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In this paper we study Bessel processes in terms of the Kingman convolution method. In particular, we propose a higher dimensional model of the Kingman convolution algebras. We show that every Bessel process started at 0 is induced by a Kingman convolution. Moreover, a new concept of increments of stochastic processes is introduced. It permits to regard Bessel processes as “stationary and independent increments processes”.
Czasopismo
Rocznik
Tom
Strony
119--134
Opis fizyczny
Bibliogr. 19 poz.
Twórcy
autor
- Department of Mathematics, International University, HCMC, Quarter Nr. 6, LinhTrung Ward, Thu Duc Distr., HoChiMinh City, Vietnam
Bibliografia
- [1] N. H. Bingham, Random walks on spheres, Z. Wahrsch. Verw. Gebiete 22 (1973), pp. 169-172.
- [2] J. C. Cox, J. E. Ingersoll Jr. and S. A. Ross, A theory of the term structure of interest rates, Econometrica 53 (2) (1985).
- [3] K. Itô and H. P. Mckean Jr., Diffusion Processes and Their Sample Paths, Springer, Berlin-Heidelberg-New York 1996.
- [4] O. Kalenberg, Random Measures, 3rd edition, Academic Press, New York 1983.
- [5] J. F. C. Kingman, Random walks with spherical symmetry, Acta Math. 109 (1963), pp. 11-53.
- [6] B. M. Levitan, Generalized Translation Operators and some of Their Applications, Israel Program for Scientific Translations, Jerusalem 1962.
- [7] V. T. Nguyen, Generalized independent increments processes, Nagoya Math. J. 133 (1994), pp. 155-175.
- [8] V. T. Nguyen, Generalized translation operators and Markov processes, Demonstratio Math. 34 (2) (2001), pp. 295-304.
- [9] V. T. Nguyen, Double-indexes Bessel diffusions, in: Proceedings of the International Symposium on “Abstract and Applied Analysis”, World Scientific 2004, pp. 563-567.
- [10] V. T. Nguyen, S. Ogawa and M. Yamazato, A convolution approach to multivariate Bessel processes, in: Proceedings of the 6th Ritsumeikan International Symposium on “Stochastic Processes and Applications to Mathematical Finance”, J. Akahori, S. Ogawa and S. Watanabe (Eds.), World Scientific 2006, pp. 233-244.
- [11] B. S. Rajput and J. Rosiński, Spectral representation of infinitely divisible processes, Probab. Theory Related Fields 82 (1989), pp. 451-487.
- [12] D. Revuz and M. Yor, Continuous Martingals and Brownian Motion, Springer, Berlin-Heidelberg 1991.
- [13] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999.
- [14] T. Shiga and S. Watanabe, Bessel diffusions as a one-parameter family of diffusion processes, Z. Warsch. Verw. Gebiete 27 (1973), pp. 34-46.
- [15] K. Urbanik, Generalized convolutions, Studia Math. 23 (1964), pp. 217-245.
- [16] K. Urbanik, Cram´er property of generalized convolutions, Bull. Polish Acad. Sci. Math. 37 (16) (1989), pp. 213-218.
- [17] V. E. Vólkovich, On symmetric stochastic convolutions, J. Theoret. Probab. 5 (3) (1992), pp. 417-430.
- [18] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge 1944.
- [19] M. Yor, Some Aspects of Brownian Motion, Part I: Some Special Functionals, Lecture Notes in Math., ETH Zurich, Birkhäuser Verlag, Basel 1992.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-39bc792d-eecd-45d4-b72a-ed27814d8cfb