Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes

Patie Pierre

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 3, page 667-684
  • ISSN: 0246-0203

Abstract

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We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any α, x>0, γ≥0 and fa smooth function on ℜ + , 𝐋 ( γ ) f ( x ) = x - α ( σ 2 x 2 f ' ' ( x ) + ( σ γ + b ) x f ' ( x ) + ∫ 0 ∞ f e - r x - f ( x ) e - r γ + x f ' ( x ) r 𝕀 { r ≤ 1 } ν ( d r ) ) , ( 0 . 1 ) where the coefficients b ∈ ℜ ,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. Gebiete22 (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.

How to cite

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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 45.3 (2009): 667-684. <http://eudml.org/doc/78038>.

@article{Pierre2009,
abstract = {We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any α, x&gt;0, γ≥0 and fa smooth function on $\mathfrak \{R\}^\{+\}$, \begin\{eqnarray*\}\mathbf \{L\}^\{(\gamma )\}f(x)=x^\{-\alpha \}\biggl (\frac\{\sigma \}\{2\}x^\{2\}f^\{\prime \prime \}(x)+(\sigma \gamma +b)xf^\{\prime \}(x)\\+\int \_\{0\}^\{\infty \}\bigl (f\bigl (\mathrm \{e\}^\{-r\}x\bigr )-f(x)\bigr )\mathrm \{e\}^\{-r\gamma \}+xf^\{\prime \}(x)r\{\mathbb \{I\}\}\_\{\lbrace r\le 1\rbrace \}\nu (\mathrm \{d\}r)\biggr ),\qquad (0.1)\end\{eqnarray*\} where the coefficients $b\in \mathfrak \{R\}$,σ≥0 and the measure ν, which satisfies the integrability condition ∫0∞(1∧r2)ν(dr)&lt;+∞, 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. Gebiete22 (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.},
author = {Pierre, Patie},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {infinite divisibility; first passage time; self-similar Markov processes; special functions},
language = {eng},
number = {3},
pages = {667-684},
publisher = {Gauthier-Villars},
title = {Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes},
url = {http://eudml.org/doc/78038},
volume = {45},
year = {2009},
}

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
PB - Gauthier-Villars
VL - 45
IS - 3
SP - 667
EP - 684
AB - We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any α, x&gt;0, γ≥0 and fa smooth function on $\mathfrak {R}^{+}$, \begin{eqnarray*}\mathbf {L}^{(\gamma )}f(x)=x^{-\alpha }\biggl (\frac{\sigma }{2}x^{2}f^{\prime \prime }(x)+(\sigma \gamma +b)xf^{\prime }(x)\\+\int _{0}^{\infty }\bigl (f\bigl (\mathrm {e}^{-r}x\bigr )-f(x)\bigr )\mathrm {e}^{-r\gamma }+xf^{\prime }(x)r{\mathbb {I}}_{\lbrace r\le 1\rbrace }\nu (\mathrm {d}r)\biggr ),\qquad (0.1)\end{eqnarray*} where the coefficients $b\in \mathfrak {R}$,σ≥0 and the measure ν, which satisfies the integrability condition ∫0∞(1∧r2)ν(dr)&lt;+∞, 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. Gebiete22 (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.
LA - eng
KW - infinite divisibility; first passage time; self-similar Markov processes; special functions
UR - http://eudml.org/doc/78038
ER -

