# Rough Paths Theory Fabrice Baudoin

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Rough Paths TheoryFabrice Baudoin

ContentsChapter 1.An overview of rough paths theory5Chapter 2. Ordinary differential equations1. Continuous paths with bounded variation2. Riemann-Stieltjes integrals and Gronwall’s lemma3. Differential equations driven by bounded variation paths4. Exponential of vector fields and solutions of differential equations99121417Chapter 3. Young’s integrals1. p-variation paths2. Young’s integrals3. Young’s differential equations21212326Chapter 4. Rough paths1. The signature of a bounded variation path2. Estimating iterated integrals3. Rough linear differential equations4. The Chen-Strichartz expansion formula5. Magnus expansion6. Free Carnot groups7. The Carnot-Carathéodory distance8. Geometric rough paths9. The Brownian motion as a rough path29293140444749515460Chapter 5. Rough differential equations1. Davie’s estimate2. The Lyons’ continuity theorem656569Chapter 6. Applications to stochastic differential equations1. Approximation of the Brownian rough path2. Signature of the Brownian motion3. Stochastic differential equations as rough differential equations4. The Stroock-Varadhan support theorem77777982843

CHAPTER 1An overview of rough paths theoryLet us consider a differential equation that writesd Z tXy(t) y0 Vi (y(s))dxi (s),i 10where the Vi ’s are vector fields on Rn and where the driving signal x(t) (x1 (t), · · · , xd (t))is a continuous bounded variation path. If the vector fields are Lipschitz continuous then,for any fixed initial condition, there is a unique solution y(t) to the equation. We cansee this solution y as a function of the driving signal x. It is an important question tounderstand for which topology, this function is continuous.A simple example shows that the topology of uniform convergence is not the correctone here. Indeed, let us consider the differential equationy1 (t) x1 (t)y2 (t) x2 (t)Z tZ ty3 (t) y2 (s)dx1 (s) y1 (s)dx2 (s)00where11cos(n2 t), x2 (t) sin(n2 t).nnA straightforward computation shows that y3 (t) t. When n , (x1 , x2 ) convergesuniformly to 0 whereas, of course, (y1 , y2 , y3 ) does not converge to 0. In this framework ,a correct topology is given by the topology of convergence in 1-variation on compact sets.To fix the ideas, let us work on the interval [0, 1]. The distance in 1-variation betweentwo continuous bounded variation paths x, x̃ : [0, 1] Rd is given byx1 (t) δ1 (x, x̃) kx(0) x̃(0)k supπn 1Xk(x(ti 1 ) x̃(ti 1 )) (x(ti ) x̃(ti ))k,k 0where the supremum is taken over all the subdivisionsπ {0 t1 · · · tn 1}.It is then a fact that is going to be proved in this class that if the Vi ’s are bounded andif xn : [0, 1] Rd is a sequence of bounded variation paths that converges in 1-variationto a continuous path x with bounded variation, then the solutions of the differential5

61.AN OVERVIEW OF ROUGH PATHS THEORYequationsny (t) y0 d ZXtVi (y n (s))dxi,n (s),0i 1converge in 1-variation to the solution ofy(t) y0 d ZXi 1tVi (y(s))dxi (s).0This type of continuity result suggests to use a topology in p-variation, p 1, to try toextend the map x y to a larger class of driving signals x. More precisely, for p 1, letus denote by Ωp (Rd ) the closure of the set of continuous with bounded variation pathsx : [0, 1] Rd with respect to the distance in p-variation which is given by!1/pn 1Xk(x(ti 1 ) x̃(ti 1 )) (x(ti ) x̃(ti ))kp.δp (x, x̃) kx(0) x̃(0)kp supπk 0We will then prove the following result:Proposition 0.1. Let p 2. If xn : [0, 1] Rd is a sequence of bounded variation pathsthat converges in p-variation to a path x Ωp (Rd ), then the solutions of the differentialequationsd Z tXny (t) y0 Vi (y n (s))dxi,n (s),pi 1 0dconverge in p-variation to some y Ω (R ). Moreover y is the solution of the differentialequationd Z tXy(t) y0 Vi (y(s))dxi (s),i 10where the integrals are understood in the sense of Young’s integration.The value p 2 is really a treshold: The result is simply wrong for p 2. Themain idea of the rough paths theory is to introduce a much stronger topology than theconvergence in p-variation. This topology, that we now explain, is related to the continuityof lifts of paths in free nilpotent Lie groups.Let GN (Rd ) be the free N -step nilpotent Lie group with d generators X1 , · · · , Xd . Ifx : [0, 1] Rd is continuous with bounded variation, the solution x of the equationd Z tX x (t) Xi (x (s))dxi (s),i 10is called the lift of x in GN (Rd ). For p 1, let us denote Ωp GN (Rd ) the closure of theset of lifted paths x : [0, 1] GN (Rd ) with respect to the distance in p-variation whichis given by!1n 1 p p XδpN (x , y ) supdN yt i (x ti ) 1 , yt i 1 (x ti 1 ) 1,πi 1

