View More View Less
  • 1 Aarhus Universitet, Bygning 530, NY Munkegade DK-8000 Aarhus C, Denmark
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

A closed-form expression is obtained for the conditional probability distribution of ∫t0R2sds given Rt, where (Rs, s ≧ 0) is a Bessel process of dimension δ > 0 started from 0, in terms of parabolic cylinder functions. This is done by inverting the following Laplace transform also known as the generalized Lévy’s stochastic area formula: [exp(λ220tRs2ds)|Rt=a]=(λtsinh(λt))δ/2exp(a22t(λtcoth(λt)1)). We also examine the joint distribution of (R2t, ∫t0R2sds.

  • [1]

    Abadir, К. М., The joint density of two functionals of Brownian motion, Math. Methods of Statist. 4 (1995), 449462. MR 97c:60200a

  • [2]

    Abadir, К. М., Correction: The joint density of two functionals of a Brownian motion, Math. Methods of Statist. 5 (1996), 124. MR 97c:60200b

    • Search Google Scholar
    • Export Citation
  • [3]

    Biane, Ph., Pitman, J. and Yor, М., Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 435465. MR 2003b:11083

    • Search Google Scholar
    • Export Citation
  • [4]

    Bismut, J. М., The Atiyah-Singer theorems: a probabilistic approach. I. The index theorem, J. Funct. Anal. 57 (1984), 5699. MR 86q:58128a

    • Search Google Scholar
    • Export Citation
  • [5]

    Bismut, J. М., The Atiyah-Singer theorems: a probabilistic approach. II. The Lefschetz fixed point formulas, J. Funct. Anal. 57 (1984), 329348. MR 86q:58128b

    • Search Google Scholar
    • Export Citation
  • [6]

    Borodin, A. N. and Salminen, P., Handbook of Brownian motion-facts and formulae, Birkhäuser, Second Edition (2002). MR 2003q:60001

  • [7]

    Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), 95153. MR 57#1574

    • Search Google Scholar
    • Export Citation
  • [8]

    Gradshteyn, I. S., Ryzhik, I. М., and Jeffrey, A., editor, Table of Integrals, Series, and Products, 6th edition, San Diego, С A: Academic Press (2000). MR 2001c:00002

    • Search Google Scholar
    • Export Citation
  • [9]

    Helmes, K. and Schwane, A., Lévy’s stochastic area formula in higher dimensions, J. Funct. Anal. 54 (1983), 177192. MR 86a:60107

  • [10]

    Levy, P.,. Wiener’s random function, and other Laplacian random functions, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (1950), 171187. MR 13,476b

    • Search Google Scholar
    • Export Citation
  • [11]

    Lipster R. S. and Shiryaev, A. N., Statistics of Random Processes II: Applications, Springer, Second Edition (2001). MR 2001k:60001b

  • [12]

    Oberhettinger, F. and Badii, L., Tables of Laplace Transforms, Springer (1973). MR 50#5375

  • [13]

    Phillips, P. С. B., Time series regression with a unit root, Econometrica 55 (1987), 277301. MR 89c:62156

  • [14]

    Pitman, J. and Yor, М., A decomposition of Bessel bridges, Z. Wahrscheinlichkeitstheorie verw. Gebiete 59 (1982), 425457. MR 84a:60091

    • Search Google Scholar
    • Export Citation
  • [15]

    Tolmatz, L., On the distribution of the square integral of the Brownian bridge, Ann. Probab. 30 (2002), 253269. MR 2003q:60064

  • [16]

    Yor, М., Interpretations in terms of Brownian and Bessel meanders of the distribution of a subordinated perpetuity, in: Levy processes Theory and applications (Barndorff-Nielsen, О. E., Mikosch, T. and Resnick S. I., eds.), Birkhäuser, (2001), pp. 361375. MR 2002e:60136

    • Search Google Scholar
    • Export Citation