hav... |

7 |
On A Triplet of Exponential Brownian Functionals
- Alili, Matsumoto, et al.
- 2000
(Show Context)
Citation Context ...tained explicit forms of the Green functions. From the point of view of probability theory along the line of [31], another explicit representation for the heat kernel qk(t, z, z ′ ) has been shown in =-=[1]-=- by using an extension of formula (1.2) and Gruet’s formula (3.6). We introduce the result in [1] together with some arguments taken from [31]. To show an explicit representation for qk(t, z, z ′ ), w... |

7 |
Intertwining of Markov semigroups, some examples, in: Séminaire de Probabilités XXIX, p
- BIANE
(Show Context)
Citation Context ... that the semigroups of the diffusion processes e (µ) , ξ (µ) and Z (µ) satisfy some intertwining properties. For some general discussions and examples of intertwinings between Markov semigroups, see =-=[6]-=-, [10], [21], [54] and [61] among others.H. Matsumoto, M. Yor/Exponential functionals of BM, II 380 a generalized inverse Gaussian random variable whose density is given by (5.6) and define the Marko... |

7 |
An ane property of the reciprocal Asian option process
- Dufresne
- 2001
(Show Context)
Citation Context ...µ−1e−x, x > 0, independent of B. Then one has the identity in law { } { } 1 (law) 1 = + 2γµ, t > 0 . (5.2) A (−µ) , t > 0 t A (µ) t The identity in law for a fixed t > 0 has been obtained by Dufresne =-=[19]-=- and the extension (5.2) at the process level has been given in [47]. Proof. We sketch a proof based on the theory of initial enlargements of filtrations (cf. Yor [59]). Another proof based on some pr... |

7 |
A version of Pitman’s 2M − X theorem for geometric Brownian motions
- Yor
- 1999
(Show Context)
Citation Context ...otion and a three-dimensional Bessel process, and the theorems give their representations in terms of the maximum process {M (0) t } of a Brownian motion B. For the proofs and related references, see =-=[44]-=-, [46], [53] et al. It should be noted that the stochastic processes {kM (µ) t −B (µ) t }, k ∈ R, are not Markovian except for these two interesting cases k = 1, 2 and the trivial case k = 0. For a ri... |

7 |
Sur l’identité de Bougerol pour les fonctionnelles exponentielles du mouvement Brownien avec drift
- Alili, Dufresne, et al.
- 1997
(Show Context)
Citation Context ...nteresting and important identity in law. It plays an important role in several domains and in the following discussions. We show it in the simplest form. Some extensions and discussions are found in =-=[2]-=- and [3]. Theorem 3.1. Let {Wt} be a one-dimensional Brownian motion starting from 0, independent of B. Then, for every fixed t > 0, ∫ t 0 exp(Bs)dWs, sinh(Bt) and WAt are identical in law. Proof. We ... |

7 |
The law of geometric Brownian motion and its integral, revisited; application to conditional moments
- Donati-Martin, Matsumoto, et al.
- 2001
(Show Context)
Citation Context ...to a rather simple one. Theorem 5.5. For any t > 0, the conditional distribution of eAt conditionally on Bt = x coincides with that of e x (cosh( √ 2et) − cosh(x)) given e > x 2 /2t. For a proof, see =-=[20]-=- or [48]. In these articles one also finds the following results, which are shown by using the expression (4.1) for the joint density of (At, Bt). Theorem 5.6. (i) For any t > 0 and λ > 0, one has [ (... |

6 |
An Explanation of A Generalized Bougerols Identity in Terms of Hyperbolic Brownian Motion
- Alili, Gruet
- 1997
(Show Context)
Citation Context ...ee also [31]. In the rest of this section, we restrict ourselves to the case n = 2 and consider two questions related to the results and formulae presented above. For other related topics, see, e.g., =-=[2]-=- and [29]. Let us consider the following Schrödinger operator Hk, k ∈ R, on H2 with a magnetic field: Hk = 1 2 y2 ( ) 2 √−1 ∂ k + − ∂x y 1 2 y2 ( ) 2 ∂ . ∂y The differential 1-form α = ky −1 dx is cal... |

6 |
A study of the Hartman-Watson distribution motivated by numerical problems related to the pricing of Asian options
- Barrieu, Rouault, et al.
(Show Context)
Citation Context ...course, a non-negative function. However, since the integral on the right hand side of (2.9) is an oscillatory one, it is not easy to carry out numerical computations, especially when t is small. See =-=[5]-=-, [36] and [44] for some results and discussions on the numerical computations. Remark 2.4. In his fundamental memoir [63], Stieltjes considered a similar integral to that on the right hand side of (2... |

6 |
Exponential Functionals and Principal Values related to Brownian Motion, A collection of research papers, Biblioteca de la Revista Matematica Iberoamericana
- Yor
- 1997
(Show Context)
Citation Context ...en [50] and the references cited therein about this. In [50] the density is derived when > 1. See also [2] and [3] about some results in special cases. For further related discussions, see [47] and =-=[64]-=-. We present a probabilistic proof taken from [4], where we alsosnd an analytic proof. Proof. The limiting distribution coincides with that for the stochastic process f X(;)t g given by X (;) t ... |

6 |
Diffusion in a one-dimensional random medium and hyperbolic Brownian motion
- Comtet, Monthus
- 1996
(Show Context)
Citation Context ...le form, only integral representations for the density (of somewhat complicated forms!) are known by the results due to Alili-Gruet [3], Comtet-Monthus-Yor [15], Dufresne [23], Yor [69]. See also [8],=-=[14]-=-, [44], [50], [52] and [61]. We mention the results obtained in [3], [15], [23] and [69] in some detail and we show some related identities, which are of independent interest. For example, the Bougero... |

5 | Stable laws arising from hitting distributions of processes on homogeneous trees and the hyperbolic half-plane
- Baldi, Tarabusi, et al.
- 2001
(Show Context)
Citation Context ...0 exp(B(−µ) s )dW (−ν) s has been much studied in the context of risk theory. See Paulsen [50] and the references cited therein about this. In [50] the density is derived when ν > 1. See also [2] and =-=[3]-=- about some results in special cases. For further related discussions, see [47] and [64]. We present a probabilistic proof taken from [4], where we also find an analytic proof. Proof. The limiting dis... |