Yet another introduction to rough paths. (English) Zbl 1198.60002

Lejay, Antoine

Donati-Martin, Catherine (ed.) et al., Séminaire de probabilités XLII. Berlin: Springer (ISBN 978-3-642-01762-9/pbk; 978-3-642-01763-6/ebook). Lecture Notes in Mathematics 1979, 1-101 (2009).

This is an expository article on the elementary theory of rough paths, where a special emphasis is put on proper definitions of rough paths from an algebraic point of view, basically in line with [A. Lejay, 37th seminar on probability. Berlin: Springer. Lect.Notes Math.1832, 1–59 (2003; Zbl 1041.60051)], or see e.g. [T. Lyons and Z. Qian, System control and rough paths. Oxford Mathematical Monographs. Oxford: Clarendon Press (2002; Zbl 1029.93001)]. Starting with simple considerations on ordinary integrals and stressing the importance of the Green-Riemann formula, the author is aiming at an introduction to another point of view on the theory of rough paths, which is suggested by the work of D. Feyel and A. de La Pradelle [Electron.J.Probab. 11, Paper No. 35, 860–892, electronic only (2006; Zbl 1110.60031)], where they use a point of view from differential geometry and emphasize the importance of the Gauss/Green-Riemann/Stokes formula to understand the need to enhance the path with more information to get a rigorous definition of rough paths. Based upon this point of view, the construction of the algebraic structures such as tensor space and Lie groups, etc. is naturally justified, which is needed in the theory of rough paths from basic considerations on integrals of differential forms. The author tries to introduce, as easily and clearly as possible, alternative ways to understand why the construction of [T. Lyons et al., Lecture Notes Math.1908. Berlin: Springer (2007; Zbl 1176.60003)] is a natural generalization of the notion of integral of differential forms, in the sense that it shares the same smooth paths when one uses the right notion of a path.
More precisely, Section 2 is an introduction to basic notation and some elementary facts about integrals of differential forms along smooth paths as well as about Hölder continuous paths. Actually it includes the class of \(\gamma\)-Lipschitz differential forms \(\text{Lip}(\gamma; {\mathbb R}^d \to {\mathbb R}^m)\), the integral of a continuous differential form \(f\) along a path \(x \in\) \(C^1( [0,T]; {\mathbb R}^d)\) \[ \int_x f = \sum_{i=1}^d \int_0^T f_i(x_s) \left. \frac{ d x^i}{dt} \right|_{t=s} ds, \] the Green-Riemann/Stokes/Gauss formula, and the notion of finite \(p\)-variation. In Section 3, following a quick review on Young integrals \({\mathcal I}(x; s,t)\) \(=\) \(\int_{ x | [s,t]} f\), properties of integrals along \(\alpha\)-Hölder continuous paths with \(\alpha > 1/2\) are shown. The section ends with a practical counter-example in the stochastic setting arising from homogenization theory, which provides the cornerstone to understand how \({\mathcal I}\) will be defined so as to deal with irregular paths. In Section 4, under the assumption that one can integrate differential forms along \(\alpha\)-Hölder continuous path with \(\alpha \in\) \(( 1/3, 1/2 ]\), it is discussed in detail how to transform this integral \({\mathcal I}\) into a continuous one with respect to the path. In Section 5, the space \(A({\mathbb R}^2)\) is constructed, which is nothing but the space \({\mathbb R}^3\) equipped with operation \(\boxplus\) as a non-commutative group. Then, considering paths taking values in \(A({\mathbb R}^2)\), the author shows how to define the integral \(\int_x f\) as a limit of ordinary integrals. Analysis of the space \(A({\mathbb R}^2)\) is continued in Section 6, and the author introduces another formulation for the integral \({\mathcal I}\) for a path \(x \in\) \(C^{\alpha}([0,T]\); \(A({\mathbb R}^2))\). The tensor space \(T({\mathbb R}^2)\) \(=\) \({\mathbb R} \oplus {\mathbb R}^2\) \(\oplus\) \(( {\mathbb R}^2 \otimes {\mathbb R}^2)\) and its subspaces \[ T_{\xi} ({\mathbb R}^2 ) = \{ ( \xi, x^1, x^2) | \, x^1 \in {\mathbb R}^2, x^2 \in {\mathbb R}^2 \otimes {\mathbb R}^2 \} \quad \text{with} \quad \xi \in \{ 0,1\} \] are introduced, and the Riemannian structure on \(T_1( {\mathbb R}^2)\) is also investigated. Section 7 is devoted to rough paths and their integrals. As a matter of fact, a rough path \(x \in\) \(C^{\alpha}([0,T]\); \(T_1({\mathbb R}^2))\) and a geometric rough path are defined. Another integral \({\mathcal I}(x)\) of \(f\) along \(x \in\) \(V^{\alpha}([0,T]\); \(T_1({\mathbb R}^2))\) is constructed, using an expression of Riemann sum type. This construction just corresponds to the original one of T. Lyons [Rev.Mat.Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)]. Some related results are given in Section 8. For example, the case of a path living in a \(d\)-dimensional space, Chen series using iterated integrals, other constructions for paths with quadratic variation, and some link with stochastic integrals are discussed. Section 9 treats the differential equation \[ y_t = y_0 + \int_0^t g(y_s) d x_s \] with an irregular path \(x\), where the theory of rough paths are applied to solve the differential equation. This article ends with an appendix which includes

