Advertisement

Journal of Theoretical Probability

, Volume 18, Issue 3, pp 567–593 | Cite as

On First Range Times of Linear Diffusions

  • Paavo SalminenEmail author
  • Pierre Vallois
Article

Abstract

In this paper we consider first range times (with randomised range level) of a linear diffusion on R. Inspired by the observation that the exponentially randomised range time has the same law as a similarly randomised first exit time from an interval, we study a large family of non-negative 2-dimensional random variables (X,X′) with this property. The defining feature of the family is F c (x,y)=F c (x+y,0), ∀ x ≥ 0, y ≥ 0, where F c (x,y):=P (X > x, X′ > y) We also explain the Markovian structure of the Brownian local time process when stopped at an exponentially randomised first range time. It is seen that squared Bessel processes with drift are serving hereby as a Markovian element.

Keywords

Bessel bridges Bessel functions Brownian motion h-transforms infimum Ray–Knight theorem supremum 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I. 1970Handbook of Mathematical Functions, 9th printingDover Publications, Inc.New YorkGoogle Scholar
  2. 2.
    Barndorff-Nielsen, O., Blaesild, P., Halgreen, C. 1978First hitting time models for generalized inverse Gaussian distributionStoch. Proc. Appl.74954CrossRefGoogle Scholar
  3. 3.
    Borodin, A.N. 1989Distributions of functionals of the Brownian local time, ITh. Probab. Appl.34385401CrossRefGoogle Scholar
  4. 4.
    Borodin, , A., N. 1999On distribution of functionals of a Brownian motion stopped at the moment inverse to the rangeZapiski Nauchn. Semin. POMI (Russian).2605072English transl. in J. Math. SciencesGoogle Scholar
  5. 5.
    Borodin, A.N., Salminen, P. 2002Handbook of Brownian Motion – Facts and Formulae2BirkhäuserBasel, Boston, BerlinGoogle Scholar
  6. 6.
    Cheng, K.S., Cowan, R., Holst, L. 2000The ruin problem and cover times of asymmetric random walks and Brownian motionsAdv. Appl. Probab.32177192CrossRefGoogle Scholar
  7. 7.
    Chosid, L., Isaac, R. 1978On the range of recurrent Markov chainsAnn. Probab.6680687Google Scholar
  8. 8.
    Chosid, L., Isaac, R. 1980Correction to “On the range of recurrent Markov chains”Ann. Probab.81000Google Scholar
  9. 9.
    Dvoretsky A., Erdös P. (1951). Some problems on random walk in space. Second Berkeley Symp. Math. Stat. 353–368Google Scholar
  10. 10.
    Feller, W. 1951The asymtotic distribution of the range of sums of independent random variablesAnn. Math. Stat.22427432Google Scholar
  11. 11.
    Fitzsimmons, P. J. (1998). Markov process with identical bridges. Electronic J. Probab., 3(12)Google Scholar
  12. 12.
    Glynn, P. 1985On the range of a regenerative sequenceStoch. Proc. Appl.20105113CrossRefGoogle Scholar
  13. 13.
    Gradshteyn, I.S., Ryzhik, I.M. 1980Table of Integrals, Series and ProductsAcademic PressNew YorkGoogle Scholar
  14. 14.
    Hooghiemstra, G. 1987On functionals of the adjusted range processJ. Appl. Probab.24252257Google Scholar
  15. 15.
    Hsu P., March P. (1988). Brownian excursions from extremes. In: Meyer P.A., Azéma J., Yor M. (ed). Séminaire de Probabilités XXII, Number 1321 in Springer Lecture Notes in Math. Berlin, Heidelberg, New York, pp. 502–507Google Scholar
  16. 16.
    Imhof, J.P. 1985On the range of Brownian motion and its inverse processAnn. Probab.1310111017Google Scholar
  17. 17.
