메뉴 건너뛰기




Volumn 2, Issue 1, 2005, Pages 191-212

Exponential functionals of Lévy processes

Author keywords

Exponential functional; L vy process; Moment problem; Self similar markov process; Subordinator

Indexed keywords


EID: 31444454079     PISSN: None     EISSN: 15495787     Source Type: Journal    
DOI: 10.1214/154957805100000122     Document Type: Article
Times cited : (167)

References (70)
  • 1
    • 0039192975 scopus 로고    scopus 로고
    • Nonsymmetric hitting distributions on the hyperbolic half-plane and subordinated perpetuities
    • MR1900896
    • P. Baldi, E.C. Tarabusi, A. Figá-Talamanca, and M. Yor (2001). Nonsymmetric hitting distributions on the hyperbolic half-plane and subordinated perpetuities. Rev. Mat. Iberoamericana 17, 587-605. MR1900896
    • (2001) Rev. Mat. Iberoamericana , vol.17 , pp. 587-605
    • Baldi, P.1    Tarabusi, E.C.2    Figá-Talamanca, A.3    Yor, M.4
  • 2
    • 84890745569 scopus 로고    scopus 로고
    • On powers of Stieltjes moment sequence I
    • to appear
    • Ch. Berg (2004). On powers of Stieltjes moment sequence I. J. Theoret. Probab. (to appear).
    • (2004) J. Theoret. Probab.
    • Berg, C.H.1
  • 3
    • 84890765904 scopus 로고    scopus 로고
    • On powers of Stieltjes moment sequence II
    • Preprint
    • Ch. Berg (2004). On powers of Stieltjes moment sequence II. Preprint.
    • (2004)
    • Berg, C.H.1
  • 4
    • 8744268149 scopus 로고    scopus 로고
    • A transformation from Hausdorff to Stieltjes moment sequence
    • Ch. Berg and A. J. Duran (2004). A transformation from Hausdorff to Stieltjes moment sequence. Ark. Math. 42, 239-257.
    • (2004) Ark. Math. , vol.42 , pp. 239-257
    • Berg, C.H.1    Duran, A.J.2
  • 5
    • 0003975247 scopus 로고    scopus 로고
    • Cambridge University Press
    • J. Bertoin (1996). Lévy Processes. Cambridge University Press.
    • (1996) Lévy Processes
    • Bertoin, J.1
  • 6
    • 0000171910 scopus 로고    scopus 로고
    • Regularity of the half-line for Lévy processes, Bull
    • MR1465812
    • J. Bertoin (1997). Regularity of the half-line for Lévy processes, Bull. Sc. Math. 121, 345-354. MR1465812
    • (1997) Sc. Math. , vol.121 , pp. 345-354
    • Bertoin, J.1
  • 7
    • 0003359661 scopus 로고    scopus 로고
    • Subordinators: Examples and Applications
    • Lect. Notes in Maths 1717, Springer, MR1746300
    • J. Bertoin (1999). Subordinators: Examples and Applications. Ecole d'été de Probabilités de St-Flour XXVII, Lect. Notes in Maths 1717, Springer, pp. 1-91. MR1746300
    • (1999) Ecole d'été de Probabilités de St-Flour , vol.27 , pp. 1-91
    • Bertoin, J.1
  • 9
    • 1442359825 scopus 로고    scopus 로고
    • The asymptotic behavior of fragmentation processes
    • MR2017852
    • J. Bertoin (2003). The asymptotic behavior of fragmentation processes. J. Europ. Math. Soc. 5, 305-416. MR2017852
    • (2003) J. Europ. Math. Soc. , vol.5 , pp. 305-416
    • Bertoin, J.1
  • 10
    • 29144462766 scopus 로고    scopus 로고
    • Poissonian exponential functionals, q-series, q-integrals, and the moment problem for log-normal distributions
    • J. Bertoin, Ph. Biane, and M. Yor (2004). Poissonian exponential functionals, q-series, q-integrals, and the moment problem for log-normal distributions. Proceedings Stochastic Analysis, Ascona, Progress in Probability vol. 58, Birkhäuser, pp. 45-56.
    • (2004) Proceedings Stochastic Analysis, Ascona, Progress in Probability vol. , vol.58 , pp. 45-56
    • Bertoin, J.1    Biane, P.H.2    Yor, M.3
  • 11
    • 1442293041 scopus 로고    scopus 로고
    • Entrance from 0+ for increasing semi-stable Markov processes
    • J. Bertoin and M.-E. Caballero (2002). Entrance from 0+ for increasing semi-stable Markov processes. Bernoulli 8, 195-205.