A.

Carnot groups and homogeneous gauges and norms,

B.

Brownian motion on the Heisenberg group, and

C.

From almost rough paths to rough paths.

This article is well written almost in self-contained manner; it succeeds in making the rough path theory (a really touchy issue) easily accessible even to non-expert readers.
For the entire collection see [Zbl 1166.60002].

Reviewer: Isamu Dôku (Saitama)

MSC:

60-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60Hxx	Stochastic analysis
60H05	Stochastic integrals
58A10	Differential forms in global analysis

Keywords:

rough path theory; integral of differential form along a irregular path; the Green-Riemann formula; controlled differential equation; Lie algebra; Lie group; Chen series

Citations:

Zbl 1041.60051; Zbl 1029.93001; Zbl 1110.60031; Zbl 1176.60003; Zbl 0923.34056

References:

[1]	Azencott, R., Géodésiques et diffusions en temps petit (1981), Paris: Société Mathématique de France, Paris · Zbl 0507.60075
[2]	Ben Arous, G., Flots et séries de Taylor stochastiques, Probab. Theory Related Fields, 81, 1, 29-77 (1989) · Zbl 0639.60062
[3]	Baker, A., Matrix groups: An introduction to Lie group theory (2002), London: Springer-Verlag London Ltd., London · Zbl 1009.22001
[4]	Baudoin, F., An introduction to the geometry of stochastic flows (2004), London: Imperial College Press, London · Zbl 1085.60002
[5]	Burago, D.; Burago, Y.; Ivanov, S., A course in metric geometry (2001), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0981.51016
[6]	Bismut, J.-M., Large deviations and the Malliavin calculus (1984), Boston, MA: Birkhäuser Boston Inc., Boston, MA · Zbl 0537.35003
[7]	A. Bensoussan, J.-L. Lions and G. Papanicolaou. Asymptotic Analysis for Periodic Structures. North-Holland, 1978. · Zbl 0404.35001
[8]	Castell, F., Asymptotic expansion of stochastic flows, Probab. Theory Related Fields, 96, 2, 225-239 (1993) · Zbl 0794.60054
[9]	Chistyakov, VV; Galkin, OE, On maps of bounded p-variations with p > 1, Positivity, 2, 1, 19-45 (1998) · Zbl 0904.26005
[10]	Chen, K.-T., Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. of Math. (2), 65, 163-178 (1957) · Zbl 0077.25301
[11]	Chen, K.-T., Integration of Paths-A Faithful Representation, of Paths by Non-commutative Formal Power Series, Trans. Amer. Math. Soc., 89, 2, 395-407 (1958) · Zbl 0097.25803
[12]	Ciesielski, Z., On the isomorphisms of the spaces H_α and m, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 8, 217-222 (1960) · Zbl 0093.12301
[13]	Coutin, L.; Lejay, A., Semi-martingales and rough paths theory, Electron. J. Probab., 10, 23, 761-785 (2005) · Zbl 1109.60035
[14]	L. Coutin. An introduction to (stochastic) calculus with respect to fractional Brownian motion. In Séminaire de Probabilités XL, vol. 1899 of Lecture Notes in Math., pp. 3-65. Springer, 2007. · Zbl 1126.60042
[15]	Coutin, L.; Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields, 122, 1, 108-140 (2002) · Zbl 1047.60029
