Semi-Martingales and Rough Paths Theory Laure Coutin, Antoine Lejay

Semi-martingales and rough paths theory Laure Coutin, Antoine Lejay

To cite this version:

Laure Coutin, Antoine Lejay. Semi-martingales and rough paths theory. Electronic Journal of Prob- ability, Institute of Mathematical Statistics (IMS), 2005, 10 (23), pp.761-785. inria-00000411

HAL Id: inria-00000411 https://hal.inria.fr/inria-00000411 Submitted on 10 Sep 2006

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. Semi-martingales and rough paths theory

Laure Coutin1 — LSP, Universit´eToulouse 3

Antoine Lejay2 — Projet OMEGA (INRIA / Institut Elie´ Cartan, Nancy)

Abstract: We prove that the theory of rough paths, which is used to define path-wise integrals and path-wise differential equations, can be used with continuous semi-martingales. We provide then an almost sure theorem of type Wong-Zakai. Moreover, we show that the conditions UT and UCV, used to prove that one can interchange limits and Itôor Stratonovich integrals, provide the same result when one uses the rough paths theory. Keywords: Semi-martingales, p-variation, iterated integrals, rough paths, Wong-Zakai theorem, conditions UT and UCV AMS Classification: 60F17, 60H05

Published in Electron. J. Probab.. 10:23, 761–785, 2005

Archives, links & reviews: ◦ MR Number: 2164030 ◦ http://www.math.washington.edu/~ejpecp/EjpVol10/ paper23.abs.html 1Current address: Laboratoire de Statistique et Probabilités UniversitéPaul Sabatier 118 route de Narbonne 31062 Toulouse cedex 4, France E-mail: t n i tu.cco fr@ci 2Current address: Projet OMEGA (INRIA/IECN) Campus scientifique B.P. 239 54506 Vandœuvre-lès-Nancy cedex, France E-mail: . y . uenL cnaA.oyaec fen-n@ijit rn

This version may be slightly diﬀerent from the published version 25 pages L. Coutin and A. Lejay / Semi-martingales and rough paths theory

1 Introduction

The theory of rough paths allows to give a meaning to integrals like

Z t zt = z0 + g(xs) dxs 0 and controlled diﬀerential equations like

Z t yt = y0 + f(ys) dxs 0 when x is a continuous, irregular path in a Banach space, and g and f are differential forms and vector fields smooth enough (See [21, 19, 16]). But for that, one needs to know the equivalent of the iterated integrals of x, that is R dx ⊗ · · · ⊗ dx , and to use the topology of p-variation, which 0≤s1≤···≤sk≤T s1 sk is defined using the semi-norm

Ã !1/p kX−1 p Varp,[0,T ](x) = sup |x(ti+1) − x(ti)| , (1) partition Π of [0,T ] i=1 where we use the convention that the points of the partitions Π are t1 ≤ t2 ≤ · · · ≤ tk. The real p defines how irregular the path x is, and there is no canonical way to define the iterated integrals of x up to the order bpc. However, for a large class of stochastic processes, it is possible to define the equivalent of the iterated integrals for the trajectories. Although the rough path theory is completely deterministic, it has been used for a large class of processes, including Brownian motion, some Gaussian processes in a Banach space [18], fractional Brownian motion [5], free Brownian motion [3], symmetric Markov process [1],... In this article, we study the use of rough paths theory for general continuous semi-martingales. Let us remark first that any one-dimensional, continuous local martingale M may be written as Mt = BhMit , where B is a Brownian motion and hMi the quadratic variation of M. Since almost every trajectory of the Brownian motion is 1/p-Hölder continuous as soon as p > 2, then it is also of finite p-variation for any p > 2. Hence, as a M- continuous time-change, see [26], does no change the p-variation, there exists a modification of M whose trajectories are of finite p-variation for any p > 2 (See Lemma 2). Moreover, if X is a semi-martingale, the following Itôand Stratonovich integrals exist