1.AN OVERVIEW OF ROUGH PATHS THEORY7where dN denotes the Carnot-Carathéodory distance on the group GN (Rd ). This is adistance that will be explained in details later. Its main property is that it is homogeneouswith respect to the natural dilation of GN (Rd ).Consider now the map I which associates with a continuous with bounded variationpath x : [0, 1] Rd the continuous path with bounded variation y : [0, 1] Rd thatsolves the ordinary differential equationd Z tXy(t) y0 Vi (y(s))dxi (s).i 10It is clear that there exists a unique map I from the set of continuous with boundedvariation lifted paths [0, 1] GN (Rd ) onto the set of continuous with bounded variationlifted paths [0, 1] GN (Rn ) which makes the following diagram commutativex x I y . yIThe fundamental theorem of Lyons is the following:Theorem 0.2. If N [p], then in the topology of δpN -variation, there exists a continuousextension of I from Ωp GN (Rd ) into Ωp GN (Rn ).In particular, we can now give a sense to differential equations driven by some continuous paths with finite p-variation, for any p 1. Indeed, let x : [0, 1] Rd which iscontinuous with a fnite p-variation and assume that there exists x Ωp GN (Rd ) whoseprojection onto Rd is x. The projection onto Rd of I (x ) is then understood as being asolution ofd Z tXy(t) y0 Vi (y(s))dxi (s).i 10An important example of application is given by the case where the driving signal is aBrownian motion (B(t))t 0 . Brownian motion has a p-finite variation for any p 2 and,as we will see, admits a canonical lift in Ωp G2 (Rd ). As a conclusion, we can consider inthe rough paths sense, solutions to the equationd Z tXy(t) y0 Vi (y(s))dB i (s).i 10It turns out that this notion of solution is exactly equivalent to solutions that are obtained by using the Stratonovitch integration theory. Therefore, the theory of stochasticdifferential equations appears as a very special case of the rough paths theory.

CHAPTER 2Ordinary differential equations1. Continuous paths with bounded variationThe first few lectures are essentially reminders of undegraduate real analysis materials.We will cover some aspects of the theory of differential equations driven by continuouspaths with bounded variation. The point is to fix some notations that will be usedthroughout the course and to stress the importance of the topology of convergence in1-variation if we are interested in stability results for solutions with respect to the drivingsignal.If s t, we will denote by [s, t], the set of subdivisions of the interval [s, t], that isΠ [ s, t] can be writtenΠ {s t0 t1 · · · tn t} .Definition 1.1. A continuous path x : [s, t] Rd is said to have a bounded variation on[s, t], if the 1-variation of x on [s, t], which is defined askxk1 var;[s,t] : supn 1Xkx(tk 1 ) x(tk )k,Π [s,t] k 0is finite. The space of continuous bounded variation paths x : [s, t] Rd , will be denotedby C 1 var ([s, t], Rd ).k · k1 var;[s,t] is not a norm, because constant functions have a zero 1-variation, but itis oviously a semi-norm. If x is continuously differentiable on [s, t], it is easily seen thatZ tkxk1 var,[s,t] kx0 (s)kds.sProposition 1.2. Let x C 1 var ([0, T ], Rd ). The function (s, t) kxk1 var,[s,t] is additive, i.e for 0 s t u T ,kxk1 var,[s,t] kxk1 var,[t,u] kxk1 var,[s,u] ,and controls x in the sense that for 0 s t T ,kx(s) x(t)k kxk1 var,[s,t] .The function s kxk1 var,[0,s] is moreover continuous and non decreasing.9

