Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Rough path analysis via fractional calculus


Authors: Yaozhong Hu and David Nualart
Journal: Trans. Amer. Math. Soc. 361 (2009), 2689-2718
MSC (2000): Primary 60H10, 60H05; Secondary 26A42, 26A33, 46E35
DOI: https://doi.org/10.1090/S0002-9947-08-04631-X
Published electronically: November 20, 2008
MathSciNet review: 2471936
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using fractional calculus we define integrals of the form $ \int_{a}^{b}f(x_{t})dy_{t}$, where $ x$ and $ y$ are vector-valued Hölder continuous functions of order $ \beta \in (\frac{1}{3}, \frac{1 }{2})$ and $ f$ is a continuously differentiable function such that $ f^{\prime }$ is $ \lambda $-Hölder continuous for some $ \lambda >\frac{1}{ \beta }-2$. Under some further smooth conditions on $ f$ the integral is a continuous functional of $ x$, $ y$, and the tensor product $ x\otimes y$ with respect to the Hölder norms. We derive some estimates for these integrals and we solve differential equations driven by the function $ y$. We discuss some applications to stochastic integrals and stochastic differential equations.


References [Enhancements On Off] (What's this?)

  • 1. Coutin, L. and Lejay, A. Semi-martingales and rough paths theory. Electron. J. Probab. 10 (2005), 761-785. MR 2164030 (2006i:60042)
  • 2. Coutin, L. and Qian, Z. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002), 108-140. MR 1883719 (2003c:60066)
  • 3. Friz, P. K. Continuity of the Itô-map for Hölder rough paths with applications to the support theorem in Hölder norm. In: Probability and partial differential equations in modern applied mathematics, 117-135, IMA Vol. Math. Appl., 140, Springer, New York, 2005. MR 2202036 (2007f:60070)
  • 4. Friz, P. and Victoir, N. Approximations of the Brownian rough path with applications to stochastic analysis. Ann. Inst. H. Poincaré 41 (2005), 703-724. MR 2144230 (2007e:60018)
  • 5. Gubinelli, M. Controlling rough paths. J. Funct. Anal. 216 (2004), 86-140. MR 2091358 (2005k:60169)
  • 6. Hu, Y. and Nualart, D. Differential equations driven by Hö lder continuous functions of order greater than $ 1/2$. Stochastic analysis and applications, 399-413, Abel symposium 2, Springer, 2007. MR 2397797
  • 7. Ledoux, M., Qian, Z., and Zhang, T. Large deviations and support theorem for diffusion processes via rough paths. Stochastic Process. Appl. 102 (2002), 265-283. MR 1935127 (2003m:60152)
  • 8. Lejay, A. An introduction to rough paths. Lecture Notes in Math. 1832 (2003), 1-59. MR 2053040 (2005e:60120)
  • 9. Lyons, T. J. Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young. Math. Res. Lett. 1 (1994), 451-464. MR 1302388 (96b:60150)
  • 10. Lyons, T. J. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), 215-310. MR 1654527 (2000c:60089)
  • 11. Lyons, T. J., Caruana, M., and Lévy, T. Differential equations driven by rough paths. Lecture Notes in Math. 1908. Springer-Verlag, 2007. MR 2314753
  • 12. Lyons, T. J. and Qian, Z. Flow equations on spaces of rough paths. J. Funct. Anal. 149 (1997), 135-159. MR 1471102 (99b:58241)
  • 13. Lyons, T. and Qian, Z.M. System Control and Rough Paths. Clarendon Press, Oxford, 2002. MR 2036784 (2005f:93001)
  • 14. Millet, A. and Sanz-Solé, M. Large deviations for rough paths of the fractional Brownian motion. Ann. Inst. Henri Poincaré 42 (2006), 245-271. MR 2199801 (2007e:60027)
  • 15. Nualart, D. and Răşcanu, A. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002), 55-81. MR 1893308 (2003f:60105)
  • 16. Samko S. G., Kilbas A. A. and Marichev O. I. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, 1993. MR 1347689 (96d:26012)
  • 17. Young, L. C. An inequality of the Hölder type connected with Stieltjes integration. Acta Math. 67 (1936) 251-282. MR 1555421
  • 18. Zähle, M.Integration with respect to fractal functions and stochastic calculus. I. Prob. Theory Relat. Fields 111 (1998) 333-374. MR 1640795 (99j:60073)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60H10, 60H05, 26A42, 26A33, 46E35

Retrieve articles in all journals with MSC (2000): 60H10, 60H05, 26A42, 26A33, 46E35


Additional Information

Yaozhong Hu
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045-2142
Email: hu@math.ku.edu

David Nualart
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045-2142
Email: nualart@math.ku.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04631-X
Keywords: Rough path, fractional calculus, integral, integration by parts, differential equation, stability, stochastic differential equation, Wong-Zakai approximation, convergence rate.
Received by editor(s): October 2, 2006
Received by editor(s) in revised form: September 6, 2007
Published electronically: November 20, 2008
Additional Notes: The work of the first author was supported in part by the National Science Foundation under Grant No. DMS0204613 and DMS0504783.
The work of the second author was partially supported by the MCyT Grant BFM2000-0598 and the NSF Grant No. DMS-0604207.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society