Z t Z t Θs,t(X) = (Xr − Xs) ⊗ dXr and Θs,t(X) = (Xr − Xs) ⊗ ◦dXr. s s

2 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

If Θ(X) and Θ(X) are of finite p/2-variation (See Proposition 1), then (X, Θ(X)) and (X, Θ(X)) are two rough paths of finite p-variations lying above X and one may use the results from the theory of rough paths, espe- cially the continuity of integrals and solutions of differential equations. The main idea of the proof is to use a X continuous random time-change ϕ such that Xϕ = Xϕ(0) + Mϕ + Vϕ remains a semi-martingale and such that Mϕ and Vϕ are Hölder continuous trajectories. But, Xϕ is then defined on b b a random time interval [0, T ]. Yet, the value of T depends on hMiT and Var1,[0,T ](V ), which are assumed to be controlled. Basically, we prove two results:

n • If (X )n∈N is a family of semi-martingales converging in distribution to X and satisfying the conditions UT (uniformly tight) or UCV (Uni- n n n n formly controlled variations), then (X , Θ(X ))n∈N and (X , Θ(X ))n∈N converge respectively in distribution to (X, Θ(X)) and (X, Θ(X)) as rough paths in the topology induced by the p-variation distance. The convergence under the topology of Skorokhod is kwnon since the papers of Mémin and SÃlomiński [22], and Jakubowski, Mémin and Pagès[11]. Hence, this is fully coherent with the results concerning interchanging limits and stochastic integrals in the semi-martingales theory.

• For almost every trajectory of the continuous semi-martingale X, there exists a family (X(n))n∈N of piecewise linear approximations of X such that (X(n), Θ(X(n)))n∈N converges almost surely in the appropriate topology to (X, Θ(X)). In other words, one obtains the almost sure convergence of the ordinary integral (resp. the ordinary diﬀerential equation)

Z t Z(n)t = Z0 + f(X(n)s) dX(n)s 0 Z t (resp. Y (n)t = Y0 + g(Y (n)s) dX(n)s) 0 to the Stratonovich integral (resp. the stochastic differential equation) R t R t Zt = Z0 + 0 f(Xs) ◦ dXs (resp. Yt = Y0 + 0 g(Ys) ◦ dXs). A similar convergence result is also given for Itô’s integrals and Itô’s SDEs. The particular case of the Brownian motion was already known (See [20, 1],...) and was developed initially in [27]. In this Ph.D. thesis, this method can be used for martingales with α-Hölder continuous trajectories for α < 1/2. Thus, these almost-sure Wong-Zakai results are not surprising. Here, we extend this result to a general continuous semi-martingales, and our proof

3 L. Coutin and A. Lejay / Semi-martingales and rough paths theory includes recent techniques in the theory of rough path, that leads to some simplification of results. Besides, these proofs show the role played by Hölder continuous paths among paths of finite p-variation, and that it is not a big deal to use Hölder continuous path instead of path of finite p-variation using a time-change. Moreover, this time-change gives us information on the way to get some uniform control for a family of paths, since we are reduced in controlling the length of the image of the time interval on which the path is initially defined. Then, the conditions UT and UCV appear naturally in this context. Recent results show that the enhanced Brownian motion, that is a multiplicative functional lying above a Brownian motion, could be manipulated as the Brownian motion regarding many of its properties: support theorem, large deviation result, ... See [20, 7, 8]. Hence, one could think that this technique of using a time-change could be applied to generalize almost immediately to continuous semi-martingales some results given for the Brownian motion, as long as only the martingale property of the Brownian motion is involved in their proofs.

2 Rough paths

We refer the reader to [21, 19] or [16] for a detailed insight on the theory of rough paths and the objects we introduce now. Let N be a fixed integer. Throughout all this article, we consider semi- martingales with values in RN , and we denote by | · | the norm |x| = k 1 N supk=1,...,N |x | for x = (x , . . . , x ). N For a continuous function x from [0,T ] to R , we denote by Varp,[0,T ](x) its p-variation defined by (1). Denote by Vp([0,T ]; RN ) the Banach space of continuous functions of p N finite p-variation with the norm Varp,[0,T ](·)+k·k∞. Note that V ([0,T ]; R ) is not separable. Set ∆+ = { 0 ≤ s ≤ t ≤ T }. + N For a continuous function x from ∆ to R , denote also by Varp,[0,T ](x) its p-variation, that is

 1/p jX−1  p Varp,[0,T ](x) = sup |x(ti, ti+1)| . partition {ti}i=1,...,j of [0,T ] i=1

If x(s, t) = y(t) − y(s) for some continuous function y, then Varp,[0,T ](x) = Varp,[0,T ](y).