    • (2002) Bernoulli , vol.8 , pp. 195-205
    • Bertoin, J.1    Caballero, M.-E.2
  • 12
    • 4644316166 scopus 로고    scopus 로고
    • Asymptotic laws for nonconservative self-similar fragmentations
    • Available at, MR2080610
    • J. Bertoin, and A. Gnedin (2004). Asymptotic laws for nonconservative self-similar fragmentations. Elect. J. Probab. 9, 575-593. Available at http://www.math.washington.edu/ejpecp/viewarticle.php?id=1463&layout=abstract. MR2080610
    • (2004) Elect. J. Probab. , vol.9 , pp. 575-593
    • Bertoin, J.1    Gnedin, A.2
  • 13
    • 0037982096 scopus 로고    scopus 로고
    • On subordinators, self-similar Markov processes,, and some factorizations of the exponential law
    • Available at, MR1871698
    • J. Bertoin, and M. Yor (2001). On subordinators, self-similar Markov processes,, and some factorizations of the exponential law. Elect. Commun. Probab. 6, 95-106. Available at http://www.math.washington.edu/ejpecp/ECP/viewarticle.php?id=1622&layout=abstract.MR1871698
    • (2001) Elect. Commun. Probab. , vol.6 , pp. 95-106
    • Bertoin, J.1    Yor, M.2
  • 14
    • 0141792413 scopus 로고    scopus 로고
    • Entrance laws of self-similar Markov processes, and exponential functionals of Lévy processes
    • MR1918243
    • J. Bertoin, and M. Yor (2002). Entrance laws of self-similar Markov processes, and exponential functionals of Lévy processes. Potential Analysis 17, 389-400. MR1918243
    • (2002) Potential Analysis , vol.17 , pp. 389-400
    • Bertoin, J.1    Yor, M.2
  • 15
    • 0037540080 scopus 로고    scopus 로고
    • On the entire moments of self-similar Markov processes, and exponential functionals of certain Lévy processes
    • J. Bertoin, and M. Yor (2002). On the entire moments of self-similar Markov processes, and exponential functionals of certain Lévy processes. Ann. Fac. Sci. Toulouse Série 6, vol. XI, 33-45.
    • (2002) Ann. Fac. Sci. Toulouse Série , vol.6-11 , pp. 33-45
    • Bertoin, J.1    Yor, M.2
  • 16
    • 0000065881 scopus 로고
    • Exemples de théorèmes locaux sur les groupes résolubles
    • Ph. Bougerol (1983). Exemples de théorèmes locaux sur les groupes résolubles. Ann. Inst. H. Poincaré 19, 369-391.
    • (1983) Ann. Inst. H. Poincaré , vol.19 , pp. 369-391
    • Bougerol, P.H.1
  • 18
    • 0001511096 scopus 로고
    • A one-dimensional diffusion process in a random medium
    • T. Brox (1986). A one-dimensional diffusion process in a random medium. Ann. Probab. 14, 1206-1218.
    • (1986) Ann. Probab. , vol.14
    • Brox, T.1
  • 19
    • 84890615932 scopus 로고    scopus 로고
    • Overshoots of Lévy processes, and weak convergence of positive self-similar Markov processes
    • M.-E. Caballero, and L. Chaumont (2004). Overshoots of Lévy processes, and weak convergence of positive self-similar Markov processes. Preprint.
    • (2004) Preprint
    • Caballero, M.-E.1    Chaumont, L.2
  • 20
    • 0041068973 scopus 로고
    • Sur les fonctionnelles exponentielles de certains processus de Lévy
    • English translation in [70]
    • P. Carmona, F. Petit, and M. Yor (1994). Sur les fonctionnelles exponentielles de certains processus de Lévy. Stochastics, and Stochastics Reports 47, 71-101. English translation in [70].
    • (1994) Stochastics, and Stochastics Reports , vol.47 , pp. 71-101
    • Carmona, P.1    Petit, F.2    Yor, M.3
  • 21
    • 0008726577 scopus 로고    scopus 로고
    • On the distribution, and asymptotic results for exponential functionals of Lévy processes
    • In: M. Yor (editor), Biblioteca de la Revista Matemática Iberoamericana
    • P. Carmona, F. Petit, and M. Yor (1997). On the distribution, and asymptotic results for exponential functionals of Lévy processes. In: M. Yor (editor) Exponential functionals, and principal values related to Brownian motion, pp. 73-121. Biblioteca de la Revista Matemática Iberoamericana.
