On Distributions Associated with the Generalized Lévy s Stochastic Area Formula

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1 On Distributions Associated with the Generalized Lévy s Stochastic Area Formula Raouf Ghomrasni Abstract A closed-form ression is obtained for the conditional probability distribution of t R s ds given R t,wherer s,s is a Bessel process of dimension >startedfrom, 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: E λ ] Rs ds R t a λt sinhλt We also examine the joint distribution of R t, R s ds. / a t λt cothλt Key words and phrases: Bessel process, density/distribution functions, parabolic cylinder functions, Laplace inversion. AMS : 6G5, 6J65.. Introduction.. If R u,u is a Bessel process of dimension > started at, then the following formula is known to be valid see e.g. [4]:. E λ ] Rs ds R t a λt sinhλt / a t λt cothλt. If anda, then. leads to the distribution of the Brownian bridge b s,s in the L norm which is identical to Smirnov s distribution for his ω -test. We recall below the relation between the integral of the square of the Brownian bridge and the supremum of the absolute value see e.g. [3]:. b s ds + b s ds law 4 π sup s bs where b s, s is an independent copy of b s, s. If, then. is the Lévy s stochastic area formula. Indeed, Lévy [] showed that if Xt,Yt is an R -valued Brownian motion, starting from,, then for any ξ R and x, y R, E iξ ] XudY u Y udxu Xt x, Y t y E ξ ξt sinhξt where R X + Y and r x + y. ] R u du Rt r r t ξtcothξt Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8 Aarhus C, Denmark, raouf@imf.au.dk

2 Lévy s area formula arises naturally in some problems in analysis licit formula for the heat kernel corresponding to the Kohn-Laplacian of the Heisenberg group, see [7], geometry a probabilistic proof of the well-known index theorems of Atiyah and Singer due to J. M. Bismut, see [4, 5] and statistical inference parameter estimation and testing of statistical hypotheses for diffusion-type processes, see chapter 7 in []. We also note the close connection between the distributions of subordinated perpetuities and generalized Lévy s formula for the stochastic area of planar Brownian motion see [6] for details. For a historical account of Lévy s area formula, we refer the interested reader to [9] and [3]... Equivalently, we can write the generalized Lévy s stochastic area formula. as follows: ] [.6 E urt v Rs ds cosh vt+ u sinh / vt]. v In the Brownian case i.e., the Laplace inversion of.6 has been undertaken by Abadir [, ] in 995 who derived the joint density and distribution functions of the following two Brownian functionals:.7 B B s db s and Bs ds where B s, s is a standard one-dimensional Brownian motion started at. These two functionals play an important role in unit root statistics see [3]..3. The paper is organized as follows. In Section we derive licitly the density of R s ds given R t in terms of parabolic cylinder functions. In Section 3 we derive the joint density of R t, R s ds.. The density associated with the generalized Lévy s area formula The following theorem offers a method to invert.; the result may be ressed in terms of parabolic cylinder functions. Theorem. The density f a, t of R s ds given R t a is given by. f a, tx t / e a t π j j a j k j + / k x β e α 4x D β+ α x where α kt+ a +jt+ t, β j +, Dνξ is a parabolic cylinder function and 4 ν k νν +...ν + k Γν + k/γν is the Pochhammer s symbol. Proof: First, according to [; p. 59], we have. p ν e a p. ν π t ν a 8 t Dν+ a t Using the relation cothx +x, then anding the onential: λt sinh λt 3/4 t / e a t 3/4 t / e a /t / a t λt coth λt a j j j a j j/ j λ {j+/} e λ{ a + jt+ t} e λt j+ k j + / k λ {j+/} e α λ the termwise inversion of the series in.5 is readily justifiable by elementary estimates.