4 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

N N We equip R ⊗ R with a norm k · kRN ⊗RN such that kx ⊗ ykRN ⊗RN ≤ |x| × |y| for any x, y ∈ RN . p N 1 2 We denote by MV ([0,T ]; R ) the space of functions (x0, xs,t, xs,t)(s,t)∈∆+ such that

1 N xs,t = xt − xs for a continuous function x from [0,T ] to R , (2a) x2 is continuous from ∆+ to RN ⊗ RN , (2b) 1 Varp,[0,T ](x ) < +∞, (2c) 2 Varp/2,[0,T ](x ) < +∞, (2d) 2,i,j 2,i,j 2,i,j 1,i 1,j xs,t = xs,u + xu,t + xs,u · xu,t for i, j ∈ { 1,...,N } (2e) for all 0 ≤ s ≤ u ≤ t ≤ T . Sometimes, it could be useful to use tensor 1 2 product notations instead of indexes. This means that x = (xs,t, xs,t) is 1 2 N N ⊗2 seen as xs,t = 1 + xs,t + xs,t ∈ R ⊕ R ⊕ (R ) (Here, the starting point x0 of x is not taken into account). Accordingly, (2e) could be rewritten as 2 2 2 1 1 xs,t = xs,u + xu,t + xs,u ⊗ xu,t. 1 When there is no ambiguity, we identify (x0, x ) with x, that is a function + 1 2 on ∆ and a starting point with a function on [0,T ]. Thus, (x0, x , x ) is also denoted by (x, x2). The elements of MVp([0,T ]; RN ) are called multiplicative functionals. The topology we used on MVp([0,T ]; RN ) is the one induced by the norm

1 2 1 2 1 2 k(x0, x , x )k = |x0| + sup |(x , x )s,t| + Varp,[0,T ](x ) + Varp/2,[0,T ](x ). (s,t)∈∆+ We end this section by a useful Lemma, that allows to estimate the distance in MVp([0,T ]; RN ) between two rough paths by using the pointwise distance of the increments between dyadic points of the diﬀerence of these paths. Lemma 1 (See for example [9, 1, 18]). Let p > 2 and γ > p/2 − 1. Let (X1,X2) and (Y 1,Y 2) be multiplicative functionals in MVp([0,T ]; RN ). Then for any partition 0 ≤ s1 ≤ ... ≤ sk ≤ T of [0,T ], there exists a n n constant C depending only on p and γ such that, for tj = jT/2

kX−1 µ X 2Xn−1 ¶1/2 2 2 p/2 γ 1 1 p |X − Y | ≤ C n |X n n − Y n n | si,si+1 si,si+1 tj ,tj+1 tj ,tj+1 i=1 n≥1 j=0 X 2Xn−1 µ ¶1/2 γ 1 p 1 p × n |X n n | + |Y n n | tj ,tj+1 tj ,tj+1 n≥1 j=0 X 2Xn−1 γ 2 2 p + n |X n n − Y n n | . tj ,tj+1 tj ,tj+1 n≥1 j=1

5 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

3 The conditions UT and UCV for semi-martingales

Let X be a continuous semi-martingale with respect to a filtration F· = N (Ft)t≥0 with values in R , which is defined on a probability space (Ω, F, P). The filtration F· may be assumed to satisfy the usual conditions. There exists a unique decomposition of X as the sum of a local martingale M and a process V of locally of finite variation, both F·-adapted. The decomposition X = X0 + M + V is called the canonical decomposition of X. n n Consider a sequence (X )n∈N of continuous F· -semi-martingales on probability spaces (Ωn, F n, Pn). To simplify the notation, denote Pn by P when there is no ambiguity. In all this section, we consider the convergence of semi-martingales in the space of continuous functions with the uniform norm. The convergence in p-variation is considered in the next sections. The following definitions and results are taken from [22] and the review article [13].

