Absolute continuity and singularity of probability measures induced by a purely discontinuous Girsanov transform of a stable process
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- by René L. Schilling and Zoran Vondraček PDF
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Abstract:
In this paper we study mutual absolute continuity and singularity of probability measures on the path space which are induced by an isotropic stable Lévy process and the purely discontinuous Girsanov transform of this process. We also look at the problem of finiteness of the relative entropy of these measures. An important tool in the paper is the question under which circumstances the a.s. finiteness of an additive functional at infinity implies the finiteness of its expected value.References
- Iddo Ben-Ari and Ross G. Pinsky, Absolute continuity/singularity and relative entropy properties for probability measures induced by diffusions on infinite time intervals, Stochastic Process. Appl. 115 (2005), no. 2, 179–206. MR 2111192, DOI 10.1016/j.spa.2004.08.005
- R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757
- Krzysztof Bogdan and Tomasz Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian, Probab. Math. Statist. 20 (2000), no. 2, Acta Univ. Wratislav. No. 2256, 293–335. MR 1825645
- Zhen-Qing Chen and Renming Song, Conditional gauge theorem for non-local Feynman-Kac transforms, Probab. Theory Related Fields 125 (2003), no. 1, 45–72. MR 1952456, DOI 10.1007/s004400200219
- Zhen-Qing Chen and Renming Song, Drift transforms and Green function estimates for discontinuous processes, J. Funct. Anal. 201 (2003), no. 1, 262–281. MR 1986161, DOI 10.1016/S0022-1236(03)00087-9
- Kai Lai Chung and Zhong Xin Zhao, From Brownian motion to Schrödinger’s equation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 312, Springer-Verlag, Berlin, 1995. MR 1329992, DOI 10.1007/978-3-642-57856-4
- H. J. Engelbert and T. Senf, On functionals of a Wiener process with drift and exponential local martingales, Stochastic processes and related topics (Georgenthal, 1990) Math. Res., vol. 61, Akademie-Verlag, Berlin, 1991, pp. 45–58. MR 1127879
- Wolfhard Hansen and Ivan Netuka, Unavoidable sets and harmonic measures living on small sets, Proc. Lond. Math. Soc. (3) 109 (2014), no. 6, 1601–1629. MR 3293159, DOI 10.1112/plms/pdu048
- Sheng Wu He, Jia Gang Wang, and Jia An Yan, Semimartingale theory and stochastic calculus, Kexue Chubanshe (Science Press), Beijing; CRC Press, Boca Raton, FL, 1992. MR 1219534
- Jean Jacod, Calcul stochastique et problèmes de martingales, Lecture Notes in Mathematics, vol. 714, Springer, Berlin, 1979 (French). MR 542115, DOI 10.1007/BFb0064907
- Davar Khoshnevisan, Paavo Salminen, and Marc Yor, A note on a.s. finiteness of perpetual integral functionals of diffusions, Electron. Comm. Probab. 11 (2006), 108–117. MR 2231738, DOI 10.1214/ECP.v11-1203
- Hiroshi Kunita and Takesi Watanabe, Markov processes and Martin boundaries. I, Illinois J. Math. 9 (1965), 485–526. MR 181010
- Ante Mimica and Zoran Vondraček, Unavoidable collections of balls for isotropic Lévy processes, Stochastic Process. Appl. 124 (2014), no. 3, 1303–1334. MR 3148015, DOI 10.1016/j.spa.2013.11.003
- Philip E. Protter, Stochastic integration and differential equations, 2nd ed., Applications of Mathematics (New York), vol. 21, Springer-Verlag, Berlin, 2004. Stochastic Modelling and Applied Probability. MR 2020294
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
- Michael Sharpe, General theory of Markov processes, Pure and Applied Mathematics, vol. 133, Academic Press, Inc., Boston, MA, 1988. MR 958914
- Renming Song, Estimates on the transition densities of Girsanov transforms of symmetric stable processes, J. Theoret. Probab. 19 (2006), no. 2, 487–507. MR 2283387, DOI 10.1007/s10959-006-0023-4
Additional Information
- René L. Schilling
- Affiliation: Institut für Mathematische Stochastik, Fachrichtung Mathematik, Technische Universität Dresden, 01062 Dresden, Germany
- Email: rene.schilling@tu-dresden.de
- Zoran Vondraček
- Affiliation: Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička c. 30, 10000 Zagreb, Croatia
- MR Author ID: 293132
- Email: vondra@math.hr
- Received by editor(s): January 31, 2015
- Received by editor(s) in revised form: February 11, 2015
- Published electronically: May 2, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 1547-1577
- MSC (2010): Primary 60J55; Secondary 60G52, 60J45, 60H10
- DOI: https://doi.org/10.1090/tran/6757
- MathSciNet review: 3581212