An analogue of Pitman's 2M - X theorem for exponential Wiener functionals: Part I: A time-inversion approach

Nagoya Mathematical Journal

Volumn 159, Issue , 2000, Pages 125-166

  Matsumoto, Hiroyuki ^a   Yor, Marc ^b  

^a NAGOYA UNIVERSITY   (Japan)

^b University of Paris VII LPSM partially supported by ERC Starting Grant 680275 MALIG   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000131536     PISSN: 00277630     EISSN: None     Source Type: Journal    
DOI: 10.1017/s0027763000007455     Document Type: Article

References (55)

1. L. Alili, D. Dufresne et M. Yor, Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement brownien avec drift, in [55].
2. L. Arnold, Random Dynamical Systems, Springer, 1998.
3. J. Bertoin, An extension of Pitman's theorem for spectrally positive Lévy processes, Ann. Prob., 20 (1993), 1463-1483.
4. P. Biane, Quelques propriétés du mouvement Brownien dans un cone, Stoch., Proc. Appl., 53 (1994), 233-240.
5. P. Biane, Intertwining of Markov semi-groups, some examples, Sém. Prob., XXIX, Lecture Notes in Math., 1613, 30-36, Springer, 1995.
6. D. R. Brillinger, A particle migrating randomly on a sphere, J. Theor. Prob., 10 (1997), 429-443.
7. P. Carmona, F. Petit and M. Yor, On the distribution and asymptotic results for exponential functionals of Lévy processes, in [55].
8. P. Carmona, F. Petit and M. Yor, Beta-Gamma variables and intertwinings of certain Markov processes, Rev. Mat. Iberoamericana, 14 (1998), 311-367.
9. L. de Haan and R. L. Karandikar, Embedding a stochastic difference equation into a continuous-time process, Stoch. Proc. Appl., 32 (1989), 225-235.
10. C.Donati-Martin and M. Yor, Some measure-valued Markov processes attached to occupation times of Brownian motion, Bernoulli, 6 (2000), 63-72.
11. C. Donati-Martin, R. Ghomrasni and M. Yor, On certain Markov processes attached to exponential functionals of Brownian motion; application to Asian options, to appear in Rev. Mat. Iberoamericana.
12. R.A.Doney, J.Warren and M.Yor, Perturbed Bessel processes, Sém. Prob., XXXII, Lecture Notes in Math., 1686, 237-249, Springer, 1998.
13. D. Dufresne, An affine property of the reciprocal Asian option process, to appear in Osaka J. Math.
14. M. Emery, private communications.
15. M. Emery and E. Perkins, La filtration de B + L, Z.W., 59 (1982), 383-390.
16. P. J. Fitzsimmons, A converse to a theorem of P. Lévy, Ann. Prob., 15 (1987), 1515-1523.
17. Y. Hu, private communications.
18. K. Itô and H. P. McKean Jr., Diffusion Processes and Their Sample Paths, Springer-Verlag, Berlin, 1965.
19. D. G. Kendall, Pole-seeking Brownian motion and bird navigation, J. Roy. Stat. Soc., 36B (1974), 365-417.
20. N. N. Lebedev, Special Functions and their Applications, Dover, New York, 1972.
21. J. F. Le Gall, Sur la mesure de Hausdorff de la courbe brownienne, Sém. Prob. XIX, Lec. Notes in Math., 1123, 297-313, Springer, 1985..
22. J. F. Le Gall, Mouvement brownien, cônes et processus stables, Prob. Th. Rel. Fields, 76 (1987), 587-627.
23. J. F. Le Gall and M. Yor, Excursions browniennes et carrés de processus de Bessel, C. R. Acad. Sci. Paris, Série I, 303 (1986), 73-76.
24. H. Matsumoto and M. Yor, On Bougerol and Dufresne's identities for exponential Brownian functionals, Proc. Japan Acad., 74, Ser.A (1998), 152-155.
25. H. Matsumoto and M. Yor, A version of Pitman's 2M - X theorem for geometric Brownian motions, C. R. Acad. Sc. Paris Série I, 328 (1999), 1067-1074.
26. H. Matsumoto and M. Yor, A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration, to appear in Osaka J. Math.