    • (1997) Exponential functionals, and principal values related to Brownian motion , pp. 73-121
    • Carmona, P.1    Petit, F.2    Yor, M.3
  • 22
    • 0041649295 scopus 로고    scopus 로고
    • Exponential functionals of Lévy processes
    • O. Barndorff-Nielsen, T. Mikosch, and S. Resnick (editors), Birkhäuser. MR1833689
    • P. Carmona, F. Petit, and M. Yor (2001). Exponential functionals of Lévy processes. O. Barndorff-Nielsen, T. Mikosch, and S. Resnick (editors). Lévy processes: theory, and applications, pp. 41-55, Birkhäuser. MR1833689
    • (2001) Lévy processes: theory, and applications, pp. , pp. 41-55
    • Carmona, P.1    Petit, F.2    Yor, M.3
  • 23
    • 0032397394 scopus 로고    scopus 로고
    • Exponential functionals of Brownian motion, and disordered systems
    • A. Comtet, C. Monthus,, and M. Yor (1998). Exponential functionals of Brownian motion, and disordered systems. J. Appl. Probab. 35, 255-271.
    • (1998) J. Appl. Probab. , vol.35 , pp. 255-271
    • Comtet, A.1    Monthus, C.2    Yor, M.3
  • 24
    • 21344490580 scopus 로고
    • A stochastic model of fragment formation when DNA replicates
    • MR1274788
    • R. Cowan, and S. N. Chiu (1994). A stochastic model of fragment formation when DNA replicates. J. Appl. Probab. 31, 301-308 MR1274788
    • (1994) J. Appl. Probab. , vol.31 , pp. 301-308
    • Cowan, R.1    Chiu, S.N.2
  • 25
    • 0036508747 scopus 로고    scopus 로고
    • A Markovian analysis of additive-increase, multiplicative-decrease (AIMD) algorithms
    • V. Dumas, F. Guillemin, and Ph. Robert (2002). A Markovian analysis of additive-increase, multiplicative-decrease (AIMD) algorithms. Adv. in Appl. Probab. 34, 85-111.
    • (2002) Adv. in Appl. Probab. , vol.34 , pp. 85-111
    • Dumas, V.1    Guillemin, F.2    Robert, P.H.3
  • 26
    • 84864849879 scopus 로고
    • The distribution of a perpetuity, with applications to risk theory, and pension funding
    • D. Dufresne (1990). The distribution of a perpetuity, with applications to risk theory, and pension funding. Scand. Actuar. J. 1-2, 39-79.
    • (1990) Scand. Actuar. J. , vol.1-2 , pp. 39-79
    • Dufresne, D.1
  • 27
    • 0001167174 scopus 로고    scopus 로고
    • An affine property of the reciprocal Asian option process
    • D. Dufresne (2001). An affine property of the reciprocal Asian option process. Osaka J. Math. 38, 379-381.
    • (2001) Osaka J. Math. , vol.38 , pp. 379-381
    • Dufresne, D.1
  • 28
    • 3843126281 scopus 로고    scopus 로고
    • Exponential functionals, and means of neutral-to-the-right priors
    • I. Epifani, A. Lijoi, and I. Prnster (2003). Exponential functionals, and means of neutral-to-the-right priors Biometrika 90, 791-808.
    • (2003) Biometrika , vol.90 , pp. 791-808
    • Epifani, I.1    Lijoi, A.2    Prnster, I.3
  • 29
    • 0003421261 scopus 로고
    • An introduction to probability theory, and its applications
    • 2nd edn, Wiley, New York
    • W. E. Feller (1971). An introduction to probability theory, and its applications, 2nd edn, vol. 2. Wiley, New York.
    • (1971) , vol.2
    • Feller, W.E.1
  • 30
    • 0001128091 scopus 로고
    • On the distribution of the sizes of particles which undergo splitting
    • A. F. Filippov (1961). On the distribution of the sizes of particles which undergo splitting. Th. Probab. Appl. 6, 275-293.
    • (1961) Th. Probab. Appl. , vol.6 , pp. 275-293
    • Filippov, A.F.1
  • 32
    • 0031592036 scopus 로고    scopus 로고
    • Present value distributions with application to ruin theory, and stochastic equations
    • H. K. Gjessing, and J. Paulsen (1997). Present value distributions with application to ruin theory, and stochastic equations. Stochastic Process. Appl. 71, 123-144.
