Lévy processes with respect to the Whittaker convolution

Sousa, Rúben; Guerra, Manuel; Yakubovich, Semyon

doi:10.1090/tran/8294

Lévy processes with respect to the Whittaker convolution

Authors: Rúben Sousa, Manuel Guerra and Semyon Yakubovich
Journal: Trans. Amer. Math. Soc. 374 (2021), 2383-2419
MSC (2020): Primary 60G51, 60B15, 47D07, 33C15
DOI: https://doi.org/10.1090/tran/8294
Published electronically: January 21, 2021
MathSciNet review: 4223020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

It is natural to ask whether it is possible to construct Lévy-like processes where actions by random elements of a given semigroup play the role of increments. Such semigroups induce a convolution-like algebra structure in the space of finite measures.

In this paper, we show that the Whittaker convolution operator, related with the Shiryaev process, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. We then introduce the class of Lévy processes with respect to the Whittaker convolution and study their basic properties. We obtain a martingale characterization of the Shiryaev process analogous to Lévy’s characterization of Brownian motion.

Our results demonstrate that a nice theory of Lévy processes with respect to generalized convolutions can be developed for differential operators whose associated convolution does not satisfy the usual compactness assumption on the support of the convolution.

References

D. Bakry and N. Huet, The Hypergroup Property and Representation of Markov Kernels, Séminaire de Probabilités XLI, Springer, Berlin, 2008, pp. 297–347, DOI 10.1007/978-3-540-77913-1_15.

C. Barrabès, V. Frolov, and R. Parentani, Metric fluctuation corrections to Hawking radiation, Phys. Rev. D (3) 59 (1999), no. 12, 124010, 14. MR 1693070, DOI https://doi.org/10.1103/PhysRevD.59.124010

Yu. M. Berezansky and A. A. Kalyuzhnyi, Harmonic analysis in hypercomplex systems, Mathematics and its Applications, vol. 434, Kluwer Academic Publishers, Dordrecht, 1998. Translated from the 1992 Russian original by P. V. Malyshev and revised by the authors. MR 1627482

C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Springer, Berlin 1975, DOI 10.1007/978-3-642-66128-0.

W. R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, Walter de Gruyter, Berlin, 1995. DOI 10.1515/9783110877595.

M. Borowiecka-Olszewska, B. H. Jasiulis-Gołdyn, J. K. Misiewicz, and J. Rosiński, Lévy processes and stochastic integrals in the sense of generalized convolutions, Bernoulli 21 (2015), no. 4, 2513–2551. MR 3378476, DOI https://doi.org/10.3150/14-BEJ653

Björn Böttcher, René Schilling, and Jian Wang, Lévy matters. III, Lecture Notes in Mathematics, vol. 2099, Springer, Cham, 2013. Lévy-type processes: construction, approximation and sample path properties; With a short biography of Paul Lévy by Jean Jacod; Lévy Matters. MR 3156646

Eric A. Carlen, Jeffrey S. Geronimo, and Michael Loss, On the Markov sequence problem for Jacobi polynomials, Adv. Math. 226 (2011), no. 4, 3426–3466. MR 2764893, DOI https://doi.org/10.1016/j.aim.2010.10.024

Houcine Chébli, Sur la positivité des opérateurs de “translation généralisée” associés à un opérateur de Sturm-Liouville sur $[0,\infty ]$, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A601–A604 (French). MR 308857

H. Chebli, Opérateus de translation généralisée et semi-groupes de convolution, Théorie du Potentiel et Analyse Harmonique, Springer, Berlin, 1974, pp. 35–59, DOI 10.1007/BFb0060609.

H. Chébli, Sturm-Liouville hypergroups, Applications of hypergroups and related measure algebras (Seattle, WA, 1993) Contemp. Math., vol. 183, Amer. Math. Soc., Providence, RI, 1995, pp. 71–88. MR 1334772, DOI https://doi.org/10.1090/conm/183/02055

William C. Connett and Alan L. Schwartz, Subsets of ${\bf R}$ which support hypergroups with polynomial characters, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 73–84. MR 1379120, DOI https://doi.org/10.1016/0377-0427%2895%2900101-8

Edward Brian Davies, One-parameter semigroups, London Mathematical Society Monographs, vol. 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 591851

J. Delsarte, Sur une extension de la formule de Taylor, J. Math. Pures Appl. 17 (9) (1938) 213–231.

Catherine Donati-Martin, Raouf Ghomrasni, and Marc Yor, On certain Markov processes attached to exponential functionals of Brownian motion; application to Asian options, Rev. Mat. Iberoamericana 17 (2001), no. 1, 179–193. MR 1846094, DOI https://doi.org/10.4171/RMI/292

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions – Vol. I, McGraw–Hill, New York, 1953.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions – Vol. II, McGraw–Hill, New York, 1953.