Deﬁnition 1 (Condition UT, Uniformly Tight). Let Hn be the class of n n n n F· -predictable, simple processes bounded by 1 on (Ω , F , P ). The sequence n (X )n∈N is said to satisfy the condition UT if for each t > 0 and for any ε > 0, there exists C large enough such that

n sup sup P [ |H · Xt | > C ] < ε. (UT) n∈N H∈Hn

n Theorem 1. Assume that (X )n∈N satisfies the condition UT. n (i) The sequence (X )n∈N is tight and any cluster point X of the sequence n X (X )n∈N is a semi-martingale with respect to the smallest filtration F· which is right continuous it generates. n n (ii) Let (H )n∈N be a sequence of F· -progressively measurable càdlàgpro- n n cesses such that (H ,X )n∈N converges to (H,X) in the Skorohod topology, then X is a semi-martingale with respect to the filtration generated by (H,X), n n n n and, when all the stochastic integrals are defined, (H ,X ,H · X )n∈N converges to (H,X,H · X). n n (iii) The sequence (X , hX i)n∈N converges to (X, hXi). From now, consider a time T > 0 and restrict only to processes on [0,T ].

Deﬁnition 2 (Condition UCV, Uniformly Controlled Variations). n The sequence (X )n∈N of semi-martingales with canonical decompositions

6 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

n n n n X = X0 + M + V is said to satisfy the condition UCV on [0,T ] if

n (hM iT )n∈N is tight, (3) n (Var1,[0,T ](V ))n∈N is tight. (4)

Remark 1. This definition is slightly different from the one in [13], but it is easily seen that both definitions are equivalent.

n Theorem 2. (i) Assume that the sequence (X )n∈N converges weakly to X. n Then (X )n∈N satisfies the condition UT if and only if it satisfies the condition UCV. n (ii) Let (M )n∈N be a sequence of local martingales converging in distribu- n tion to M. Then M is a local martingale and (M )n∈N satisfies the conditions n UT and UCV. In particular, (hM iT )n∈N is tight. n (iii) Let (X )n∈N be a sequence of continuous semi-martingales converging in distribution to X and satisfying the equivalent conditions UT or UCV. n n n n n Assume that X = X0 + M + V is the canonical decomposition of X . Then there exists a filtration F· such that X is a F·-semi-martingale and there exists a local F·-martingale M and a term locally of finite variation V n n n F·-adapted such that (X ,M ,V )n∈N converges in distribution to (X,M,V ). Counter-example 1. Of course, any sequence of semi-martingale does not necessarily satisfy the conditions UT and UCV. Let V be a smooth function from R to R which is one-periodic, and set

Z t ε 1 ε Xt = Bt + b(Xs /ε) ds, (5) ε 0 where B is a Brownian motion and b = ∇V . Then, it is well known that ε R t X is equal in distribution to εX·/ε, where Xt = Bt + 0 b(Xs) ds, and that Xε converges in distribution to σβ, where σ is a constant matrix which is not in general the identity matrix, and β is a Brownian motion (This is the ε homogenization problem: See for example [2]). Hence (X )ε>0 cannot satisfy the conditions UT and UCV, since it contradicts Theorem 1-(iii) (See also [15] and [17]). In facts, it is easy to construct family of semi-martingales that does not satisfy the conditions UT and UCV. For that, one has consider semi- martingales where the term of ﬁnite-variation becomes a martingale when one passes to the limit. The example above comes from an homogenization problem and appears in a completely natural situation.

7 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

4 Semi-martingales and p-variation

For any one-dimensional, continuous local martingale M on [0,T ], there exists a one-dimensional Brownian motion B such that Mt = BhMit for all t ∈ [0,T ]. Let p > 2 be ﬁxed. Using the scaling property of the Brownian motion, for any T > 0, h i 1/2 E Varp,[0,T ](B) ≤ CpT (6) with Cp = E[ Varp,[0,1](B)]. It follows that for any continuous local martingale N i i i M with values in R and any C > 0, if B is such that B i = M , hM it t " # h i i i P Varp,[0,T ](M) > C =P sup Varp,[0,hM iT ](B ) > C i=1,...,N XN h i i i ≤ P Varp,[0,K](B ) > C; hM iT ≤ K i=1 (7) XN h i i i i + P Varp,[0,hM iT ](B ) > C; hM iT > K i=1 XN h i CK,p 1/2 i ≤ K + P hM iT > K . C i=1 Remark that if V is of ﬁnite variations, then for N = 1 and for all s ≤ t ≤ u,

µZ t ¶p µZ u ¶p µZ u ¶p p p |Vt − Vs| + |Vu − Vt| ≤ |dVr| + |dVr| ≤ |dVr| . s t s Clearly, Varp,[0,T ] V ≤ Var1,[0,T ](V ). (8) The next lemma follows immediately from (7) and (8).