    • (1997) Stochastic Process. Appl. , vol.71 , pp. 123-144
    • Gjessing, H.K.1    Paulsen, J.2
  • 33
    • 17044405972 scopus 로고    scopus 로고
    • Regenerative composition structures
    • A. Gnedin, and J. Pitman (2005). Regenerative composition structures. Ann. Probab. 33, 445-479.
    • (2005) Ann. Probab. , vol.33 , pp. 445-479
    • Gnedin, A.1    Pitman, J.2
  • 34
    • 84890653487 scopus 로고    scopus 로고
    • Asymptotic laws for compositions derived from transformed subordinators
    • Preprint available via
    • A. Gnedin, J. Pitman, and M. Yor (2004). Asymptotic laws for compositions derived from transformed subordinators. Preprint available via http://front.math.ucdavis.edu/math.PR/0403438.
    • (2004)
    • Gnedin, A.1    Pitman, J.2    Yor, M.3
  • 35
    • 0000732230 scopus 로고
    • Implicit renewal theory, and tails of solutions of random equations
    • C. M. Goldie (1991). Implicit renewal theory, and tails of solutions of random equations. Ann. Appl. Probab. 1, 126{166.
    • (1991) Ann. Appl. Probab. , vol.1 , pp. 126-166
    • Goldie, C.M.1
  • 36
    • 26844563763 scopus 로고    scopus 로고
    • AIMD algorithms, and exponential functionals
    • F. Guillemin, Ph. Robert,, and B. Zwart (2004). AIMD algorithms, and exponential functionals. Ann. Appl. Probab. 14, 90-117.
    • (2004) Ann. Appl. Probab. , vol.14 , pp. 90-117
    • Guillemin, F.1    Robert, P.H.2    Zwart, B.3
  • 37
    • 0037524295 scopus 로고    scopus 로고
    • Loss of mass in deterministic, and random fragmentations
    • B. Haas (2003). Loss of mass in deterministic, and random fragmentations. Stochastic Process. Appl. 106, 245-277.
    • (2003) Stochastic Process. Appl. , vol.106 , pp. 245-277
    • Haas, B.1
  • 38
    • 2942593709 scopus 로고    scopus 로고
    • Regularity of formation of dust in self-similar fragmentations
    • B. Haas (2004). Regularity of formation of dust in self-similar fragmentations. Ann. Inst. Henri Poincaré 40, 411-438.
    • (2004) Ann. Inst. Henri Poincaré , vol.40 , pp. 411-438
    • Haas, B.1
  • 39
    • 33646915441 scopus 로고    scopus 로고
    • Rates of convergence of diffusions with drifted Brownian potentials
    • Y. Hu, Z. Shi,, and M. Yor (1999). Rates of convergence of diffusions with drifted Brownian potentials. Trans. Amer. Math. Soc. 351, 3915-3934.
    • (1999) Trans. Amer. Math. Soc. , vol.351 , pp. 3915-3934
    • Hu, Y.1    Shi, Z.2    Yor, M.3
  • 40
    • 0003065227 scopus 로고    scopus 로고
    • Brownian motion on the hyperbolic plane, and Selberg trace formula
    • N. Ikeda, and H. Matsumoto (1999). Brownian motion on the hyperbolic plane, and Selberg trace formula. J. Funct. Anal. 163, 63-110.
    • (1999) J. Funct. Anal. , vol.163 , pp. 63-110
    • Ikeda, N.1    Matsumoto, H.2
  • 41
    • 0036065413 scopus 로고    scopus 로고
    • Self-similar processes with independent increments associated with Lévy, and Bessel
    • MR1919614
    • M. Jeanblanc, J. Pitman,, and M. Yor (2002). Self-similar processes with independent increments associated with Lévy, and Bessel. Stochastic Process. Appl. 100, 223-231. MR1919614
    • (2002) Stochastic Process. Appl. , vol.100 , pp. 223-231
    • Jeanblanc, M.1    Pitman, J.2    Yor, M.3
  • 42
    • 0010823278 scopus 로고
    • On the maximum of a diffusion process in a drifted Brownian environment
    • In: Séminaire de Probabilités, Springer, Berlin
    • K. Kawazu, and H. Tanaka (1993). On the maximum of a diffusion process in a drifted Brownian environment. In: Séminaire de Probabilités XXVII pp. 78-85, Lecture Notes in Math. 1557, Springer, Berlin.
