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

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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.

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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.