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Bessel processes and hyperbolic Brownian motions stopped at different random times

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  • D'Ovidio, Mirko
  • Orsingher, Enzo

Abstract

Iterated Bessel processes R[gamma](t),t>0,[gamma]>0 and their counterparts on hyperbolic spaces, i.e. hyperbolic Brownian motions Bhp(t),t>0 are examined and their probability laws derived. The higher-order partial differential equations governing the distributions of and are obtained and discussed. Processes of the form R[gamma](Tt),t>0,Bhp(Tt), t>0 where are examined and numerous probability laws derived, including the Student law, the arcsine laws (also their asymmetric versions), the Lamperti distribution of the ratio of independent positively skewed stable random variables and others. For the random variable (where and B[mu] is a Brownian motion with drift [mu]), the explicit probability law and the governing equation are obtained. For the hyperbolic Brownian motions on the Poincaré half-spaces , (of respective dimensions 2,3) we study Bhp(Tt),t>0 and the corresponding governing equation. Iterated processes are useful in modelling motions of particles on fractures idealized as Bessel processes (in Euclidean spaces) or as hyperbolic Brownian motions (in non-Euclidean spaces).

Suggested Citation

  • D'Ovidio, Mirko & Orsingher, Enzo, 2011. "Bessel processes and hyperbolic Brownian motions stopped at different random times," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 441-465, March.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:3:p:441-465
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    References listed on IDEAS

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    1. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(6), pages 1157-1160, December.
    2. Beghin, L. & Orsingher, E. & Ragozina, T., 2001. "Joint distributions of the maximum and the process for higher-order diffusions," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 71-93, July.
    3. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(5), pages 1025-1031, October.
    4. Lachal, Aimé, 2008. "First hitting time and place for pseudo-processes driven by the equation subject to a linear drift," Stochastic Processes and their Applications, Elsevier, vol. 118(1), pages 1-27, January.
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    Cited by:

    1. D’Ovidio, Mirko, 2012. "From Sturm–Liouville problems to fractional and anomalous diffusions," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3513-3544.

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