×

zbMATH — the first resource for mathematics

Rough path analysis via fractional calculus. (English) Zbl 1175.60061
In the present paper, the authors analyse the dynamical system \[ dx_t=f(x_t) dy_t,\;x(0)=x_0,\tag{1} \] where \(x\) and \(y\) are Hölder continuous of order \(\beta\in(\tfrac 13,\tfrac 12)\). First, the authors define the integrals of the form \[ \int^b_af(x_t)dy_t \] using fractional calculus and obtain an explicit formula for this integral. Secondly, they prove the existence, uniqueness and continuous dependence of a solution for (1) in the input parameters \(x_0,y\) and \(y\otimes y\). Finally, the authors discuss the applications of these results to stochastic integrals and stochastic differential equations.

MSC:
60L20 Rough paths
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
26A33 Fractional derivatives and integrals
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Laure Coutin and Antoine Lejay, Semi-martingales and rough paths theory, Electron. J. Probab. 10 (2005), no. 23, 761 – 785. · Zbl 1109.60035 · doi:10.1214/EJP.v10-162 · doi.org
[2] Laure Coutin and Zhongmin Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields 122 (2002), no. 1, 108 – 140. · Zbl 1047.60029 · doi:10.1007/s004400100158 · doi.org
[3] 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. · Zbl 1090.60038 · doi:10.1007/978-0-387-29371-4_8 · doi.org
[4] 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). · Zbl 1080.60021 · doi:10.1016/j.anihpb.2004.05.003 · doi.org
[5] M. Gubinelli, Controlling rough paths, J. Funct. Anal. 216 (2004), no. 1, 86 – 140. · Zbl 1058.60037 · doi:10.1016/j.jfa.2004.01.002 · doi.org
[6] 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. · Zbl 1144.34038 · doi:10.1007/978-3-540-70847-6_17 · doi.org
[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. · Zbl 1075.60510 · doi:10.1016/S0304-4149(02)00176-X · doi.org
[8] Antoine Lejay, An introduction to rough paths, Séminaire de Probabilités XXXVII, Lecture Notes in Math., vol. 1832, Springer, Berlin, 2003, pp. 1 – 59. · Zbl 1041.60051 · doi:10.1007/978-3-540-40004-2_1 · doi.org
[9] 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. · Zbl 0835.34004 · doi:10.4310/MRL.1994.v1.n4.a5 · doi.org
[10] Terry J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), no. 2, 215 – 310. · Zbl 0923.34056 · doi:10.4171/RMI/240 · doi.org
[11] 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. · Zbl 1176.60002
[12] Terry Lyons and Zhongmin Qian, Flow equations on spaces of rough paths, J. Funct. Anal. 149 (1997), no. 1, 135 – 159. · Zbl 0890.58090 · doi:10.1006/jfan.1996.3088 · doi.org
[13] Terry Lyons and Zhongmin Qian, System control and rough paths, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2002. Oxford Science Publications. · Zbl 1029.93001
[14] 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). · Zbl 1087.60035 · doi:10.1016/j.anihpb.2005.04.003 · doi.org
[15] David Nualart and Aurel Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55 – 81. · Zbl 1018.60057
[16] 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\(^{\prime}\)skiĭ; Translated from the 1987 Russian original; Revised by the authors.
[17] L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), no. 1, 251 – 282. · Zbl 0016.10404 · doi:10.1007/BF02401743 · doi.org
[18] M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields 111 (1998), no. 3, 333 – 374. · Zbl 0918.60037 · doi:10.1007/s004400050171 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.