[16]	Davie, AM, Differential Equations Driven by Rough Signals: an Approach via Discrete Approximation, Appl. Math. Res. Express. AMRX, 2, 40 (2007)
[17]	Duistermaat, J. J.; Kolk, J. A. C., Lie groups (2000), Berlin: Universitext. Springer-Verlag, Berlin · Zbl 0955.22001
[18]	Dudley, RM; Norvaiša, R., An introduction to p-variation and Young integrals—with emphasis on sample functions of stochastic processes (1998), University of Aarhus: Lecture given at the Centre for Mathematical Physics and Stochastics, Department of Mathematical Sciences, University of Aarhus · Zbl 0937.28001
[19]	Doss, H., Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré Sect. B (N.S.), 13, 2, 99-125 (1977) · Zbl 0359.60087
[20]	Dodson, C. T. J.; Poston, T., Tensor geometry: The geometric viewpoint and its uses (1991), Berlin: Springer-Verlag, Berlin · Zbl 0732.53002
[21]	Feyel, D.; de La Pradelle, A., Curvilinear integrals along enriched paths, Electron. J. Probab., 11, 35, 860-892 (2006) · Zbl 1110.60031
[22]	Feyel, D.; de La Pradelle, A.; Mokobodzki, G., A non-commutative sewing lemma, Electron. Commun. Probab., 13, 24-34 (2008) · Zbl 1186.26009
[23]	Fliess, M., Fonctionnelles causales non linéaires et indéterminées non communtatives, Bull. Soc. Math. France, 109, 1, 3-40 (1981) · Zbl 0476.93021
[24]	Fliess, M.; Normand-Cyrot, D., Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K. T. Chen, Séminaire de Probabilités, XVI, 257-267 (1982), Berlin: Springer, Berlin · Zbl 0495.60064
[25]	Foll, G. B., Harmonic analysis in phase space (1989), Princeton, NJ: Princeton University Press, Princeton, NJ
[26]	P. Friz. Continuity of the Itâ-Map for Hölder rough paths with applications to the support theorem in Hölder norm. In Probability and Partial Differential Equations in Modern Applied Mathematics, vol. 140 of IMA Volumes in Mathematics and its Applications, pp. 117-135. Springer, 2005. · Zbl 1090.60038
[27]	G. B. Folland and E. M. Stein. Hardy spaces on homogeneous groups, vol. 28 of Mathematical Notes. Princeton University Press, 1982. · Zbl 0508.42025
[28]	P. Friz and N. Victoir. Multidimensional Stochastic Processes as Rough Paths. Theory and Applications. Cambridge University Press, 2009. · Zbl 1193.60053
[29]	Friz, P.; Victoir, N., A Note on the Notion of Geometric Rough Paths, Probab. Theory Related Fields, 136, 3, 395-416 (2006) · Zbl 1108.34052
[30]	P. Friz and N. Victoir. Differential Equations Driven by Gaussian Signals I. <ArXiv: 0707.0313>, Cambridge University (preprint), 2007. · Zbl 1202.60058
[31]	P. Friz and N. Victoir. Differential Equations Driven by Gaussian Signals II. <ArXiv: 0711.0668>, Cambridge University (preprint), 2007. · Zbl 1202.60058
[32]	Friz, P.; Victoir, N., Euler Estimates of Rough Differential Equations, J. Differential Equations, 244, 2, 388-412 (2008) · Zbl 1140.60037
[33]	Gaveau, B., Principle de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math., 139, 1-2, 95-153 (1977) · Zbl 0366.22010
[34]	Goodman, R., Filtrations and asymptotic automorphisms on nilpotent Lie groups, J. Differential Geometry, 12, 2, 183-196 (1977) · Zbl 0389.22015
