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Transactions of the American Mathematical Society
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
Published electronically: November 20, 2008
MathSciNet review: 2471936
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-08-04631-X
PII: S 0002-9947(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.