Some explicit Krein representations of certain subordinators, including the Gamma process

Publications of the Research Institute for Mathematical Sciences

Volumn 42, Issue 4, 2006, Pages 879-895

  Donati Martin, Catherine ^a   Yor, Marc ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33847003670     PISSN: 00345318     EISSN: 16634926     Source Type: Journal    
DOI: 10.2977/prims/1166642190     Document Type: Article

References (36)

1. Bertoin, J., Applications de la théorie spectrale des cordes vibrantes aux fonctionnelles additives principales d'un mouvement brownien réfléchi, Ann. Inst. H. Poincaré Probab. Statist., 25 (1989), 307-323.
2. _, Subordinators: examples and applications, Lectures on Probability Theory and Statistics, Ecole d'Eté de Probabilités de Saint Flour XXVH-1997, Lecture Notes in Math., 1717, Springer, 1999.
3. Biane, P. and Yor, M., Valeurs principales associées aux temps locaux browniens, Bull. Sci. Math., 111 (1987), 23-101.
4. Donati-Martin, C. and Yor, M., On a Krein representation of the variance-gamma processes in terms of the BES(0,β↓) processes. In preparation (2005).
5. Donati-Martin, C., Roynette, B., Vallois, P. and Yor, M., On constants related to the choice of the local time at 0 and the corresponding Itô measure for Bessel processes with dimension ci = 2(1 -α),0 < α < 1, Prépublication de l'Institut E. Cartan, 2005. To appear in Studia Sci. Math. Hungar.
6. Emery, M. and Yor, M., A parallel between Brownian bridges and gamma bridges, Publ. RIMS, Kyoto Univ., 40 (2004), 669-688.
7. Esscher, F., On the probability function in the collective theory of risk, Scandinavisk Aktuarietidskrift, 15 (1932), 175-195.
8. Ito, K. and McKean, H. P., Diffusion Processes and Their Sample Paths, Springer, Berlin, 1965.
9. Karatzas, I. and Shreve, S. E., Brownian Motion and Stochastic Calculus, Second edition, Springer-Verlag, New York, 1991.
10. Knight, F. B., Characterization of the Lévy measure of inverse local times of gap diffusion, Seminar on Stochastic Processes, 1981, 53-78, Prog. Prob. Statist., 1, Birkhäuser, 1981.
11. Kotani, S. and Watanabe, S., Krein's spectral theory of strings and generalized diffusion processes, Functional Analysis in Markov processes (Katata/Kyoto 1981), 235-259, LNM, 923, Springer, 1982.
12. Küchler, U., On sojourn times, excursions and spectral measures connected with quasidiffusions, J. Math. Kyoto Univ., 26 (1986), 403-421.
13. Küchler, U. and Salminen, P., On spectral measures of strings and excursions of quasi diffusions, Séminaire de Probabilités, XXIII, 490-502, Lecture Notes in Math., 1372, Springer, Berlin, 1989.
14. Lebedev, N. N., Special Functions and Their Applications, Dover Publications, 1972.
15. Madan, D. B., Carr, P. P. and Chang, E. C., The variance gamma process and option pricing, European Finance Review, 2 (1998), 79-105.
16. Madan, D. B. and Milne, F., Option pricing with variance gamma martingale components. Math. Finance, 1 (1991), 39-56.
17. Madan, D. B. and Seneta, E., The variance gamma model for share markets returns, J. Bus., 63 (1990), 511-524.
18. Matsumoto, H. and Yor, M., An analogue of Pitman's 2M -X theorem for exponential Wiener functionals. II. The role of the generalized inverse Gaussian laws, Nagoya Math. J., 162 (2001), 65-86.
19. McKean, H. P., Brownian local times. Adv. Math., 16 (1975), 91-111.
20. Molchanov, S. A. and Ostrovski, E., Symmetric stable processes as traces of degenerate diffusion processes, Theory Probab. Appl., 14 (1969), 128-131.
21. Pitman, J. and Yor, M., Bessel processes and infinitely divisible laws, Stochastic integrals, Lecture Notes in Math., 851, Springer, Berlin, 1981.
22. _, Decomposition at the maximum for excursions and bridges of one-dimensional diffusions, Itô's Stochastic calculus and Probability theory (Ikeda, N., Watanabe, S., Fukushima, M. and Kunita, H., eds.), Springer, 1996.
23. _, The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator, Ann. Probab., 25 (1997), 855-900.
24. _, On the distribution of ranked heights of excursions of a Brownian bridge, Ann. Probab., 29 (2001), 361-384.
25. Rogers, L. C. G and Williams, D., Diffusions, Markov Processes and Martingales, vol 2: Itô calculus, Wiley and Sons, New York, 1987.
26. Shiryaev, A. N., Essentials of Stochastic Finance, World Scientific, Singapore, 1999.
27. Vershik, A. and Yor, M., Multiplicativité du processus gamma et étude asymptotique des lois stables d'indice α, lorsque a tend vers 0, Prépublication 289, Laboratoire de Probabilités, Paris 6, 1995.
28. Tsilevich, N. and Vershik, A., Quasi-invariance of the gamma process and multiplicative properties of the Poisson-Dirichlet measures, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 163-168.
29. Tsilevich, N., Vershik, A. and Yor, M., An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process, J. Funct. Anal., 185 (2001), 274-296.
30. _, The Markov-Krein identity and the quasi-invariance of the gamma process, (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 283 (2001),
31. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 6, 21-36, 258;
32. translation in J. Math. Sci. (N. Y.), 121 (2004), 2303-2310.
33. Watanabe, S., On time inversion of one-dimensional diffusion processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31 (1974/75), 115-124.
34. _, Invariants of one-dimensional diffusion processes and applications. International Conference on Probability Theory and its Applications (Taejon, 1998), J. Korean Math. Soc., 35 (1998), 637-658.
35. _, Bilateral Bessel diffusion processes with drift and time inversion, preprint.
36. Williams, D., Markov properties of Brownian local time, Bull. Amer. Math. Soc., 75 (1969), 1035-1036.