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On First Range Times of Linear Diffusions

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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 Fc(x,y)=Fc(x+y,0), ∀ x ≥ 0, y ≥ 0, where Fc(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.

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Authors and Affiliations

  1. Mathematical Department, Åbo Akademi University, FIN-20500, Åbo, Finland

    Paavo Salminen

  2. Département de Mathématique, Université Henri Poincaré, F-54506, Vandoeuvre-les-Nancy, France

    Pierre Vallois

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  1. Paavo Salminen
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  2. Pierre Vallois
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Correspondence to Paavo Salminen.

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Salminen, P., Vallois, P. On First Range Times of Linear Diffusions. J Theor Probab 18, 567–593 (2005). https://doi.org/10.1007/s10959-005-3519-4

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  • DOI: https://doi.org/10.1007/s10959-005-3519-4

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