Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 3, pp. 667-684.

We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any α, x>0, γ≥0 and f a smooth function on + ,

𝐋 (γ) f(x)=x -α (σ 2x 2 f '' (x)+(σγ+b)xf ' (x)+ 0 f e -r x - f (x)e -rγ +xf ' (x)r𝕀 {r1} ν(dr)),(0.1)
where the coefficients b, σ≥0 and the measure ν, which satisfies the integrability condition 0(1∧r2)ν(dr)<+∞, are uniquely determined by the distribution of a spectrally negative, infinitely divisible random variable, with characteristic exponent ψ. L(γ) is known to be the infinitesimal generator of a positive α-self-similar Feller process, which has been introduced by Lamperti [Z. Wahrsch. Verw. Gebiete 22 (1972) 205-225]. The eigenfunctions are expressed in terms of a new family of power series which includes, for instance, the modified Bessel functions of the first kind and some generalizations of the Mittag-Leffler function. Then, we show that some specific combinations of these functions are Laplace transforms of self-decomposable or infinitely divisible distributions concentrated on the positive line with respect to the main argument, and, more surprisingly, with respect to the parameter ψ(γ). In particular, this generalizes a result of Hartman [Ann. Sc. Norm. Super. Pisa Cl. Sci. IV-III (1976) 267-287] for the increasing solution of the Bessel differential equation. Finally, we compute, for some cases, the associated decreasing eigenfunctions and derive the Laplace transform of the exponential functionals of some spectrally negative Lévy processes with a negative first moment.

Nous commençons par caractériser les fonctions propres croissantes, au sens strict, de la famille d'opérateurs intégro-différentiels (0.1), pour tout α>0, γ≥0, f une function définie sur + et suffissament régulière, et où les coefficients b, σ≥0 et la mesure ν, qui satisfait la condition d'intégrabilité 0(1∧r2)ν(dr)<+∞, sont données, de manière unique, par la distribution d'une variable aléatoire infiniment divisible et spectralement négative dont on écrit ψ son exposant caractéristique. L(γ) est le générateur infinitésimal d'un processus positif Fellerien α-auto-similaire, introduit par Lamperti [Z. Wahrsch. Verw. Gebiete 22 (1972) 205-225]. Les fonctions propres sont définies en terme d'une nouvelle famille de séries entières qui contient, par exemple, les fonctions de Bessel modifiées du premier ordre et des généralisations des fonctions de Mittag-Leffler. Nous continuons par montrer que des combinaisons particulières de ces séries entières correspondent à des transformées de Laplace de variables aléatoires positives auto-décomposables ou infiniment divisibles, par rapport à la valeur propre associée mais aussi par rapport au paramètre ψ(γ), ce qui est plus surprenant. En particulier, ceci généralise un résultat de Hartman [Ann. Sc. Norm. Super. Pisa Cl. Sci. IV-III (1976) 267-287] sur les fonctions de Bessel modifiées. Finalement, nous calculons, dans certains cas, les fonctions propres décroissantes, ce qui nous permet de caractériser la loi, par le biais de sa transformée de Laplace, de la fonctionnelle exponentielle de certains processus de Lévy spectralement négatifs ayant un premier moment négatif.

DOI: 10.1214/08-AIHP182
Classification: 31C05,  60G18,  33E12,  20C20
Keywords: infinite divisibility, first passage time, self-similar Markov processes, special functions
@article{AIHPB_2009__45_3_667_0,
     author = {Pierre, Patie},
     title = {Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of {L\'evy} processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {667--684},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     doi = {10.1214/08-AIHP182},
     zbl = {1180.31010},
     mrnumber = {2548498},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/08-AIHP182/}
}
TY  - JOUR
AU  - Pierre, Patie
TI  - Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
DA  - 2009///
SP  - 667
EP  - 684
VL  - 45
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/08-AIHP182/
UR  - https://zbmath.org/?q=an%3A1180.31010
UR  - https://www.ams.org/mathscinet-getitem?mr=2548498
UR  - https://doi.org/10.1214/08-AIHP182
DO  - 10.1214/08-AIHP182
LA  - en
ID  - AIHPB_2009__45_3_667_0
ER  - 
%0 Journal Article
%A Pierre, Patie
%T Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 667-684
%V 45
%N 3
%I Gauthier-Villars
%U https://doi.org/10.1214/08-AIHP182
%R 10.1214/08-AIHP182
%G en
%F AIHPB_2009__45_3_667_0
Pierre, Patie. Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 3, pp. 667-684. doi : 10.1214/08-AIHP182. http://www.numdam.org/articles/10.1214/08-AIHP182/

[1] R. P. Agarwal. A propos d'une note de M. Pierre Humbert. C. R. Math. Acad. Sci. Paris 236 (1953) 2031-2032. | MR | Zbl

[2] V. Bally and L. Stoica. A class of Markov processes which admit a local time. Ann. Probab. 15 (1987) 241-262. | MR | Zbl

[3] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl

[4] J. Bertoin and M. Yor. The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17 (2002) 389-400. | MR | Zbl

[5] J. Bertoin and M. Yor. On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes. Ann. Fac. Sci. Toulouse Math. 11 (2002) 19-32. | Numdam | MR | Zbl

[6] J. Bertoin and M. Yor. Exponential functionals of Lévy processes. Probab. Surv. 2 (2005) 191-212. | MR