3 Corollary. The density f,t of R s ds given R t is given by.6 f,tx t / π x 4 k k k + e /4 t kt + t/ x D + x where D νξ is a parabolic cylinder function and ν k νν+...ν+k Γν+k/Γν is the Pochhammer s symbol. Remark: L. Tolmatz [5] determined the density.6 in the particular case for. 3. The joint density of R t, R s ds Theorem 3. The joint distribution g t of R t, R s ds is given by 3. g tx, y π Γ j x j+ y j 4 j k j + k e 4y {k+j+ 4 t+ x k + j + } D t + x 4 +j+ y where D νξ is a parabolic cylinder function and ν k νν+...ν+k Γν+k/Γν is the Pochhammer s symbol. Proof: Two methods lead to the same result 3.. The first method follows from Theorem. by integrating the conditional density f a,t with respect to the law of Rt P R t dx t / Γ x/ e x/t dx. This leads immediately to 3. and the details will be omitted. The second method is based on inverting the Laplace transform.6 and this can be done as follows. Set X Rt and Y R s ds. Using formula.6, the joint density of X and Y is found to be given by g tx, y 4π We note that 3. β+i γ+i β i γ i e xu+yv[ cosh vt+ [ cosh vt+ u sinh ] / vt v k u v sinh vt] / dudv. k / /e v vkt+ t u v k u +. v k+ Then, according to [; p. 39], we have 3.3 p a ν p + a µ. Γµ ν tµ ν e at F ν; µ ν;at for Rµ ν > so it follows that [ cosh vt+ u sinh ] / vt v 3.4 dx e xu k k By anding Kummer s function: 3.5 F k; ; x v / v e vkt+ t+ x Γ x / F k; ; x v. j k j x j j v j 3

4 we conclude as in the proof of Theorem.: 3.6 g tx, y π Γ j j xj+ y j 4 i i i! i j e 4y {i+ 4 t+ x i + } D t + x 4 +j+. y To show the equivalence between 3.6 and 3., let us compare the coefficients of these ressions. Since i j for i<jwe see that the second summation in 3.6 takes place only over i j, so that by setting k i j the coefficients in 3. and 3.6 respectively become: Cj, k j j + k Dj, k j+k j + kj. j j + It is easily verified that: Cj, k + j + k j Γ + j Dj, k. Remarks:. A. Borodin kindly informed us that a similar ression for g t appears in the new edition of [6] see.9.8 p Abadir [] has derived the joint density of Bs dbs, B s ds B, B s ds which correspond to the case. Acknowledgments It is a pleasure to thank Goran Peskir for his kind invitation to Aarhus university and many fruitful discussions. The author thanks the anonymous referee for a careful reading of the manuscript and helpful suggestions. References [] Abadir, K. M The joint density of two functionals of Brownian motion. Math. Methods of Statist [] Abadir, K. M Correction: The joint density of two functionals of a Brownian motion. Math. Methods of Statist [3] Biane, Ph., Pitman, J. and Yor, M.. Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull.Amer.Math.Soc.N.S [4] Bismut, J. M The Atiyah-Singer theorems: a probabilistic approach. I. The index theorem. J. Funct. Anal [5] Bismut, J. M The Atiyah-Singer theorems: a probabilistic approach. II. The Lefschetz fixed point formulas. J. Funct. Anal [6] Borodin, A.N., Salminen, P.. Handbook of Brownian motion-facts and formulae Birkhäuser, Second Edition. [7] Gaveau, B Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math [8] Gradshteyn, I.S., I.M. Ryzhik; Alan Jeffrey, Editor. Table of Integrals, Series, and Products, 6th edition, San Diego, CA: Academic Press. [9] Helmes, K. and Schwane A Lévy s stochastic area formula in higher dimensions. J. Funct. Anal

5 [] Lévy, P. 95. Wiener s random function, and other Laplacian random functions. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability [] Lipster R. S. and Shiryaev A. N.. Statistics of Random Processes II: Applications. Springer, Second Edition. [] Oberhettinger, F. and Badii L Tables of Laplace Transforms. Springer. [3] Phillips, P. C. B Time series regression with a unit root. Econometrica [4] Pitman, J. and Yor, M. 98. A decomposition of Bessel bridges. Z. Wahrscheinlichkeitstheorie verw. Gebiete [5] Tolmatz, L.. On the distribution of the square integral of the Brownian bridge. Ann. Probab [6] Yor, M.. Interpretations in terms of Brownian and Bessel meanders of the distribution of a subordinated perpetuity. In: Barndorff-Nielsen, O. E., Mikosch, T. and Resnick S. I. Eds, Lévy processes Theory and applications. Birkhäuser,