102. ORDINARY DIFFERENTIAL EQUATIONSProof. If Π1 [s, t] and Π2 [t, u], then Π1 Π2 [s, u]. As a consequence,we obtainsupn 1Xkx(tk 1 ) x(tk )k supΠ1 [s,t] k 0n 1Xkx(tk 1 ) x(tk )k Π2 [t,u] k 0supn 1Xkx(tk 1 ) x(tk )k,Π [s,u] k 0thuskxk1 var,[s,t] kxk1 var,[t,u] kxk1 var,[s,u] .Let now Π [s, u]:Π {s t0 t1 · · · tn t} .Let k max{j, tj t}. By the triangle inequality, we haven 1Xj 0kx(tj 1 ) x(tj )k k 1Xkx(tj 1 ) x(tj )k j 0n 1Xkx(tj 1 ) x(tj )kj k kxk1 var,[s,t] kxk1 var,[t,u] .Taking the sup of Π [s, u] giveskxk1 var,[s,t] kxk1 var,[t,u] kxk1 var,[s,u] ,which completes the proof. The proof of the continuity and monoticity of s kxk1 var,[0,s]is let to the reader. This control of the path by the 1-variation norm is an illustration of the notion ofcontrolled path which is very useful in rough paths theory.Definition 1.3. A map ω : {0 s t T } [0, ) is called superadditive if for alls t u,ω(s, t) ω(t, u) ω(s, u).If, in adition, ω is continuous and ω(t, t) 0, we call ω a control. We say that a pathx : [0, T ] R is controlled by a control ω, if there exists a constant C 0, such that forevery 0 s t T ,kx(t) x(s)k Cω(s, t).Obviously, Lipschitzfunctions have a bounded variation. The converse is of course not true: t t has a bounded variation on [0, 1] but is not Lipschitz. However, anycontinuous path with bounded variation is the reparametrization of a Lipschitz path inthe following sense.Proposition 1.4. Let x C 1 var ([0, T ], Rd ). There exist a Lipschitz function y : [0, 1] Rd , and a continuous and non-decreasing function φ : [0, T ] [0, 1] such that x y φ.Proof. We assume kxk1 var,[0,T ] 6 0 and considerφ(t) kxk1 var,[0,t].kxk1 var,[0,T ]

1. CONTINUOUS PATHS WITH BOUNDED VARIATION11It is continuous and non decreasing. There exists a function y such that x y φ becauseφ(t1 ) φ(t2 ) implies x(t1 ) x(t2 ). We have then, for s t,ky(φ(t)) y(φ(s))k kx(t) x(s)k kxk1 var,[s,t] kxk1 var,[0,T ] (φ(t) φ(s)). The next result shows that the set of continuous paths with bounded variation is aBanach space.Theorem 1.5. The space C 1 var ([0, T ], Rd ) endowed with the norm kx(0)k kxk1 var,[0,T ]is a Banach space.Proof. Let xn C 1 var ([0, T ], Rd ) be a Cauchy sequence. It is clear thatkxn xm k kxn (0) xm (0)k kxn xm k1 var,[0,T ] .Thus, xn converges uniformly to a continuous path x : [0, T ] R. We need to prove thatx has a bounded variation. LetΠ {0 t0 t1 · · · tn T }be a a subdivision of [0, T ]. There is m 0, such that kx xm k n 1Xkx(tk 1 ) x(tk )k k 0n 1Xkx(tk 1 ) xm (tk )k n 1X1,2nthuskxm (tk ) x(tk )k kxm k1 var,[0,T ]k 0k 0n 1 sup kx k1 var,[0,T ] .nThus, we havekxk1 var,[0,T ] 1 sup kxn k1 var,[0,T ] .n For approximations purposes, it is important to observe that the set of smooth paths isnot dense in C 1 var ([0, T ], Rd ) for the 1-variation convergence topology. The closure of theset of smooth paths in the 1-variation norm, which shall be denoted by C 0,1 var ([0, T ], Rd )is the set of absolutely continuous paths.Proposition 1.6. Let x C 1 var ([0, T ], Rd ). Then, x C 0,1 var ([0, T ], Rd ) if and onlyif there exists y L1 ([0, T ]) such that,Z tx(t) x(0) y(s)ds.0Proof. First let us assume thatZx(t) x(0) ty(s)ds,0for some y L1 ([0, T ]). Since smooth paths are dense in L1 ([0, T ]), we can find a sequencey n in L1 ([0, T ]) such that ky y n k1 0. Define then,Z tnx (t) x(0) y n (s)ds.0

