Excursions of the integral of the brownian motion

Emmanuel Jacob

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 3, page 869-887
  • ISSN: 0246-0203

Abstract

top
The integrated brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first construct a stationary Langevin process and then determine explicitly its stationary excursion measure. This is then used to provide new descriptions of Itô’s excursion measure of the Langevin process reflected at a completely inelastic boundary, which has been introduced recently by Bertoin.

How to cite

top

Jacob, Emmanuel. "Excursions of the integral of the brownian motion." Annales de l'I.H.P. Probabilités et statistiques 46.3 (2010): 869-887. <http://eudml.org/doc/242758>.

@article{Jacob2010,
abstract = {The integrated brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first construct a stationary Langevin process and then determine explicitly its stationary excursion measure. This is then used to provide new descriptions of Itô’s excursion measure of the Langevin process reflected at a completely inelastic boundary, which has been introduced recently by Bertoin.},
author = {Jacob, Emmanuel},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Langevin process; stationary process; excursion measure; time-reversal; h-transform; -transform},
language = {eng},
number = {3},
pages = {869-887},
publisher = {Gauthier-Villars},
title = {Excursions of the integral of the brownian motion},
url = {http://eudml.org/doc/242758},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Jacob, Emmanuel
TI - Excursions of the integral of the brownian motion
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 3
SP - 869
EP - 887
AB - The integrated brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first construct a stationary Langevin process and then determine explicitly its stationary excursion measure. This is then used to provide new descriptions of Itô’s excursion measure of the Langevin process reflected at a completely inelastic boundary, which has been introduced recently by Bertoin.
LA - eng
KW - Langevin process; stationary process; excursion measure; time-reversal; h-transform; -transform
UR - http://eudml.org/doc/242758
ER -

References

top
  1. [1] J. Azéma. Théorie générale des processus et retournement du temps. Ann. Sci. École Norm. Sup. (4) 6 (1973) 459–519. Zbl0303.60061MR365725
  2. [2] J. Bertoin. Reflecting a Langevin process at an absorbing boundary. Ann. Probab. 35 (2007) 2021–2037. Zbl1132.60057MR2353380
  3. [3] J. Bertoin. A second order SDE for the Langevin process reflected at a completely inelastic boundary. J. Eur. Math. Soc. (JEMS) 10 (2008) 625–639. Zbl1169.60009MR2421156
  4. [4] R. K. Getoor. Excursions of a Markov process. Ann. Probab. 7 (1979) 244–266. Zbl0399.60069MR525052
  5. [5] J. P. Gor’kov. A formula for the solution of a certain boundary value problem for the stationary equation of Brownian motion. Dokl. Akad. Nauk SSSR 223 (1975) 525–528. Zbl0324.35013MR386033
  6. [6] A. Lachal. Sur le premier instant de passage de l’intégrale du mouvement brownien. Ann. Inst. H. Poincaré Probab. Statist. 27 (1991) 385–405. Zbl0747.60075MR1131839
  7. [7] A. Lachal. Application de la théorie des excursions à l’intégrale du mouvement brownien. In Séminaire de Probabilités XXXVII 109–195. Lecture Notes in Math. 1832. Springer, Berlin, 2003. Zbl1045.60084MR2053045
  8. [8] M. Lefebvre. First-passage densities of a two-dimensional process. SIAM J. Appl. Math. 49 (1989) 1514–1523. Zbl0681.60084MR1015076
  9. [9] B. Maury. Direct simulation of aggregation phenomena. Commun. Math. Sci. 2 (2004) 1–11. Zbl1086.76072MR2119870
  10. [10] H. P. McKean Jr.A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 (1963) 227–235. Zbl0119.34701MR156389
  11. [11] J. Pitman. Stationary excursions. In Séminaire de Probabilités, XXI 289–302. Lecture Notes in Math. 1247. Springer, Berlin, 1987. Zbl0619.60040MR941992

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.