    • (1993) Lecture Notes in Math. , vol.27
    • Kawazu, K.1    Tanaka, H.2
  • 43
    • 33344475302 scopus 로고    scopus 로고
    • A diffusion process in a Brownian environment with drift
    • K. Kawazu, and H. Tanaka (1997). A diffusion process in a Brownian environment with drift. J. Math. Soc. Japan 49, 189-211.
    • (1997) J. Math. Soc. Japan , vol.49 , pp. 189-211
    • Kawazu, K.1    Tanaka, H.2
  • 44
    • 0037675743 scopus 로고    scopus 로고
    • Quantum Calculus
    • Springer, Universitext, New York
    • V. Kac, and P. Cheung (2002). Quantum Calculus. Springer, Universitext, New York.
    • (2002)
    • Kac, V.1    Cheung, P.2
  • 45
    • 2942672026 scopus 로고
    • Random difference equations, and renewal theory for products of random matrices
    • H. Kesten (1973). Random difference equations, and renewal theory for products of random matrices. Acta Math. 131, 207-248.
    • (1973) Acta Math. , vol.131 , pp. 207-248
    • Kesten, H.1
  • 46
    • 3943085588 scopus 로고    scopus 로고
    • Some probability distributions in modelling DNA replication
    • A. Lachal (2003). Some probability distributions in modelling DNA replication. Ann. Appl. Probab. 13, 1207-1230.
    • (2003) Ann. Appl. Probab. , vol.13 , pp. 1207-1230
    • Lachal, A.1
  • 48
    • 37649027722 scopus 로고    scopus 로고
    • Single molecule statistics, and the polynucleotide unzipping transition
    • D. K. Lubensky, and D. R. Nelson (2002). Single molecule statistics, and the polynucleotide unzipping transition. Phys. Rev. E 65 03917.
    • (2002) Phys. Rev. E , vol.65 , pp. 03917
    • Lubensky, D.K.1    Nelson, D.R.2
  • 49
    • 0000131536 scopus 로고    scopus 로고
    • An analogue of Pitman's 2M - X theorem for exponential Wiener functionals
    • H. Matsumoto, and M. Yor (2000). An analogue of Pitman's 2M - X theorem for exponential Wiener functionals. I. A time-inversion approach. Nagoya Math. J. 159, 125-166.
    • (2000) I. A time-inversion approach. Nagoya Math. J. , vol.159 , pp. 125-166
    • Matsumoto, H.1    Yor, M.2
  • 51
    • 0037357658 scopus 로고    scopus 로고
    • On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts
    • H. Matsumoto, and M. Yor (2003). On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts. Adv. in Appl. Probab. 35, 184-206.
    • (2003) Adv. in Appl. Probab. , vol.35 , pp. 184-206
    • Matsumoto, H.1    Yor, M.2
  • 52
    • 84890688263 scopus 로고    scopus 로고
    • Tail asymptotics for exponential functionals of Lévy processes
    • to appear
    • K. Maulik, and B. Zwart (2004). Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process. Appl. (to appear).
    • (2004) Stochastic Process. Appl.
    • Maulik, K.1    Zwart, B.2
  • 53
    • 26144455410 scopus 로고    scopus 로고
    • Equations aux différences aléatoires: des fonctionnelles exponentielles de Lévy aux polymères dirigés en environnement aléatoire
    • Thèse de Doctorat de l'Université Paul Sabatier Toulouse III
    • O. Méjane (2003). Equations aux différences aléatoires: des fonctionnelles exponentielles de Lévy aux polymères dirigés en environnement aléatoire. Thèse de Doctorat de l'Université Paul Sabatier Toulouse III.
    • (2003)
    • Méjane, O.1
  • 54
    • 0030078157 scopus 로고    scopus 로고
    • On the distribution of a randomly discounted compound Poisson process
    • MR1386179
    • T. Nilsen, and J. Paulsen (1996). On the distribution of a randomly discounted compound Poisson process. Stochastic Process. Appl. 61, 305-310. MR1386179
    • (1996) Stochastic Process. Appl. , vol.61 , pp. 305-310
    • Nilsen, T.1    Paulsen, J.2
  • 55
    • 38249002913 scopus 로고
    • Risk theory in a stochastic economic environment
    • J. Paulsen (1993). Risk theory in a stochastic economic environment. Stochastic Process. Appl. 46, 327-361.
    • (1993) Stochastic Process. Appl. , vol.46 , pp. 327-361
    • Paulsen, J.1
  • 56
    • 0031349836 scopus 로고    scopus 로고
    • Ruin theory with stochastic return on investments
    • J. Paulsen, and H. K. Gjessing (1997). Ruin theory with stochastic return on investments. Adv. in Appl. Probab. 29, 965-985.
