Rough path analysis via fractional calculus

Hu, Yaozhong; Nualart, David

doi:10.1090/S0002-9947-08-04631-X

Rough path analysis via fractional calculus
HTML articles powered by AMS MathViewer

by Yaozhong Hu and David Nualart PDF

Trans. Amer. Math. Soc. 361 (2009), 2689-2718 Request permission

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

Laure Coutin and Antoine Lejay, Semi-martingales and rough paths theory, Electron. J. Probab. 10 (2005), no. 23, 761–785. MR 2164030, DOI 10.1214/EJP.v10-162
Laure Coutin and Zhongmin Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields 122 (2002), no. 1, 108–140. MR 1883719, DOI 10.1007/s004400100158
Peter K. Friz, Continuity of the Itô-map for Hölder rough paths with applications to the support theorem in Hölder norm, Probability and partial differential equations in modern applied mathematics, IMA Vol. Math. Appl., vol. 140, Springer, New York, 2005, pp. 117–135. MR 2202036, DOI 10.1007/978-0-387-29371-4_{8}
Peter Friz and Nicolas Victoir, Approximations of the Brownian rough path with applications to stochastic analysis, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 4, 703–724 (English, with English and French summaries). MR 2144230, DOI 10.1016/j.anihpb.2004.05.003
M. Gubinelli, Controlling rough paths, J. Funct. Anal. 216 (2004), no. 1, 86–140. MR 2091358, DOI 10.1016/j.jfa.2004.01.002
Yaozhong Hu and David Nualart, Differential equations driven by Hölder continuous functions of order greater than 1/2, Stochastic analysis and applications, Abel Symp., vol. 2, Springer, Berlin, 2007, pp. 399–413. MR 2397797, DOI 10.1007/978-3-540-70847-6_{1}7
M. Ledoux, Z. Qian, and T. Zhang, Large deviations and support theorem for diffusion processes via rough paths, Stochastic Process. Appl. 102 (2002), no. 2, 265–283. MR 1935127, DOI 10.1016/S0304-4149(02)00176-X
Antoine Lejay, An introduction to rough paths, Séminaire de Probabilités XXXVII, Lecture Notes in Math., vol. 1832, Springer, Berlin, 2003, pp. 1–59. MR 2053040, DOI 10.1007/978-3-540-40004-2_{1}
Terry Lyons, Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young, Math. Res. Lett. 1 (1994), no. 4, 451–464. MR 1302388, DOI 10.4310/MRL.1994.v1.n4.a5
Terry J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), no. 2, 215–310. MR 1654527, DOI 10.4171/RMI/240
Terry J. Lyons, Michael Caruana, and Thierry Lévy, Differential equations driven by rough paths, Lecture Notes in Mathematics, vol. 1908, Springer, Berlin, 2007. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004; With an introduction concerning the Summer School by Jean Picard. MR 2314753, DOI 10.1007/978-3-540-71285-5
Terry Lyons and Zhongmin Qian, Flow equations on spaces of rough paths, J. Funct. Anal. 149 (1997), no. 1, 135–159. MR 1471102, DOI 10.1006/jfan.1996.3088
Terry Lyons and Zhongmin Qian, System control and rough paths, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2002. Oxford Science Publications. MR 2036784, DOI 10.1093/acprof:oso/9780198506485.001.0001
Annie Millet and Marta Sanz-Solé, Large deviations for rough paths of the fractional Brownian motion, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 2, 245–271 (English, with English and French summaries). MR 2199801, DOI 10.1016/j.anihpb.2005.04.003
David Nualart and Aurel Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308
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
L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), no. 1, 251–282. MR 1555421, DOI 10.1007/BF02401743
M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields 111 (1998), no. 3, 333–374. MR 1640795, DOI 10.1007/s004400050171