Shift Harnack inequality and integration by parts formula for functional SDEs driven by fractional Brownian motion
HTML articles powered by AMS MathViewer
- by Zhi Li
- Proc. Amer. Math. Soc. 144 (2016), 2651-2659
- DOI: https://doi.org/10.1090/proc/12915
- Published electronically: December 22, 2015
- PDF | Request permission
Abstract:
The shift Harnack inequality and the integration by parts formula for functional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $\frac 12 <H<1$ are established by using a transformation formula for fractional Brownian motion and a new coupling argument.References
- Elisa Alòs, Olivier Mazet, and David Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab. 29 (2001), no. 2, 766–801. MR 1849177, DOI 10.1214/aop/1008956692
- Jean-Michel Bismut, Large deviations and the Malliavin calculus, Progress in Mathematics, vol. 45, Birkhäuser Boston, Inc., Boston, MA, 1984. MR 755001
- B. Boufoussi and S. Hajji, Functional differential equations driven by a fractional Brownian motion, Comput. Math. Appl. 62 (2011), no. 2, 746–754. MR 2817911, DOI 10.1016/j.camwa.2011.05.055
- Bruce K. Driver, Integration by parts for heat kernel measures revisited, J. Math. Pures Appl. (9) 76 (1997), no. 8, 703–737. MR 1470881, DOI 10.1016/S0021-7824(97)89966-7
- L. Decreusefond and A. S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1999), no. 2, 177–214. MR 1677455, DOI 10.1023/A:1008634027843
- K. D. Elworthy and X.-M. Li, Formulae for the derivatives of heat semigroups, J. Funct. Anal. 125 (1994), no. 1, 252–286. MR 1297021, DOI 10.1006/jfan.1994.1124
- XiLiang Fan, Harnack inequality and derivative formula for SDE driven by fractional Brownian motion, Sci. China Math. 56 (2013), no. 3, 515–524. MR 3017935, DOI 10.1007/s11425-013-4569-1
- Xiliang Fan, Integration by parts formula and applications for SDEs driven by fractional Brownian motions, Stoch. Anal. Appl. 33 (2015), no. 2, 199–212. MR 3305466, DOI 10.1080/07362994.2014.975819
- Céline Jost, Transformation formulas for fractional Brownian motion, Stochastic Process. Appl. 116 (2006), no. 10, 1341–1357. MR 2260738, DOI 10.1016/j.spa.2006.02.006
- David Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
- David Nualart and Youssef Ouknine, Regularization of differential equations by fractional noise, Stochastic Process. Appl. 102 (2002), no. 1, 103–116. MR 1934157, DOI 10.1016/S0304-4149(02)00155-2
- Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
- Feng-Yu Wang, Harnack inequalities on manifolds with boundary and applications, J. Math. Pures Appl. (9) 94 (2010), no. 3, 304–321 (English, with English and French summaries). MR 2679029, DOI 10.1016/j.matpur.2010.03.001
- Feng-Yu Wang, Integration by parts formula and shift Harnack inequality for stochastic equations, Ann. Probab. 42 (2014), no. 3, 994–1019. MR 3189064, DOI 10.1214/13-AOP875
- Feng-Yu Wang and Xi-Cheng Zhang, Derivative formula and applications for degenerate diffusion semigroups, J. Math. Pures Appl. (9) 99 (2013), no. 6, 726–740 (English, with English and French summaries). MR 3055216, DOI 10.1016/j.matpur.2012.10.007
- Xicheng Zhang, Stochastic flows and Bismut formulas for stochastic Hamiltonian systems, Stochastic Process. Appl. 120 (2010), no. 10, 1929–1949. MR 2673982, DOI 10.1016/j.spa.2010.05.015
- S. Q. Zhang, Shift Harnack inequality and integration by part formula for semilinear SPDE, arXiv:1208.2425v3, 2012.
Bibliographic Information
- Zhi Li
- Affiliation: School of Information and Mathematics, Yangtze University, Jingzhou 434023, People’s Republic of China
- Received by editor(s): May 4, 2015
- Received by editor(s) in revised form: July 13, 2015
- Published electronically: December 22, 2015
- Communicated by: David Levin
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2651-2659
- MSC (2010): Primary 60H15; Secondary 60G15, 60H05
- DOI: https://doi.org/10.1090/proc/12915
- MathSciNet review: 3477083