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References

  1. J. Abate and W. Whitt. Transient behavior of regulated Brownian motion, I: Starting at the origin. Adv. Appl. Probab., 19:560-598, 1987. Math. Review 0903538 (88m:60204a)
  2. L. Alili, D. Dufresne, and M. Yor. Sur l'identitÈ de Bougerol pour les fonctionnelles exponentielles du mouvement brownien avec drift. In M.Yor, editor, Exponential functionals and principal values related to Brownian motion, Revista Matematic· Iberoamericana, pages 3-14, 1997. Math. Review 1648654 (2000f:60125)
  3. Ph. Biane. Comparaison entre temps d'atteinte et temps de sÈjour de certaines diffusions rÈelles. In J. AzÈma and M. Yor, editors, SÈminaire de ProbabilitÈs XIX, number 1123 in Springer Lecture Notes in Mathematics, pages 291-296, Berlin, Heidelberg, New York, 1985. Math. Review 0889490 (88k:60144)
  4. A.N. Borodin and P. Salminen. Handbook of Brownian Motion - Facts and Formulae, 2nd edition. Birkh‰user, Basel, Boston, Berlin, 2002. Math. Review 1912205 (2003g:60001)
  5. A.N. Borodin and P. Salminen. On some exponential integral functionals of BM(µ) and BES(3). Zap. Nauchn. Semin. POMI, 311:51-78, 2004. Preprint available in http://arxiv.org/abs/math.PR/0408367. Math. Review number not available.
  6. Z.Ciesielski and S.J. Taylor. First passage times and sojourn density for Brownian motion in space and exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc., 103:434-450, 1962. Math. Review 0143257 (26 #816)
  7. P. Deheuvels and G. Martynov. Karhunen-Loeve expansions for weighted Wiener processes and Brownian bridges via Bessel functions. In J.Hoffmann-Joergensen et al., editor, Third international conference on high dimensional probability, volume55 of Progress in Probability, pages 57-93, Birkhauser Verlag, Basel, 2003. Math. Review 2033881
  8. C.Donati-Martin, H. Matsumoto, and M. Yor. On positive and negative moments of the integrals of geometric Brownian motions. Stat. Probab. Lett., 49:45-52, 2000. Math. Review 1789663 (2002f:60156)
  9. C.Donati-Martin, H.Matsumoto, and M. Yor. On striking identities about the exponential functionals of the Brownian bridge and Brownian motion. Periodica Math. Hung., 41(1-2):103-120, 2000. Math. Review 1812799 (2003d:60158)
  10. C. Donati-Martin, H. Matsumoto, and M. Yor. The law of geometric Brownian motion, and its integral, revisited; application to conditional moments. In H. Geman, D. Madan, S. Pliska, and T.Vorst, editors, Mathematical Finance - Bachelier Congress, 2000, Springer Finance, pages 221-244, Berlin, 2001. Math. Review 1960566 (2004k:60225)
  11. C. Donati-Martin, H. Matsumoto, and M. Yor. Some absolute continuity relationships for certain anticipative transformations of geometric Brownian motion. Publ. RIMS, Kyoto, 37(3):295-326, 2001. Math. Review 1855425 (2002h:60170)
  12. C. Donati-Martin and M. Yor. Fubini's theorem for double Wiener integrals and the variance of the Brownian paths. Ann. Inst. H.P., 27:181-200, 1991. Math. Review 1118933 (92m:60072)
  13. C. Donati-Martin and M. Yor. Some Brownian functionals and their laws. Ann. Probab., 25:1011-1058, 1997. Math. Review 1457611 (98e:60135)