References

top
  1. [1] R. P. Agarwal. A propos d’une note de M. Pierre Humbert. C. R. Math. Acad. Sci. Paris 236 (1953) 2031–2032. Zbl0051.30801MR55502
  2. [2] V. Bally and L. Stoica. A class of Markov processes which admit a local time. Ann. Probab. 15 (1987) 241–262. Zbl0615.60069MR877600
  3. [3] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996. Zbl0861.60003MR1406564
  4. [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. Zbl1004.60046MR1918243
  5. [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. Zbl1031.60038MR1986381
  6. [6] J. Bertoin and M. Yor. Exponential functionals of Lévy processes. Probab. Surv. 2 (2005) 191–212. Zbl1189.60096MR2178044
  7. [7] Ph. Biane and M. Yor. Variations sur une formule de Paul Lévy. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987) 359–377. Zbl0623.60099MR898500
  8. [8] R. M. Blumenthal. On construction of Markov processes. Z. Wahrsch. Verw. Gebiete 63 (1983) 433–444. Zbl0494.60071MR705615
  9. [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.) Zbl0830.60072MR1787143
  10. [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. Zbl1098.60038MR2243877
  11. [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. Zbl0121.13003MR143257
  12. [12] E. B. Dynkin. Markov Processes I, II. Die Grundlehren der Mathematischen Wissenschaften, Bände 121 122. Academic Press, New York, 1965. Zbl0132.37901
  13. [13] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi. Higher Transcendental Functions 3. McGraw-Hill, New York, 1955. Zbl0064.06302MR66496
  14. [14] W. E. Feller. An Introduction to Probability Theory and Its Applications 2, 2nd edition. Wiley, New York, 1971. Zbl0219.60003
  15. [15] I. I. Gikhman and A. V. Skorokhod. The Theory of Stochastic Processes II. Springer, Berlin, 1975. Zbl0296.00013MR2058260
  16. [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. Zbl0386.34016MR404760
  17. [17] P. Hartman and G. S. Watson. “Normal”“Normal” distribution functions on spheres and the modified Bessel functions. Ann. Probab. 2 (1974) 593–607. Zbl0305.60033MR370687
  18. [18] P. Humbert. Quelques résultats relatifs à la fonction de Mittag-Leffler. C. R. Math. Acad. Sci. Paris 236 (1953) 1467–1468. Zbl0050.10404MR54107
  19. [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. Zbl1059.60052MR1919614
  20. [20] J. Kent. Some probabilistic properties of Bessel functions. Ann. Probab. 6 (1978) 760–770. Zbl0402.60080MR501378
  21. [21] A. A. Kilbas and J. J. Trujillo. Differential equations of fractional orders: Methods, results and problems. Appl. Anal. 78 (2001) 153–192. Zbl1031.34002MR1887959
  22. [22] A. A. Kilbas and M. Saigo. On solution of integral equations of Abel–Volterra type. Differential Integral Equations 8 (1995) 993–1011. Zbl0823.45002MR1325543
  23. [23] J. Lamperti. Semi-stable Markov processes. Z. Wahrsch. Verw. Gebiete 22 (1972) 205–225. Zbl0274.60052MR307358
  24. [24] N. N. Lebedev. Special Functions and Their Applications. Dover, New York, 1972. Zbl0271.33001MR350075
  25. [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. Zbl0044.13802MR44774
  26. [26] K. Maulik and B. Zwart. Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process. Appl. 116 (2006) 156–177. Zbl1090.60046MR2197972
  27. [27] P. A. Meyer. Processus à accroissements indépendants et positifs. Séminaire de probabilités de Strasbourg 3 (1969) 175–189. Zbl0181.44901MR270446
  28. [28] G. Mittag-Leffler. Sur la nouvelle function Eα(x). C. R. Math. Acad. Sci. Paris 137 (1903) 554–558. Zbl34.0435.01JFM34.0435.01
  29. [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. Zbl1171.60009
  30. [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. Zbl0469.60076MR620995
  31. [31] V. Rivero. Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli 11 (2005) 471–509. Zbl1077.60055MR2146891
  32. [32] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge, 1999. Zbl0973.60001MR1739520
  33. [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. Zbl1063.60001MR2011862
  34. [34] J. Vuolle-Apiala. Itô excursion theory for self-similar Markov processes. Ann. Probab. 22 (1994) 546–565. Zbl0810.60067MR1288123
  35. [35] S. J. Wolfe. On the unimodality of L functions. Ann. Math. Stat. 42 (1971) 912–918. Zbl0219.60026MR278357
  36. [36] S. J. Wolfe. On a continuous analogue of the stochastic difference equation Xn=ρXn−1+Bn. Stochastic Process. Appl. 12 (1982) 301–312. Zbl0482.60062MR656279
  37. [37] M. Yor. Loi de l’indice du lacet brownien et distribution de Hartman–Watson. Z. Wahrsch. Verw. Gebiete 53 (1980) 71–95. Zbl0436.60057MR576898
  38. [38] M. Yor. Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin, 2001. Zbl0999.60004MR1854494

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