[35]	M. Gromov. Carnot-Carathéodory spaces seen from within. In Sub-Riemannian geometry, vol. 144 of Progr. Math., pp. 79-323. Birkhäuser, 1996. · Zbl 0864.53025
[36]	Gubinelli, M., Controlling rough paths, J. Funct. Anal., 216, 1, 86-140 (2004) · Zbl 1058.60037
[37]	Hall, B. C., Lie groups, Lie algebras, and representations (2003), New York: Springer-Verlag, New York · Zbl 1026.22001
[38]	B.M. Hambly and T.J. Lyons. Uniqueness for the signature of a path of bounded variation and continuous analogues for the free group. Oxford University (preprint), 2006. · Zbl 1276.58012
[39]	Isidori, A., Nonlinear control systems (1995), Berlin: Springer-Verlag, Berlin · Zbl 0878.93001
[40]	N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes, vol. 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Second edition, 1989. · Zbl 0684.60040
[41]	Kawski, M., Nonlinear control and combinatorics of words, Geometry of feedback and optimal control, 305-346 (1998), New York: Dekker, New York · Zbl 0925.93368
[42]	Lyons, T.; Caruana, M.; Lévy, T.; Picard, J., Differential Equations Driven by Rough Paths, École d’été de probabilités de Saint-Flour XXXIV—2004 (2007), Berlin: Springer, Berlin
[43]	Lejay, A., On the convergence of stochastic integrals driven by processes converging on account of a homogenization property, Electron. J. Probab., 7, 18, 1-18 (2002) · Zbl 1007.60018
[44]	A. Lejay. An introduction to rough paths. In Séminaire de probabilités, XXXVII, vol. 1832 of Lecture Notes in Mathematics, pp. 1-59. Springer-Verlag, 2003. · Zbl 1041.60051
[45]	Lejay, A., Stochastic Differential Equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem, ESAIM Probab. Stat., 10, 356-379 (2006) · Zbl 1181.60085
[46]	Lejay, A., Stochastic Differential Equations driven by processes generated by divergence form operators II: convergence results, ESAIM Probab. Stat., 12, 387-411 (2008) · Zbl 1185.60061
[47]	Lévy, P., Processus stochastiques et mouvement brownien (1965), Paris: Gauthier-Villars & Cie, Paris · Zbl 0137.11602
[48]	A. Lejay and T. Lyons. On the importance of the Lévy area for systems controlled by converging stochastic processes. Application to homogenization. In New Trends in Potential Theory, Conference Proceedings, Bucharest, September 2002 and 2003, edited by D. Bakry, L. Beznea, Gh. Bucur and M. Röckner, pp. 63-84. The Theta Foundation, 2006. · Zbl 1199.60292
[49]	Li, XD; Lyons, T. J., Smoothness of Itâ maps and diffusion processes on path spaces. I, Ann. Sci. École Norm. Sup., 39, 4, 649-677 (2006) · Zbl 1127.60033
[50]	Ledoux, M.; Lyons, T.; Qian, Z., Lévy area of Wiener processes in Banach spaces, Ann. Probab., 30, 2, 546-578 (2002) · Zbl 1016.60071
[51]	Lyons, T.; Qian, Z., Flow of diffeomorphisms induced by a geometric multiplicative functional, Proba. Theory Related Fields, 112, 1, 91-119 (1998) · Zbl 0918.60009
[52]	Lyons, T.; Qian, Z., System Control and Rough Paths (2002), Oxford Mathematical Monographs: Oxford University Press, Oxford Mathematical Monographs · Zbl 1029.93001