    Imhof, J.P. 1992A construction of the Brownian motion path from BES(3) piecesStoch. Proc. Appl.43345353CrossRefGoogle Scholar
  18. 18.
    Jain, N.C., Orey, S. 1968On the range of random walkIsrael J. Math.6373380Google Scholar
  19. 19.
    Jain, N.C., Pruitt, W.E. 1972The range of random walkSixth Berkeley Symp. Math. Stat. Probab.33150Google Scholar
  20. 20.
    McKean, H.P.,Jr. 1963Excursions of a non-singular diffusionZ. Wahrscheinlichkeitstheorie verw. Gebiete.1230239CrossRefGoogle Scholar
  21. 21.
    Pitman J., Yor M. (1981). Bessel processes and infinitely divisible laws. In: Williams D. (ed). Stochastic Integrals, Number 851 in Springer Lecture Notes in Math., Berlin, Heidelberg, New York, pp. 285–370Google Scholar
  22. 22.
    Pitman, J., Yor, M. 1982A decomposition of Bessel bridges. Z. Wahrscheinlichkeitstheorie verwGebiete59425457CrossRefGoogle Scholar
  23. 23.
    Pitman, J., Yor, M. 1993Dilatation d’espace temps réarrangments des trajectoires browniennes et quelques extensions d’une identité de KnightC.R. Acad. Sci. Paris Série I.316723726Google Scholar
  24. 24.
    Salminen, P. 1984Brownian excursions revisitedChung, K.L.Cinlar, E.Getoor, R.K. eds. Seminar in stochastic processes, 1983, Progress in probability and statistics.BirkhauserBoston, Basel, Stuttgart161188Google Scholar
  25. 25.
    Salminen, P. 1984One-dimensional diffusions and their exit spacesMath. Scand.54209220Google Scholar
  26. 26.
    Salminen, P. 1997On last exit decomposition of linear diffusionsStudia Sci. Math. Hungar.33251262Google Scholar
  27. 27.
    Sharpe, M. 1988General Theory of Markov ProcessesAcademic PressSan DiegoGoogle Scholar
  28. 28.
    Tapiero, S., Vallois, P. 1995Moments of the amplitude process in a random walk and approximations: computations and applicationsOperat. Res.29117Google Scholar
  29. 29.
    Troutman, B.M. 1983Weak convergence of the adjusted range of cumulative sums of exchangeable random variablesJ. Appl. Probab.20297304Google Scholar
  30. 30.
    Vallois, P. 1991La loi gaussienne inverse généralisée comme premier ou dernier temps de passage de diffusionsBull. Sc. Math. 2e Série.115301368Google Scholar
  31. 31.
    Vallois, P. 1991Une extension des théorèmes de Ray et Knight sur les temps locaux browniensProbab. Th. Rel. Fields.88445482CrossRefGoogle Scholar
  32. 32.
    Vallois, P. 1993Diffusion arrêtée au premier instant oùl’amplitude atteint un niveau donnéStochastics and Stochastics Reports4393115Google Scholar
  33. 33.
    Vallois, P. 1995Decomposing the Brownian path via the range processStochastic Processes and their Applications55211226CrossRefGoogle Scholar
  34. 34.
    Vallois, P. 1996The range of a simple random walk on ZAdv. Appl. Probab.2810141033Google Scholar
  35. 35.
    Watanabe, S. 1975On time inversion of one-dimensional diffusion processes. Z. Wahrscheinlichkeitstheorie verwGebiete31115124CrossRefGoogle Scholar
  36. 36.
    Watson G., N. 1948A Treatise on the Theory of Bessel Functions2Cambridge University PressLondonGoogle Scholar
  37. 37.
    Williams, D. 1974Path decompositions and continuity of local time for one-dimensional diffusionsProc. London Math. Soc.28738768Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Mathematical DepartmentÅbo Akademi UniversityÅboFinland
  2. 2.Département de MathématiqueUniversité Henri PoincaréVandoeuvre-les-NancyFrance

Personalised recommendations