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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Using fractional calculus we define integrals of the form , where and are vector-valued Hölder continuous functions of order and is a continuously differentiable function such that is -Hölder continuous for some . Under some further smooth conditions on the integral is a continuous functional of , , and the tensor product with respect to the Hölder norms. We derive some estimates for these integrals and we solve differential equations driven by the function . We discuss some applications to stochastic integrals and stochastic differential equations.

**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 .*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)**

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.