  14. R.A. Doney and D.R. Grey. Some remarks on Brownian motion with drift. J. Appl. Probab., 26:659-663, 1989. Math. Review 1010955 (90i:60055)
  15. D. Dufresne. The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J., 1-2:39-79, 1990. Math. Review 1129194 (92i:62195)
  16. D. Dufresne. The integral of geometric Brownian motion. Adv. Appl. Probab., 33:223-241, 2001. Math. Review 1825324 (2002c:60132)
  17. R. Durrett. Brownian Motion and Martingales in Analysis. Wadsworth Inc., Belmont, California, 1984. Math. Review 0750829 (87a:60054)
  18. E.B. Dynkin. Markov Processes, Vol. I and II. Springer Verlag, Berlin, Gˆttingen, Heidelberg, 1965. Math. Review 0193671 (33 #1887)
  19. H.J. Engelbert and T. Senf. On functionals of Wiener process with drift and exponential local martingales. In M. Dozzi, H.J. Engelbert, and D. Nualart, editors, Stochastic processes and related topics. Proc. Wintersch. Stochastic Processes, Optim. Control, Georgenthal/Ger. 1990, number 61 in Math. Res., Academic Verlag, pages 45-58, Berlin, 1991. Math. Review 1127879 (92h:60072)
  20. R.K. Getoor and M.J. Sharpe. Excursions of Brownian motion and Bessel processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 47:83-106, 1979. Math. Review 0521534 (80b:60104)
  21. J.-C. Gruet. Windings of hyperbolic Brownian motion. In M. Yor, editor, Exponential functionals and principal values related to Brownian motion, Revista Matematic'a Iberoamericana, pages 35-72, 1997. Math. Review 1648656 (2000a:60153)
  22. Y. Hariya. Personal communication.
  23. J-P. Imhof. On the time spent above a level by Brownian motion with negative drift. Adv. Appl. Probab., 18:1017-1018, 1986. Math. Review 0867098 (88b:60182)
  24. K. ItÙ and H.P. McKean. Diffusion Processes and Their Sample Paths. Springer Verlag, Berlin, Heidelberg, 1974. Math. Review 0345224 (49 #9963)
  25. M. Jeanblanc, J. Pitman, and M. Yor. The Feynman-Kac formula and decomposition of Brownian paths. Comput. App. Math., 16:27-52, 1997. Math. Review 1458521 (98f:60156)
  26. E. Kamke. Differentialgleichungen, Lˆsungsmethoden und Lˆsungen. Akademische Verlagsgesellschaft, Leipzig, 1943. Math. Review 0466672 (57 #6549)
  27. I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Springer Verlag, Berlin, Heidelberg, 1991. Math. Review 1121940 (92h:60127)
  28. J. Kent. Some probabilistic properties of Bessel functions. Ann. Probab., 6:760-770, 1978. Math. Review 0501378 (58 #18750)
  29. J. Lamperti. Semi-stable Markov processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 22:205-225, 1972. Math. Review 0307358 (46 #6478)
  30. J.F. LeGall. Sur la mesure de Hausdorff de la courbe brownienne. In J. AzÈma and M. Yor, editors, SÈminaire de ProbabilitÈs XIX, number 1123 in Springer Lecture Notes in Mathematics, pages 297-313, Berlin, Heidelberg, New York, 1985. Math. Review 0889491 (88i:60127)
  31. R. Mansuy. An interpretation and some generalizations of the Anderson-Darling statistics in terms of squared Bessel bridges. Stat. Probab. Letters (to appear); PMA-prÈpublication 827, UniversitÈ Paris VI & VII, 2003. Math. Review number not available.