Lemma 2. Let X be a F·-semi-martingale with decomposition X = X0+M + V . Then X is almost surely of ﬁnite p-variation as soon as p > 2. Moreover, n if (X )n∈N is a sequence of semi-martingales satisfying the conditions UCV, then for any ε > 0 there exists C large enough such that h i n P Varp,[0,T ](X ) > C < ε. (9)

n N This implies that if (X )n∈N converges in distribution in C([0,T ]; R ) to X n p N (which is then a semi-martingale), then (X )n∈N converges in V ([0,T ]; R ) to X for any p > 2.

8 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

i,j For a semi-martingale X, deﬁne Θs,t(X) by

Z t i,j i i j Θs,t(X) = (Xr − Xs) dXr s

i,j and Θ(X) = (Θs,t(X); 0 ≤ s ≤ t ≤ T, i, j ∈ { 1,...,N }). Deﬁne also

Z t i,j i i j Θs,t(X) = (Xr − Xs)◦dXr s 1 ³ ´ = Θi,j (X) + hXi,Xji − hXi,Xji . s,t 2 t s Clearly, (X, Θ(X)) and (X, Θ(X)) satisfy (2a), (2b) and (2e). We have seen in Lemma 2 that they also satisfy (2c). To prove that they belong to MVp([0,T ]; RN ), it remains to prove that Θ(X) and Θ(X) satisfy (2d).

Proposition 1. Let X be a F·-semi-martingale. Then Θ(X) and Θ(X) are almost surely of ﬁnite p/2-variation for all p > 2. Consequently, both (X, Θ(X)) and (X, Θ(X)) belong to MVp([0,T ]; RN ) for all p > 2. n N Furthermore, if (X )n∈N converges in distribution to X in C([0,T ]; R ) n n n n and satisﬁes the conditions UCV or UT, then (X , Θ(X ))n∈N and (X , Θ(X ))n∈N converge respectively to (X, Θ(X)) and (X, Θ(X)) in MVp([0,T ]; RN ).

Proof. Let us decompose the semi-martingale X as X = X0 + M + V . Besides, let V + and V − be the increasing, continuous functions such that V = V + − V −. We consider now the process (M,V +,V −) in R3N , and de- + − note by Varp,[s,t](M,V ,V ) its p-variation. Associate to the semi-martingale X the function ϕ deﬁned by n o + − p ϕ(t) = inf s > 0 Varp,[0,s](M,V ,V ) > t .

The process t 7→ Varp,[0,t](X) is continuous and F·-adapted. Assume that ± ϕ(t) = ϕ(s) for some s < t. Then for y = M or y = V , Varp,[s,t](y) = 0 p p p since Varp,[0,s](y) + Varp,[s,t](y) ≤ Varp,[0,t](y) . Consequently, y is constant on [s, t], and thus that X, M and V are constant on the intervals [ϕ(t−), ϕ(t)] for all t ∈ [0,T ]. ϕ ϕ ϕ Let F· = (Ft )t≥0 be the filtration defined by Ft = Fϕ(t). Because M and V are constant on the intervals [ϕ(t−), ϕ(t)], Mϕ = (Mϕ(t))t∈[0,T ] is a ϕ local F· -martingale. Moreover, (Vϕ(t))t∈[0,T ] is locally of finite variation (See Propositions V.1.4 and V.1.5 in [26]). b + − p Let us set T = Varp,[0,T ](M,V ,V ) . Then

+ − + − Varp,[0,ϕ(Tb)](M,V ,V ) = Varp,[0,T ](M,V ,V ) (10)

9 L. Coutin and A. Lejay / Semi-martingales and rough paths theory and

 2/p nX−1  p/2 Varp/2,[0,T ] Θ(X) = sup Θϕ(ti),ϕ(ti+1)(X) . t0≤···≤tn partition of [0,Tb] i=0

According to Propositions V.1.4 and V.1.5 in [26],

Z ϕ(t) Z t i i j i i j (Xr − Xϕ(s)) dXr = (Xϕ(r) − Xϕ(s)) dXϕ(r) ϕ(s) s for i, j ∈ { 1,...,N }. Thus, Θs,t(Xϕ) = Θϕ(s),ϕ(t)(X) and

