Properties of perpetual integral functionals of brownian motion with drift

Paavo Salminen; Marc Yor

Annales de l'I.H.P. Probabilités et statistiques (2005)

  • Volume: 41, Issue: 3, page 335-347
  • ISSN: 0246-0203

How to cite


Salminen, Paavo, and Yor, Marc. "Properties of perpetual integral functionals of brownian motion with drift." Annales de l'I.H.P. Probabilités et statistiques 41.3 (2005): 335-347. <>.

author = {Salminen, Paavo, Yor, Marc},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Local time; Green function; Kac's moment formula; Khas'minskii's lemma; Last exit time},
language = {eng},
number = {3},
pages = {335-347},
publisher = {Elsevier},
title = {Properties of perpetual integral functionals of brownian motion with drift},
url = {},
volume = {41},
year = {2005},

AU - Salminen, Paavo
AU - Yor, Marc
TI - Properties of perpetual integral functionals of brownian motion with drift
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2005
PB - Elsevier
VL - 41
IS - 3
SP - 335
EP - 347
LA - eng
KW - Local time; Green function; Kac's moment formula; Khas'minskii's lemma; Last exit time
UR -
ER -


  1. [1] J. Bertoin, Subordinators: Examples and applications, in: Bertoin J., Martinelli F., Peres Y. (Eds.), École d'eté de Probabilités de Saint-Flour XXVII-1997, Lecture Notes in Math., vol. 1717, Springer, Berlin, 1999, pp. 1-91. Zbl0955.60046MR1746300
  2. [2] A.N. Borodin, P. Salminen, Handbook of Brownian Motion – Facts and Formulae, Birkhäuser, Basel, 2002. Zbl1012.60003MR1912205
  3. [3] A.S. Cherny, Convergence of some integrals associated with Bessel processes, Theoret. Probab. Appl.45 (2000) 251-267. Zbl0982.60077MR1967756
  4. [4] K.L. Chung, Z. Zhao, From Brownian Motion to Schrödinger's Equation, Springer-Verlag, Berlin, 1995. Zbl0819.60068MR1329992
  5. [5] M. Csörgö, L. Horváth, Q.M. Shao, Convergence of integrals of uniform empirical and quantile processes, Stochastic Process. Appl.45 (1993) 278-294. Zbl0784.60038MR1208874
  6. [6] C. Dellacherie, P.A. Meyer, Probabilités et Potentiel, vols. I–V, Hermann, Paris, 1975. Zbl0323.60039MR488194
  7. [7] D. Dufresne, The distribution of a perpetuity, with applications to risk theory and pension funding, Scand. Actuar. J.1–2 (1990) 39-79. Zbl0743.62101MR1129194
  8. [8] R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Belmont, CA, 1984. Zbl0554.60075MR750829
  9. [9] M. Emery, Une définition faible de BMO, Ann. Inst. H. Poincare21 (1985) 59-72. Zbl0611.60043MR791270
  10. [10] H.J. Engelbert, W. Schmidt, On the behaviour of certain functionals of the Wiener process and applications to stochastic differential equations, in: Stochastic Differential Systems. Proc. 3rd IFIP-WG 7/1 Working Conf., Lecture Notes in Control and Inform. Sci., vol. 36, Springer-Verlag, Berlin, 1981, pp. 47-55. Zbl0468.60077MR653645
  11. [11] H.J. Engelbert, W. Schmidt, On exponential local martingales connected with diffusion processes, Math. Nachr.119 (1984) 97-115. Zbl0565.60063MR774179
  12. [12] H.J. Engelbert, W. Schmidt, On the behaviour of certain Bessel functionals. An application to a class of stochastic differential equations, Math. Nachr.131 (1987) 219-234. Zbl0627.60070MR908813
  13. [13] H.J. Engelbert, T. Senf, On functionals of Wiener process with drift and exponential local martingales, in: Dozzi M., Engelbert H.J., Nualart D. (Eds.), Stochastic Processes and Related Topics, Proc. Wintersch. Stochastic Processes, Optim. Control, Georgenthal/Ger. 1990, Math. Res., vol. 61, Academic-Verlag, Berlin, 1991, pp. 45-58. Zbl0744.60098MR1127879
  14. [14] P. Fitzsimmons, J. Pitman, Kac's moment formula and the Feynman–Kac formula for additive functionals of a Markov process, Stochastic Process. Appl.79 (1999) 117-134. Zbl0962.60067MR1670526
  15. [15] T. Jeulin, Semimartingales et grossissement d'une filtration, Lecture Notes in Math., vol. 833, Springer-Verlag, Berlin, 1980. Zbl0444.60002MR604176
  16. [16] T. Jeulin, Sur la convergence absolue de certaines intégrales, in: Azéma J., Yor M. (Eds.), Séminaire de Probabilités XVI, Lecture Notes in Math., vol. 920, Springer, Berlin, 1982, pp. 248-256. Zbl0483.60020MR658688
  17. [17] N. Kazamaki, Continuous Exponential Martigales and BMO, Lecture Notes in Math., vol. 1579, Springer-Verlag, Berlin, 1994. Zbl0806.60033MR1299529
  18. [18] N. Kazamaki, T. Sekiguchi, On the transformation of some classes of martingales by a change of law, Tôhoku Math. J.31 (1979) 261-279. Zbl0438.60040MR547641
  19. [19] R. Khas'minskii, On positive solutions of the equation A u + V u = 0 , Theoret. Probab. Appl.4 (1959) 309-318. Zbl0089.34501
  20. [20] P.A. Meyer, Probabilités et potential, Hermann (Editions Scientifiques), Paris, 1966. MR205287
  21. [21] J. Pitman, M. Yor, Bessel processes and infinitely divisible laws, in: Williams D. (Ed.), Stochastic Integrals, Lecture Notes in Math., vol. 851, Springer-Verlag, Berlin, 1981, pp. 285-370. Zbl0469.60076MR620995
  22. [22] J. Pitman, M. Yor, A decomposition of Bessel bridges, Z. Wahrsch. Verw. Gebiete59 (1982) 425-457. Zbl0484.60062MR656509
  23. [23] J. Pitman, M. Yor, Some divergent integrals of Brownian motion, Adv. Appl. Probab. (Supplement) (1986) 109-116. Zbl0618.60074MR868512
  24. [24] P. Salminen, M. Yor, Perpetual integral functionals as hitting times, Elec. J. Prob., in press. Zbl1110.60078
  25. [25] L.A. Shepp, Radon–Nikodym derivatives of Gaussian measures, Ann. Math. Statist.37 (1966) 321-354. Zbl0142.13901MR190999
  26. [26] B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, 1979. Zbl0434.28013MR544188
  27. [27] W. Stummer, K.-T. Sturm, On exponentials of additive functionals of Markov processes, Stochastic Process. Appl.85 (2000) 45-60. Zbl0996.60090MR1730619
  28. [28] X.-X. Xue, A zero-one law for integral functionals of the Bessel process, in: Azéma J., Meyer P.A., Yor M. (Eds.), Séminaire de Probabilités XXIV, Lecture Notes in Math., vol. 1426, Springer, Berlin, 1990, pp. 137-153. Zbl0704.60082MR1071537
  29. [29] M. Yor, Sur certaines fonctionnelles exponentielles du mouvement brownien réel, J. Appl. Probab.29 (1992) 202-208. Zbl0758.60085MR1147781
  30. [30] M. Yor, Exponential Functionals of Brownian Motion and Related Processes, Springer Finance, Springer-Verlag, Berlin, 2001. Zbl0999.60004MR1854494

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