Achim Klenke, Probability theory, Translation from the German edition, Universitext, Springer, London, 2014. A comprehensive course. MR 3112259

Robert I. Jewett, Spaces with an abstract convolution of measures, Advances in Math. 18 (1975), no. 1, 1–101. MR 394034, DOI https://doi.org/10.1016/0001-8708%2875%2990002-X

B. Lewitan, Normed rings generated by the generalized operation of translation, C. R. (Doklady) Acad. Sci. URSS (N. S.) 47 (1945), 3–6. MR 0013236

B. M. Levitan, Generalized translation operators and some of their applications, Israel Program for Scientific Translations, Jerusalem; Daniel Davey & Co., Inc., 1964. Translated by Z. Lerman; edited by Don Goelman. MR 0172118

Vadim Linetsky, Spectral expansions for Asian (average price) options, Oper. Res. 52 (2004), no. 6, 856–867. MR 2104142, DOI https://doi.org/10.1287/opre.1040.0113

Ju. V. Linnik and Ĭ. V. Ostrovs′kiĭ, Decomposition of random variables and vectors, American Mathematical Society, Providence, R. I., 1977. Translated from the Russian; Translations of Mathematical Monographs, Vol. 48. MR 0428382

Y. L. Luke, The special functions and their approximations – Vol. I, Academic Press, New York and London, 1969, DOI 10.1016/S0076-5392(08)62632-6.

Lutz Mattner, Complex differentiation under the integral, Nieuw Arch. Wiskd. (5) 2 (2001), no. 1, 32–35. MR 1823156

Henry P. McKean Jr., Elementary solutions for certain parabolic partial differential equations, Trans. Amer. Math. Soc. 82 (1956), 519–548. MR 87012, DOI https://doi.org/10.1090/S0002-9947-1956-0087012-3

Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248

K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR 0226684

Goran Peskir, On the fundamental solution of the Kolmogorov-Shiryaev equation, From stochastic calculus to mathematical finance, Springer, Berlin, 2006, pp. 535–546. MR 2234289, DOI https://doi.org/10.1007/978-3-540-30788-4_26

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 1, Gordon & Breach Science Publishers, New York, 1986. Elementary functions; Translated from the Russian and with a preface by N. M. Queen. MR 874986

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 2, Gordon & Breach Science Publishers, New York, 1986. Special functions; Translated from the Russian by N. M. Queen. MR 874987

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 3, Gordon and Breach Science Publishers, New York, 1990. More special functions; Translated from the Russian by G. G. Gould. MR 1054647

Christian Rentzsch and Michael Voit, Lévy processes on commutative hypergroups, Probability on algebraic structures (Gainesville, FL, 1999) Contemp. Math., vol. 261, Amer. Math. Soc., Providence, RI, 2000, pp. 83–105. MR 1788113, DOI https://doi.org/10.1090/conm/261/04135

Ken-iti Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original; Revised by the author. MR 1739520

René L. Schilling, When does a càdlàg process have continuous sample paths?, Exposition. Math. 12 (1994), no. 3, 255–261. MR 1295708

A. N. Shiryaev, The problem of the most rapid detection of a disturbance in a stationary process, Soviet Math. Dokl. 2 (1961) 795–799.

Rúben Sousa, Manuel Guerra, and Semyon Yakubovich, On the product formula and convolution associated with the index Whittaker transform, J. Math. Anal. Appl. 475 (2019), no. 1, 939–965. MR 3944355, DOI https://doi.org/10.1016/j.jmaa.2019.03.009

Rúben Sousa and Semyon Yakubovich, The spectral expansion approach to index transforms and connections with the theory of diffusion processes, Commun. Pure Appl. Anal. 17 (2018), no. 6, 2351–2378. MR 3814379, DOI https://doi.org/10.3934/cpaa.2018112

H. M. Srivastava, Yu. V. Vasil′ev, and S. B. Yakubovich, A class of index transforms with Whittaker’s function as the kernel, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 195, 375–394. MR 1645572, DOI https://doi.org/10.1093/qjmath/49.195.375

L. Székelyhidi, Functional Equations on Hypergroups, World Scientific, Singapore, 2013, DOI 10.1142/9789814407014_fmatter.

K. Urbanik, Generalized convolutions, Studia Math. 23 (1963/64), 217–245. MR 160267, DOI https://doi.org/10.4064/sm-23-3-217-245

K. Urbanik, Generalized convolutions. II, Studia Math. 45 (1973), 57–70. MR 328991, DOI https://doi.org/10.4064/sm-45-1-57-70

V. È. Vol′kovich, Infinitely divisible distributions in algebras with stochastic convolution, J. Soviet Math. 40 (1988), no. 4, 459–467. Stability problems of stochastic models. MR 957099, DOI https://doi.org/10.1007/BF01083639

V. Vol′kovich, On symmetric stochastic convolutions, J. Theoret. Probab. 5 (1992), no. 3, 417–430. MR 1176429, DOI https://doi.org/10.1007/BF01060427

Jet Wimp, A class of integral transforms, Proc. Edinburgh Math. Soc. (2) 14 (1964/65), 33–40. MR 164204, DOI https://doi.org/10.1017/S0013091500011202

Semen B. Yakubovich and Yurii F. Luchko, The hypergeometric approach to integral transforms and convolutions, Mathematics and its Applications, vol. 287, Kluwer Academic Publishers Group, Dordrecht, 1994. MR 1304259

S. Yakubovich, Index Transforms, World Scientific, Singapore, 1996, DOI 10.1142/9789812831064_fmatter.

H. Zeuner, Laws of Large Numbers of Hypergroups on $\mathbb {R}^+$, Math. Ann. 283 (1989) 657–678, DOI 10.1007/BF01442859.

Hansmartin Zeuner, Moment functions and laws of large numbers on hypergroups, Math. Z. 211 (1992), no. 3, 369–407. MR 1190219, DOI https://doi.org/10.1007/BF02571436

Hm. Zeuner, Domains of attraction with inner norming on Sturm-Liouville hypergroups, J. Appl. Anal. 1 (1995), no. 2, 213–221. MR 1395275, DOI https://doi.org/10.1515/JAA.1995.213