[7] Ph. Biane and M. Yor. Variations sur une formule de Paul Lévy. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987) 359-377. | Numdam | MR | Zbl

[8] R. M. Blumenthal. On construction of Markov processes. Z. Wahrsch. Verw. Gebiete 63 (1983) 433-444. | MR | Zbl

[9] Ph. Carmona, F. Petit and M. Yor. Sur les fonctionnelles exponentielles de certains processus de Lévy. Stoch. Stoch. Rep. 47 (1994) 71-101. (English version in [38], p. 139-171.) | MR | Zbl

[10] M. E. Caballero and L. Chaumont. Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Probab. 34 (2006) 1012-1034. | MR | Zbl

[11] Z. Ciesielski and S. J. Taylor. First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103 (1962) 434-450. | MR | Zbl

[12] E. B. Dynkin. Markov Processes I, II. Die Grundlehren der Mathematischen Wissenschaften, Bände 121 122. Academic Press, New York, 1965. | MR | Zbl

[13] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi. Higher Transcendental Functions 3. McGraw-Hill, New York, 1955. | MR | Zbl

[14] W. E. Feller. An Introduction to Probability Theory and Its Applications 2, 2nd edition. Wiley, New York, 1971. | Zbl

[15] I. I. Gikhman and A. V. Skorokhod. The Theory of Stochastic Processes II. Springer, Berlin, 1975. | MR | Zbl

[16] P. Hartman. Completely monotone families of solutions of n-th order linear differential equations and infinitely divisible distributions. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV-III (1976) 267-287. | EuDML | Numdam | MR | Zbl

[17] P. Hartman and G. S. Watson. “Normal”“Normal” distribution functions on spheres and the modified Bessel functions. Ann. Probab. 2 (1974) 593-607. | MR | Zbl

[18] P. Humbert. Quelques résultats relatifs à la fonction de Mittag-Leffler. C. R. Math. Acad. Sci. Paris 236 (1953) 1467-1468. | MR | Zbl

[19] M. Jeanblanc, J. Pitman and M. Yor. Self-similar processes with independent increments associated with Lévy and Bessel processes. Stochastic Process. Appl. 100 (2002) 223-232. | MR | Zbl

[20] J. Kent. Some probabilistic properties of Bessel functions. Ann. Probab. 6 (1978) 760-770. | MR | Zbl

[21] A. A. Kilbas and J. J. Trujillo. Differential equations of fractional orders: Methods, results and problems. Appl. Anal. 78 (2001) 153-192. | MR | Zbl

[22] A. A. Kilbas and M. Saigo. On solution of integral equations of Abel-Volterra type. Differential Integral Equations 8 (1995) 993-1011. | MR | Zbl

[23] J. Lamperti. Semi-stable Markov processes. Z. Wahrsch. Verw. Gebiete 22 (1972) 205-225. | MR | Zbl

[24] N. N. Lebedev. Special Functions and Their Applications. Dover, New York, 1972. | MR | Zbl

[25] P. Lévy. Wiener's random functions, and other Laplacian random functions. In Proc. Sec. Berkeley Symp. Math. Statist. Probab., 1950 II. 171-187. California Univ. Press, Berkeley, 1951. | MR | Zbl

[26] K. Maulik and B. Zwart. Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process. Appl. 116 (2006) 156-177. | MR | Zbl

[27] P. A. Meyer. Processus à accroissements indépendants et positifs. Séminaire de probabilités de Strasbourg 3 (1969) 175-189. | EuDML | Numdam | MR | Zbl

[28] G. Mittag-Leffler. Sur la nouvelle function Eα(x). C. R. Math. Acad. Sci. Paris 137 (1903) 554-558. | JFM

[29] P. Patie. Exponential functional of one-sided Lévy processes and self-similar continuous state branching processes with immigration. Bull. Sci. Math. In press, 2008. | MR | Zbl

[30] J. Pitman and M. Yor. Bessel processes and infinitely divisible laws. In Stochastic Integrals (In Proc. Sympos. Univ. Durham, Durham, 1980) 285-370. D. Williams (ed.). Lecture Notes in Math. 851. Springer, Berlin, 1981. | MR | Zbl

[31] V. Rivero. Recurrent extensions of self-similar Markov processes and Cramér's condition. Bernoulli 11 (2005) 471-509. | MR | Zbl

[32] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge, 1999. | MR | Zbl

[33] F. W. Steutel and K. Van Harn. Infinite divisibility of probability distributions on the real line. Monographs and Textbooks in Pure and Applied Mathematics 259. Marcel Dekker Inc., New York, 2004. | MR | Zbl

[34] J. Vuolle-Apiala. Itô excursion theory for self-similar Markov processes. Ann. Probab. 22 (1994) 546-565. | MR | Zbl

[35] S. J. Wolfe. On the unimodality of L functions. Ann. Math. Stat. 42 (1971) 912-918. | MR | Zbl

[36] S. J. Wolfe. On a continuous analogue of the stochastic difference equation Xn=ρXn−1+Bn. Stochastic Process. Appl. 12 (1982) 301-312. | MR | Zbl

[37] M. Yor. Loi de l'indice du lacet brownien et distribution de Hartman-Watson. Z. Wahrsch. Verw. Gebiete 53 (1980) 71-95. | MR | Zbl

[38] M. Yor. Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin, 2001. | MR | Zbl

Cited by Sources: