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An Introduction to Rough Paths Antoine Lejay An Introduction to Rough Paths Antoine Lejay To cite this version: Antoine Lejay. An Introduction to Rough Paths. J. Azéma; M. Ledoux; M. Emery; M. Yor. Seminaire de Probabilités XXXVII, 1832 (1832), Springer Berlin / Heidelberg, pp.1-59, 2003, Lecture Notes in Mathematics, 978-3-540-20520-3. 10.1007/b94376. inria-00102184 HAL Id: inria-00102184 https://hal.inria.fr/inria-00102184 Submitted on 25 Feb 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. An Introduction to Rough Paths Antoine Lejay1 | Projet Tosca (INRIA / IECN) Abstract: This article aims to be an introduction to the theory of rough paths, in which integrals of di®erential forms against irregu- lar paths and di®erential equations controlled by irregular paths are de¯ned. This theory makes use of an extension of the notion of iter- ated integrals of the paths, whose algebraic properties appear to be fundamental. This theory is well-suited for stochastic processes. Keywords: controlled di®erential equations, integration against irregular paths, p-variation, stochastic processes, iterated integrals, Chen series, geometric multiplicative functional AMS Classi¯cation: Primary 60H10, 34A10; Secondary 34F05 Published in S¶eminaire de probabilit¶esXXXVII. vol. 1832 of Lecture Notes in Mathematics, pp. 1{59. Springer-Verlag, 2003 Archives, links & reviews: ± mr number: 2053040 ± hal: inria-00102184 ± doi: 10.1007/b94376 1Current address: Projet TOSCA (INRIA / IECN ), IECN, Campus scienti¯que, BP 239, 54506 Vandœuvre-l`es-Nancy CEDEX, France E-mail: An ino et Le. j ay@i ecn . -n anu cy f. r 1 A. Lejay / An Introduction to Rough Paths 1 Introduction This article is an introduction to the theory of rough paths, which has been de- veloped by T. Lyons and his co-authors since the early '90s. The main results presented here are borrowed from [32, 36]. This theory concerns di®erential equa- tions controlled by irregular paths and integration of di®erential forms against irregular trajectories. Here, x is a continuous function from [0; 1] to Rd, and the notion of irregularity we use is that of p-variation, as de¯ned by N. Wiener. This means that for some p > 1, kX¡1 j ¡ jp 1 sup xti+1 xti < + : > 6 6¢¢¢6 6 k 1; 0 t0 tk 1 i=0 partition of [0;1] As we will see, the integer bpc plays an important role in this theory. In probability theory, most stochastic processes are not of ¯nite variation, but are of ¯nite p-variation for some p > 2. We show in Sect. 10 how to apply this theory to Brownian motion. But the theory of rough paths could be used for many types of processes, as presented in Sect. 12. Firstly, we give a meaning to the integral Z Z t f(xs) dxs, or equivalently, f (1.1) 0 x([0;t]) for a di®erential form Xd i f(x) = fi(x) dx : (1.2) i=1 We are also interested in solving the controlled di®erential equation dyt = f(yt) dxt; (1.3) where f is the vector ¯eld Xd @ f(y) = fi(y) : i=1 @xi This will be done using Picard's iteration principle, from the result on integration of one-forms. Using the terminology of controlled di®erential equations, x is called a control. The theory of rough paths also provided some results on the continuity of the map x 7! y, where y is given either by (1.1) or (1.3). The theory of rough paths may be seen as a combination of two families of results: 2 A. Lejay / An Introduction to Rough Paths (1) Integration of functions of ¯nite q-variation against functions of ¯nite p- variation with 1=p + 1=q > 1 as de¯ned by L.C. Young in [52]. (2) Representation of the solutions of (1.3) using iterated integrals of x: this approach is in fact an algebraic one, much more than an analytical one. Let us give a short review of these notions. (1) Young's integral Let x and y be two continuous functions respectively 1=pR and 1=q-HÄolder contin- t uous with θ = 1=p + 1=q > 1.P Then, Young's integral s yr dxr of y against