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On First Range Times of Linear Diffusions

Author

Listed:
  • Paavo Salminen

    (Åbo Akademi University)

  • Pierre Vallois

    (Université Henri Poincaré)

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.

Suggested Citation

  • Paavo Salminen & Pierre Vallois, 2005. "On First Range Times of Linear Diffusions," Journal of Theoretical Probability, Springer, vol. 18(3), pages 567-593, July.
    • Handle: RePEc:spr:jotpro:v:18:y:2005:i:3:d:10.1007_s10959-005-3519-4
    DOI: 10.1007/s10959-005-3519-4
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    Cited by:

    1. Etienne Tanré & Pierre Vallois, 2006. "Range of Brownian Motion with Drift," Journal of Theoretical Probability, Springer, vol. 19(1), pages 45-69, January.

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