Tanaka formula for strictly stable processes

• Autorzy: Tsukada Hiroshi

Identyfikatory

DOI

10.19195/0208-4147.39.1.3

• Języki publikacji: EN

Abstrakty

EN

For symmetric Lévy processes, if the local times exist, the Tanaka formula has already been constructed via the techniques in the potential theory by Salminen and Yor (2007). In this paper, we study the Tanaka formula for arbitrary strictly stable processes with index α ϵ (1, 2), including spectrally positive and negative cases in a framework of Itô’s stochastic calculus. Our approach to the existence of local times for such processes is different from that of Bertoin (1996).

Słowa kluczowe

EN

local time; stable process; Itô’s stochastic calculus

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2019
• Tom: Vol. 39, Fasc. 1
• Strony: 39--60
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Tsukada Hiroshi

• Graduate School of Science, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

Bibliografia

• [1] D. Applebaum, Lévy Processes and Stochastic Calculus, second edition, Cambridge Stud. Adv. Math., Vol. 116, Cambridge 2009.
• [2] M. T. Barlow, Necessary and sufficient conditions for the continuity of local time of Lévy processes, Ann. Probab. 16 (4) (1988), pp. 1389-1427.
• [3] S. M. Berman, Local times and sample function properties of stationary Gaussian processes, Trans. Amer. Math. Soc. 137 (1969), pp. 277-299.
• [4] J. Bertoin, Lévy Processes, Cambridge Tracts in Math., Vol. 121, Cambridge 1996.
• [5] E. S. Boylan, Local times for a class of Markov processes, Illinois J. Math. 8 (1964), pp. 19-39.
• [6] K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, second edition, Birkhäuser, Boston 1990.
• [7] H.-J. Engelbert, V. P. Kurenok, and A. Zalinescu, On existence and uniqueness of reflected solutions of stochastic equations driven by symmetric stable processes, in: From Stochastic Calculus to Mathematical Finance, Yu. Kabanov, R. Liptser, and J. Stoyanov (Eds.), Springer, Berlin-Heidelberg 2006, pp. 227-248.
• [8] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, second edition, North-Holland Math. Library, Vol. 24, Amsterdam-Tokyo 1989.
• [9] T. Komatsu, On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type, Proc. Japan Acad. Ser. A Math. Sci. 58 (8) (1982), pp. 353-356.
• [10] P. Salminen and M. Yor, Tanaka formula for symmetric Lévy processes, in: Séminaire de Probabilités XL, Lecture Notes in Math., Vol. 1899, Springer, Berlin 2007, pp. 265-285.
• [11] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Stud. Adv. Math., Vol. 68, Cambridge 1999.
• [12] H. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J. 9 (1978), pp. 163-177.
• [13] H. F. Trotter, A property of Brownian motion paths, Illinois J. Math. 2 (1958), pp. 425-433.
• [14] S. Watanabe, On stable processes with boundary conditions, J. Math. Soc. Japan. 14 (2)(1962), pp. 170-198.
• [15] K. Yamada, Fractional derivatives of local times of α-stable Lévy processes as the limits of occupation time problems, in: Limit Theorems in Probability and Statistics, Vol. II (Balatonlelle, 1999), I. Berkes, E. Csáki, and M. Csörgő (Eds.), János Bolyai Math. Soc., Budapest 2002, pp. 553-573.
• [16] K. Yano, On harmonic function for the killed process upon hitting zero of asymmetric Lévy processes, J. Math-for-Ind. 5A (2013), pp. 17-24.