122. ORDINARY DIFFERENTIAL EQUATIONSWe havekx xn k1 var,[0,T ] ky y n k1 .This implies that x C 0,1 var ([0, T ], Rd ). Conversely, if x C 0,1 var ([0, T ], Rd ), thereexists a sequence of smooth paths xn that converges in the 1-variation topology to x.Each xn can be written as,Z tnny n (s)ds.x (t) x (0) 0We still havekxm xn k1 var,[0,T ] ky m y n k1 ,so that y n converges to some y in L1 . It is then clear thatZ tx(t) x(0) y(s)ds,0 Exercise 1.7. Let x C 1 var ([0, T ], Rd ). Show that x is the limit in 1-variation ofpiecewise linear interpolations if and only if x C 0,1 var ([0, T ], Rd ).2. Riemann-Stieltjes integrals and Gronwall’s lemmaLet y : [0, T ] Re d be a piecewise continuous path and x C 1 var ([0, T ], Rd ). Itis well-known that we can integrate y against x by using the Riemann-Stieltjes integralwhich is a natural extension of the Riemann integral. The idea is to use the Riemannsumsn 1Xy(tk )(x(tk 1 ) x(tk )),k 0where Π {0 t0 t1 · · · tn T }. It is easy to prove that, when the mesh of thesubdivision Π goes to 0, the Riemann sums converge to a limit which is independentfromRTthe sequence of subdivisions that was chosen. The limit is then denoted 0 y(t)dx(t) andcalled the Riemann-Stieltjes integral of y against x. Since x has a bounded variation, itis easy to see that, more generally,n 1Xy(ξk )(x(tk 1 ) x(tk )),k 0with tk ξk tk 1 would also converge toRTy(t)dx(t). IfZ tx(t) x(0) g(s)ds00is an absolutely continuous path, then it is not difficult to prove that we haveZ TZ Ty(t)dx(t) y(t)g(t)dt,00where the integral on the right hand side is understood in Riemann’s sense.

2. RIEMANN-STIELTJES INTEGRALS AND GRONWALL’S LEMMA13We haven 1Xy(tk )(x(tk 1 ) x(tk )) k 0n 1Xky(tk )kk(x(tk 1 ) x(tk ))kk 0 n 1Xky(tk )kk(x(tk 1 ) x(tk ))kk 0 n 1Xky(tk )kkxk1 var,[tk ,tk 1 ] .k 0Thus, by taking the limit when the mesh of the subdivision goes to 0, we obtain theestimateZ TZ Tky(t)kkdx(t)k kyk ,[0,T ] kxk1 var,[0,T ] ,y(t)dx(t) 00RTwhere 0 ky(t)kkdx(t)k is the notation for the Riemann-Stieltjes integral of kyk againstthe bounded variation path l(t) kxk1 var,[0,t] . We ca

main idea of the rough paths theory is to introduce a much stronger topology than the convergence in p-variation. This topology, that we now explain, is related to the continuity of lifts of paths in free nilpotent Lie groups. Let G N(Rd) be the free N-step nilpotent Lie group with dgenerators X 1; ;X d. If x: [0;1] !Rd is continuous with bounded variation, the solution x of the equation x(t .

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