x is k¡1 ¡ de¯ned as the limit of Is;t(¦) = i=0 yti (xti+1 xti ) when the mesh of the partition ¦ = f ti s 6 t0 6 ¢ ¢ ¢ 6 tk 6 t g of [s; t] goes to zero (see for example [12, 52]). It is possible to choose a point tj in ¦ such that 1 jI (¦) ¡ I (¦ n f t g)j 6 Cjt ¡ sjθ s;t s;t j (Card ¦)θ for some constant C that depends only on the HÄolder norm of x and y. Whatever j j 6 j ¡ j j ¡ jθ Pthe size of the partition ¦ is, Is;t(¦) ys(xt xs) + t s ³(θ), where ³(θ) = θ n>1 1=n . The limit of Is;t(¦) as the mesh of ¦ goes to 0 may be considered. One may be tempted to replace y by f(x), where the regularity of f depends on the irregularity of x. But to apply directly the proof of L.C. Young, one has to assume that f is ®-HÄolder continuous with ® > p ¡ 1, which is too restrictive as > 2 Rd Psoon asR p 2. To bypass this limitation, we construct when xt the integral d t j j=1 s fj(xr) dxr as à kX¡1 Xd Xd @f j ¡ j j1 i;(j2;j1) lim fj(xti )(xt xt ) + (xti )xt ;t mesh(¦)!0 i+1 i i i+1 i=0 j=1 j ;j =1 @xj2 1 2 ! Xd bpc¡1 @ f bpc;(j ;:::;j ) + ¢ ¢ ¢ + j1 (x )x bpc 1 (1.4) ¢ ¢ ¢ ti ti;ti+1 @xjb c @xj j1;:::;jbpc=1 p 2 with formally Z i;(j ;:::;j ) x i 1 = dxji ¢ ¢ ¢ dxj1 : (1.5) s;t si s1 s6si6¢¢¢6s16t This expression (1.4) is provided by the Taylor formula on f and the more x is irregular, i.e., the larger p is, the more regular f needs. What makes the previous de¯nition formal is that the \iterated integrals" of x have to be de¯ned, and there is no general procedure to construct them, nor are they unique. The terms xk;(i1;:::;ik) for k = 2;:::; bpc are limits of iterated integrals 3 A. Lejay / An Introduction to Rough Paths of piecewise smooth approximations of x, but they are sensitive to the way the path x is approximated. Due to this property, the general principle in the theory of rough paths is: P R d t j The integral j=1 s fj(xr) dxr is not driven by x but, if it exists, by 1;(i1) 2;(i1;i2) bpc;(i1;:::;ibpc) x = (x ; x ;:::; x )i1;:::;ibpc=1;:::;d corresponding for- mally to (1.5). (2) Formal solutions of di®erential equations k;(i1;:::;ik) Assume now that x is smooth, and let xs;t be its iterated integrals de¯ned by (1.5). Given some indeterminates X1;:::;Xd, we consider the formal non- commutative power series: X X i1 ik k;(i1;:::;ik) ©([s; t]; x) = 1 + X ¢ ¢ ¢ X xs;t : k>1 k (i1;:::;ik)2f 1;:::;d g As ¯rst proved by K.T. Chen in [6], ©([s; t]; x) fully characterizes the path x, and for all s 6 u 6 t, ©([s; u]; x)©([u; t]; x) = ©([s; t]; x): (1.6) This relation between iterated integrals is also used to prove that the limit in (1.4) exists. If exp is the non-commutative exponential (de¯ned by a power series), then there exists a formal series ª([s; t]; x) such that ©([s; t]; x) = exp(ª([s; t]; x)) and X X 1 d k;(i1;:::;ik) ª([s; t]; x) = F(i1;:::;id)(X ;:::;X )xs;t k>1 k (i1;:::;ik)2f 1;:::;d g 1 d where F(i1;:::;id)(X ;:::;X ) belongs to the Lie algebra generated by the indetermi- nates X1;:::;Xd, i.e., the smallest submodule containing X1;:::;Xd and closed under the Lie brackets [Y; Z] = YZ ¡ ZY . If f = (f1; : : : ; fd) and each of the fi is linear, i.e., fi(y) = Ciy where Ci is a matrix, then the solution y of (1.3) is equal to ( ¶ b yt = exp ª([s; t]; x) ys; b i where ª([s; t]; x) is equal to ª([s; t]; x) in which X was replaced by the matrix Ci. If f is not linear, but is for example a left-invariant vector ¯eld on a Lie group, i then a similar relation holds, where X is replaced by fi, and the Lie brackets [¢; ¢] are replaced by the Lie bracket between vector ¯elds. Here, the exponential is replaced by the map de¯ning a left-invariant vector ¯eld from a vector in the Lie algebra, i.e., the tangent space at 0 (see for example [13]). 4 A. Lejay / An Introduction to Rough Paths This result suggests that when one knows x, he can compute its iterated in- tegrals and then formally solve (1.3) by replacing the indeterminates by f.
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