A closed-form expression is obtained for the conditional probability distribution of ∫t 0 R 2 s ds given R t, where (R s, 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: . We also examine the joint distribution of (R 2 t, ∫t 0 R 2 s ds.
Abadir, К. М., The joint density of two functionals of Brownian motion, Math. Methods of Statist. 4 (1995), 449–462. MR 97c:60200a
Abadir, К. М., Correction: The joint density of two functionals of a Brownian motion, Math. Methods of Statist. 5 (1996), 124. MR 97c:60200b
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), 435–465. MR 2003b:11083
Bismut, J. М., The Atiyah-Singer theorems: a probabilistic approach. I. The index theorem, J. Funct. Anal. 57 (1984), 56–99. MR 86q:58128a
Bismut, J. М., The Atiyah-Singer theorems: a probabilistic approach. II. The Lefschetz fixed point formulas, J. Funct. Anal. 57 (1984), 329–348. MR 86q:58128b
Borodin, A. N. and Salminen, P., Handbook of Brownian motion-facts and formulae, Birkhäuser, Second Edition (2002). MR 2003q:60001
Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), 95–153. MR 57#1574
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
Helmes, K. and Schwane, A., Lévy’s stochastic area formula in higher dimensions, J. Funct. Anal. 54 (1983), 177–192. MR 86a:60107
Levy, P.,. Wiener’s random function, and other Laplacian random functions, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (1950), 171–187. MR 13,476b
Lipster R. S. and Shiryaev, A. N., Statistics of Random Processes II: Applications, Springer, Second Edition (2001). MR 2001k:60001b
Oberhettinger, F. and Badii, L., Tables of Laplace Transforms, Springer (1973). MR 50#5375
Phillips, P. С. B., Time series regression with a unit root, Econometrica 55 (1987), 277–301. MR 89c:62156
Pitman, J. and Yor, М., A decomposition of Bessel bridges, Z. Wahrscheinlichkeitstheorie verw. Gebiete 59 (1982), 425–457. MR 84a:60091
Tolmatz, L., On the distribution of the square integral of the Brownian bridge, Ann. Probab. 30 (2002), 253–269. MR 2003q:60064
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. 361–375. MR 2002e:60136