27. H. Matsumoto and M. Yor, An analogue of Pitman's 2M - X theorem for exponential Brownian functional Part II: the role of generalized inverse Gaussian distributions, to appear in Nagoya Math. J.
28. M. Nagasawa, Time reversal of Markov processes, Nagoya Math. J., 24 (1964), 177-204.
29. Y. Ogura, One-dimensional bi-generalized diffusion processes, J. Math. Soc. Japan, 41 (1989), 213-242.
30. J. W. Pitman, One-dimensional Brownian motion and the three-dimensional Bessel process, Adv. Appl. Prob., 7 (1975), 511-526.
31. J. W. Pitman and M. Yor, Processus de Bessel, et mouvement brownien, avec ≪Drift≫, C. R. Acad. Sc. Paris Série I, 291 (1980), 151-153.
32. J. W. Pitman and M. Yor, Bessel processes and infinitely divisible laws, Stochastic Integrals, ed. by D.Williams, Lecture Notes Math., 851, 285-370, Springer-Verlag, Berlin, 1981.
33. B. Rauscher, Some remarks on Pitman's theorem, Sém. Prob. XXXI, 266-271, Lecture Notes Math., 1655, Springer, Berlin, 1997.
34. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd. Ed., Springer-Verlag, Berlin, 1999.
35. L. C. G. Rogers, Characterizing all diffusions with the 2M - X property, Ann. Prob., 9 (1981), 561-572.
36. L. C. G. Rogers and J. W. Pitman, Markov functions, Ann. Prob., 9 (1981), 573-582.
37. Y. Saisho and H. Tanemura, Pitman type theorem for one-dimensional diffusion processes, Tokyo J. Math., 13 (1990), 429-440.
38. V. Seshadri, The Inverse Gaussian Distributions, Oxford Univ. Press, Oxford, 1993.
39. L. Shepp, A first passage problem for the Wiener process, Ann. Math. Stat., 38 (1967), 1912-1914.
40. K. Takaoka, On the martingale obtained by an extension due to Saisho, Tanemura and Yor of Piman's theorem, Sém. Prob. XXXI, 256-265, Lecture Notes Math., 1655, Springer, Berlin, 1997.
41. H. Tanaka, Time reversal of random walks in one dimension, Tokyo J. Math., 12 (1989), 159-174.
42. d, Tokyo J. Math., 13 (1989), 375-389.
43. S. Watanabe, On time inversion of one-dimensional diffusion processes, Z.W., 31 (1975), 115-124.
44. S. Watanabe, Invariants of one-dimensional diffusion processes and applications, J. Korean Math. Soc., 35 (1998), 637-658.
45. S. Watanabe, Bilateral Bessel diffusion processes with drift and time inversion, preprint.
46. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1944.
47. D. Williams, Diffusions, Markov Processes, & Martingales, Volume 1: Foundations, John Wiley & Sons, New York, 1979.
48. M. Yor, Some Aspects of Brownian Motion, Part I:Some Special Functional, Lectures in Math., ETH Zürich, Birkhäuser, Basel, 1992.
49. M. Yor, Some Aspects of Brownian Motion, Part II: Some Recent Martingale Problems, Lectures in Math., ETH Zürich, Birkhäuser, Basel, 1997.
50. M. Yor, Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson, Z.W., 53 (1980), 71-95.
51. M. Yor, On square-root boundaries for Bessel processes and pole seeking Brownian motions, Stochastic Analysis and Applications, ed. by A. Truman and D. Williams, 100-107, Lecture Notes Math., 1095, Springer, Berlin, 1984.
52. M. Yor, Une extension markovienne de l'algèbre des lois béta-gamma, C. R. Acad. Sci. Paris Série I, 308 (1989), 257-260.
53. M. Yor, On some exponential functionals of Brownian motion, Adv. Appl. Prob., 24 (1992), 509-531.
54. M. Yor, Interpretation in terms of Brownian and Bessel meanders of the distribution of a subordinated perpetuity, to appear in a Birkhäser volume on Lévy processes, ed. by T. Mikosh.
55. M. Yor (ed.), Exponential Functionals and Principal Values related to Brownian Motion, A collection of research papers, Biblioteca de la Revista Matemática Iberoamericana, 1997.