[53]	Lejay, A.; Victoir, N., On (p, q)-rough paths, J. Differential Equations, 225, 1, 103-133 (2006) · Zbl 1097.60048
[54]	Lyons, T.; Victoir, N., An Extension Theorem to Rough Paths, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24, 5, 835-847 (2007) · Zbl 1134.60047
[55]	Lyons, TJ, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14, 2, 215-310 (1998) · Zbl 0923.34056
[56]	Lyons, TJ; Zhang, T., Decomposition of Dirichlet Processes and its Application, Ann. Probab., 22, 1, 494-524 (1994) · Zbl 0804.60044
[57]	Magnus, W., On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., 7, 649-673 (1954) · Zbl 0056.34102
[58]	Mahony, R.; Manton, JH, The Geometry of the Newton Method on Non-Compact Lie Groups, J. Global Optim., 23, 309-327 (2002) · Zbl 1019.22005
[59]	E. J. McShane. Stochastic differential equations and models of random processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pp. 263-294. Univ. California Press, 1972.
[60]	R. Montgomery. A tour of subriemannian geometries, their geodesics and applications, vol. 91 of Mathematical Surveys and Monographs. American Mathematical Society, 2002. · Zbl 1044.53022
[61]	Musielak, J.; Semadeni, Z., Some classes of Banach spaces depending on a parameter, Studia Math., 20, 271-284 (1961) · Zbl 0099.09401
[62]	Millet, A.; Sanz-solé, M., Approximation of rough paths of fractional Brownian motion, Seminar on Stochastic Analysis, Random Fields and Applications V, 275-303 (2008), Birkhäuser: Progr. Probab., Birkhäuser · Zbl 1142.60027
[63]	C. Reutenauer. Free Lie algebras, vol. 7 of London Mathematical Society Monographs. New Series. Oxford University Press, 1993. · Zbl 0798.17001
[64]	R.A. Ryan. Introduction to Tensor Products of Banach Spaces. Springer-Verlag, 2002. · Zbl 1090.46001
[65]	E.-M. Sipiläinen. A pathwise view of solutions of stochastic differential equations. PhD thesis, University of Edinburgh, 1993.
[66]	Strichartz, R. S., The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations, J. Funct. Anal., 72, 2, 320-345 (1987) · Zbl 0623.34058
[67]	Sattinger, D. H.; Weaver, O. L., Lie groups and algebras with applications to physics, geometry, and mechanics (1993), New York: Springer-Verlag, New York · Zbl 0595.22017
[68]	K. Tapp. Matrix groups for undergraduates, vol. 29 of Student Mathematical Library. American Mathematical Society, 2005.
[69]	Varadarajan, V. S., Lie groups, Lie algebras, and their representations (1984), New York: Springer-Verlag, New York · Zbl 0955.22500
[70]	Victoir, N., Lévy area for the free Brownian motion: existence and non-existence, J. Funct. Anal., 208, 1, 107-121 (2004) · Zbl 1062.46055
[71]	Warner, FW, Foundations of differentiable manifolds and Lie groups (1983), New York: Springer-Verlag, New York · Zbl 0516.58001
[72]	Yamato, Y., Stochastic differential equations and nilpotent Lie algebras, Z. Wahrsch. Verw. Gebiete, 47, 2, 213-229 (1979) · Zbl 0427.60069
[73]	Young, LC, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67, 251-282 (1936) · JFM 62.0250.02