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We study continuous additive functionals of zero quadratic variation of strong Markov continuous local martingales by means of stochastic calculus. We show that they admit a representation as a stochastic integral with respect to local time in the sense of Bouleau and Yor.
Czasopismo
Rocznik
Tom
Strony
385--397
Opis fizyczny
Bibliogr. 26 poz.
Twórcy
autor
- Friedrich- Schiller-Universität, Fakultät für Mathematik und Informatik, Institut für Stochastik, 07740 Jena, Germany
Bibliografia
- [1] J. Bertoin, Les processus de Dirichlet et tant qu'espace de Banach, Stochastics 18 (1988), pp. 155-168.
- [2] R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Academic Press, 1968.
- [3] N. Bouleau and M. Yor, Sur la variation quadratique des temps locaux de certaines semimartingales, C. R. Acad. Sci. Paris Sér. I 292 (1981), pp. 491-494.
- [4] C. Dellacherie, Capacites et processus stochastiques, Springer, Berlin-Heidelberg-New York 1972.
- [5] — and P. A. Meyer, Probabilities and Potential B, North-Holland, Amsterdam-New York-Oxford 1982.
- [6] E. B. Dynkin, Markov Processes, Fizmatgiz, Moscow 1963.
- [7] H. J. Engelbert and W. Schmidt, Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations. I, Math. Nachr. 143 (1989), pp. 167-184.
- [8] — Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations. II, ibidem 144 (1989), pp. 241-281.
- [9] — On the representation theorem for additive functionals, in: Ma/Röckner/Yan, Dirichlet Forms and Stochastic Processes, Walter de Gruyter, Berlin-New York 1995.
- [10] H. J. Engelbert and J. Wolf, Dirichlet functions of reflected Brownian motion, Mathematica Bohémica, to appear in 1999.
- [11] — On the structure of continuous strong Markov local Dirichlet processes, to appear in: Proceedings of the 7th Vilnius Conference on Probability and Mathematical Statistics, Vilnius, August 12-18, 1998.
- [12] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin-New York 1994.
- [13] H. P. McKean and H. Tanaka, Additive functionals of the Brownian path, Mem. Coll. Sci. Univ. Kyoto A 33 (1961), pp. 479-506.
- [14] P. A. Meyer, Fonctionnelles multipticatives et additives de Markov, Ann. Inst. Fourier (Grenoble) 12 (1962), pp. 125-230.
- [15] D. Revuz, Mesures associées aux fonctionnelles additives de Markov. I, Trans. Amer. Math. Soc. 148 (1970),. pp. 501-531.
- [16] — and M. Yor, Continuous Martingales and Brownian Motion, 2nd edition, Springer, Berlin-New York 1994.
- [17] F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Related Fields 97 (1993), pp. 403-421.
- [18] — The generalized covariation process and ltô formula, Stochastic Process. Appl. 59 (1995), pp. 81-104.
- [19] — ltô formula for C1- functions of semimartingales, Probab. Theory Related Fields 104 (1996), pp. 27-41.
- [20] M. Sharpe, General Theory of Markov Processes, Academic Press, San Diego 1988.
- [21] H. Tanaka, Note on continuous additive functionals of the 1 -dimensional Brownian path, Z. Wahrscheinlichkeitstheorie 1 (1963), pp. 251-257.
- [22] V. A. Volkonski, Random time change of strong Markov processes, Teor. Veroyatnost. i Primenen. 3 (1958), pp. 332-350.
- [23] — Additive functionals of Markov processes, Trudy Moscow Math. Society 9 (1960), pp. 143-189.
- [24] A. T. Wang, Generalized Itô's formula and additive functionals of Brownian motion, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 41 (1977), pp. 153-159.
- [25] J. Wolf, Transformations of semimartingales and local Dirichlet processes, Stochastics and Stochastics Reports 62 (1997), pp. 65-101.
- [26] — An ltô formula for local Dirichlet processes, ibidem 62 (1997), pp. 103-115.
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Bibliografia
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