    • (1997) Adv. in Appl. Probab. , vol.29 , pp. 965-985
    • Paulsen, J.1    Gjessing, H.K.2
  • 57
    • 0001677581 scopus 로고
    • A diffusion process, and its applications to detecting a change in the drift of Brownian motion
    • M. Pollak, and D. Siegmund (1985). A diffusion process, and its applications to detecting a change in the drift of Brownian motion. Biometrika 72, 267- 280.
    • (1985) Biometrika , vol.72
    • Pollak, M.1    Siegmund, D.2
  • 58
    • 24144488921 scopus 로고    scopus 로고
    • Recurrent extensions of self-similar Markov processes, and Cramer's condition
    • V. Rivero (2005). Recurrent extensions of self-similar Markov processes, and Cramer's condition. Bernoulli 11, 471-509.
    • (2005) Bernoulli , vol.11 , pp. 471-509
    • Rivero, V.1
  • 59
    • 0001027466 scopus 로고
    • Self-similar processes with independent increments
    • MR1113220
    • K.-I. Sato (1991). Self-similar processes with independent increments. Probab. Theory Related Fields 89, 285-300. MR1113220
    • (1991) Probab. Theory Related Fields , vol.89 , pp. 285-300
    • Sato, K.-I.1
  • 61
    • 0041319950 scopus 로고    scopus 로고
    • Sinai's walk via stochastic calculus
    • In: Milieux aléatoires , F. Comets et al. (Eds)
    • Z. Shi (2001). Sinai's walk via stochastic calculus. In: Milieux aléatoires , F. Comets et al. (Eds) Panoramas et Synthèses 12, pp. 53-74.
    • (2001) Panoramas et Synthèses , vol.12 , pp. 53-74
    • Shi, Z.1
  • 62
    • 0002672593 scopus 로고    scopus 로고
    • The classical moment problem as a self adjoint finite difference operator
    • B. Simon (1998). The classical moment problem as a self adjoint finite difference operator. Adv. in Maths. 137, 82-203.
    • (1998) Adv. in Maths. , vol.137 , pp. 82-203
    • Simon, B.1
  • 63
    • 0002593008 scopus 로고
    • Recherches sur les fractions continues
    • Reprinted in : Ann. Fac. Sci. Toulouse Math. (6) 4 (1995),A5-A47
    • T. J. Stieltjes (1894-95) Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8, J76-J122, and 9, A5-A47. Reprinted in : Ann. Fac. Sci. Toulouse Math. (6) 4 (1995).
    • (1894) Ann. Fac. Sci. Toulouse , vol.8-9
    • Stieltjes, T.J.1
  • 64
    • 0012038363 scopus 로고    scopus 로고
    • Krein condition in probabilistic moment problems
    • J. Stoyanov (2000). Krein condition in probabilistic moment problems. Bernoulli 6, 939-949.
    • (2000) Bernoulli , vol.6 , pp. 939-949
    • Stoyanov, J.1
  • 65
    • 0041064778 scopus 로고
    • Functionals of transient stochastic processes with independent increments
    • K. Urbanik (1992). Functionals of transient stochastic processes with independent increments. Studia Math. 103 (3), 299-315.
    • (1992) Studia Math. , vol.103 , Issue.3 , pp. 299-315
    • Urbanik, K.1
  • 66
    • 0001565579 scopus 로고
    • On a stochastic difference equation, and a representation of nonnegative infinitely divisible random variables
    • W. Vervaat (1979). On a stochastic difference equation, and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11, 750-783.
    • (1979) Adv. in Appl. Probab. , vol.11 , pp. 750-783
    • Vervaat, W.1
  • 67
    • 0003617670 scopus 로고
    • Diffusions, Markov processes,, and martingales
    • Foundations. Wiley, New York
    • D. Williams (1979). Diffusions, Markov processes,, and martingales vol. 1: Foundations. Wiley, New York.
    • (1979) , vol.1
    • Williams, D.1
  • 68
    • 0000300111 scopus 로고
    • Sur certaines fonctionnelles exponentielles du mouvement brownien réel
    • English translation in [70]
    • M. Yor (1992). Sur certaines fonctionnelles exponentielles du mouvement brownien réel. J. Appl. Probab. 29 , 202-208. English translation in [70].
    • (1992) J. Appl. Probab. , vol.29 , pp. 202-208
    • Yor, M.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.