  32. R. Mansuy. On a one-parameter generalization of the Brownian bridge and associated quadratic functionals. J. Theor. Probab., 17:1021-1029, 2004. Math. Review 2105746
  33. H. Matsumoto and M. Yor. Some changes of probabilities related to a geometric Brownian motion versions of Pitman's 2M-X theorem. Elect. Comm. Probab., 4:15-23, 1999. Math. Review 1703607 (2000e:60130)
  34. H. Matsumoto and M. Yor. A version of Pitman's 2M-X theorem for geometric Brownian motions. C.R. Acad. Sci. Paris, 328, Serie I:1067-1074, 1999. Math. Review 1696208 (2000d:60134)
  35. H. Matsumoto and M. Yor. An analogue of Pitman's 2M-X theorem for exponential Brownian functionals, Part I - A time-inversion approach. Nagoya Math. J., 159:125-166, 2000. Math. Review 1783567 (2001j:60145)
  36. H. Matsumoto and M. Yor. An analogue of Pitman's 2M-X theorem for exponential Brownian functionals, Part II - The role of the generalized inverse Gaussian laws. Nagoya Math. J., 162:65-86, 2001. Math. Review 1836133 (2002d:60070)
  37. H. Matsumoto and M. Yor. On Dufresne's relation between the probability laws of exponential functionals of Brownian motion with different drifts. Adv. Appl. Probab., 36:184-206, 2003. Math. Review 1975510 (2004c:60233)
  38. H.P. McKean. Brownian motion and general diffusion: scale & clock. In S. Pliska, H. Geman, D. Madan and T. Vorst, editors, Mathematical Finance - Bachelier Congress 2000, Springer Finance, pages 248-256, Berlin, Heidelberg, New York, 2001. Springer Verlag. Math. Review 1960559 (2004b:60194)
  39. J. Pitman and M. Yor. Bessel processes and infinitely divisible laws. In D. Williams, editor, Stochastic Integrals, volume 851 of Springer Lecture Notes in Mathematics, pages 285-370, Berlin, Heidelberg, 1981. Springer Verlag. Math. Review 0620995 (82j:60149)
  40. J. Pitman and M. Yor. A decomposition of Bessel bridges. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 59:425-457, 1982. Math. Review 0656509 (84a:60091)
  41. D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer Verlag, Berlin, Heidelberg, 2001. Math. Review 1725357 (2000h:60050)
  42. P. Salminen. Optimal stopping of one-dimensional diffusions. Math. Nachr., 124:85-101, 1985. Math. Review 0827892 (87m:60177)
  43. P. Salminen and I. Norros. On busy periods of the unbounded Brownian storage. Queueing Systems, 39:317-333, 2001. Math. Review 1885742 (2002k:60198)
  44. P. Salminen and M. Yor. On Dufresne's perpetuity, translated and reflected. In J. Akahori, S. Ogawa, and S. Watanabe, editors, Proceedings of Ritsumeikan International Symposium: Stochastic Processes and Applications to Mathematical Finance, Singapore, 2004. World Scientific Publishing Co. Math. Review number not available.
  45. P. Salminen and M. Yor. Properties of perpetual integral functionals of Brownian motion with drift. To appear in Ann. I.H.P., in the volume dedicated to Paul-AndrÈ Meyer, 2005. Math. Review number not available.
  46. T. Szabados and B. Szekely. An exponential functional of random walks. J. Appl. Probab., 40:413-426, 2003. Math. Review 1978100 (2004c:60099)
  47. I.V. Vagurina. On diffusion processes corresponding to hypergeometric equation. Zap. Nauchn. Semin. POMI, 311:79-91, 2004. Math. Review number not available.
  48. D. Williams. Diffusions, Markov processes, & martingales. Volume 1: Foundations. John Wiley & Sons, New York, 1979. Math. Review 0531031(80i:60100)
  49. M. Yor. Une explication du thÈorËme de Ciesielski-Taylor. Ann. I.H.P., 27:201-213, 1991. Math. Review 1118934 (92g:60109)
  50. M. Yor. Some Aspects of Brownian Motion. Part I: Some special functionals. Birkh‰user Verlag, Basel, 1992. Math. Review 1193919 (93i:60155)
  51. M. Yor. Sur certaines fonctionnelles exponentielles du mouvement brownien rÈel. J. Appl. Probab., 29:202-208, 1992, (translated in English in 53). Math. Review 1147781 (93g:60179)
  52. M. Yor. From planar Brownian windings to Asian options. Insurance: Mathematics and Economics, 13:23-34, 1993, (also found in 53). Math. Review 1242683 (94j:60153)
  53. M. Yor. Exponential functionals of Brownian motion and related processes (in series Springer Finance). Springer Verlag, Berlin, Heidelberg, New York, 2001. Math. Review 1854494 (2003f:60147)
  54. M. Yor (ed.). Exponential functionals and principal values related to Brownian motion; a collection of research papers. Revista Matematic· Iberoamericana, 1997. Math. Review 1648653 (99d:60003)
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