Varp/2,[0,T ] Θ(X) = Varp/2,[0,Tb] Θ(Xϕ). (11)

Assume that the semi-martingale X satisﬁes h i bq/p + − q/p E[ T ] = E Varp,[0,T ](M,V ,V ) < +∞ (12) for some q > p > 2. The Burkholder-Davis-Gundy inequality may be applied: There exists some constant C1 such that  ¯ ¯  ¯Z ϕ(t) ¯q/2  ¯ i i j¯  E ¯ (Xr − Xϕ(s)) dMr ¯ ¯ ϕ(s) ¯  ¯ ¯  ¯Z ϕ(t) ¯q/4  ¯ i i 2 j ¯  ≤ C1E ¯ |Xr − Xϕ(s)| dhM ir¯ ¯ ϕ(s) ¯ " # h i C1 i i q C1 j j q/2 ≤ E sup |Xr − Xϕ(s)| + E |hM iϕ(t) − hM iϕ(s)| . 2 r∈[ϕ(s),ϕ(t)] 2

Applying the other side of the Burkholder-Davis-Gundy inequality, there exists some constant C2 such that " # h i j j q/2 j j q E |hM iϕ(t) − hM iϕ(s)| ≤ C2E sup |Mr − Mϕ(s)| . r∈[ϕ(s),ϕ(t)]

Hence, it follows that for some constant C > 0,  ¯ ¯  ¯Z ϕ(t) ¯q/2 h i  ¯ i i j¯  + − q E ¯ (Xr − Xϕ(s)) dMr ¯ ≤ CE Varq,[ϕ(s),ϕ(t)](M,V ,V ) . ¯ ϕ(s) ¯

10 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

Still using the inequality ab ≤ (a2 + b2)/2 for any a, b > 0,  ¯ ¯  ¯Z ϕ(t) ¯q/2  ¯ i i ±,j¯  E ¯ (Xr − Xϕ(s)) dVr ¯ ¯ ϕ(s) ¯ " # " ¯ ¯ # ¯Z ϕ(t) ¯q 1 i i q 1 ¯ ±,j ¯ ≤ E sup |Xr − Xϕ(s)| + E ¯ |dVr |¯ . 2 r∈[ϕ(s),ϕ(t)] 2 ¯ ϕ(s) ¯ For any continuous function y and all 0 ≤ s ≤ t,

q q sup |yr − ys| ≤ Varq,[s,t](y) . (13) r∈[s,t]

1−1/q + − Moreover, let us remark that Varq,[s,t](X) ≤ 3 Varq,[s,t](M,V ,V ). Besides, for any continuous, increasing function y of ﬁnite variation,

µZ t ¶q q dys ≤ Varq,[s,t](y) . (14) s Applying (14) to V + and V − and using (13) to M and X, one gets that for some constant C depending only on q, h i h i i,j q/2 + − q E |Θϕ(s),ϕ(t)(X)| ≤ CE Varq,[ϕ(s),ϕ(t)](M,V ,V ) .

+ − p Since Varp,[0,ϕ(s)](M,V ,V ) = s for all s ≥ 0 and

p p p Varp,[0,s](y) + Varp,[s,t](y) ≤ Varp,[0,t](y) for all 0 ≤ s ≤ t and any continuous function y of ﬁnite p-variation,

+ − p Varp,[ϕ(s),ϕ(t)](M,V ,V ) ≤ |t − s|. (15)

q q q As Varq,[s,t](y) ≤ Varp,[s,t](y) and a + b ≤ (a + b) for any q ≥ 1 and all a, b > 0, it follows from (15) that

+ − q q/p Varq,[ϕ(s),ϕ(t)](M,V ,V ) ≤ |t − s| . It follows that  ¯ ¯  ¯Z ϕ(t) ¯q/2  ¯ i i j¯  q/p E ¯ (Xr − Xϕ(s)) dXr ¯ ≤ K|t − s| ¯ ϕ(s) ¯ for some constant K that depends only on q. It follows that   n ¯ ¯q/2 2X−1 ¯Z ϕ(Tb(k+1)/2n) ¯  ¯ i i j¯  n(1−q/p) bq/p E ¯ (Xr − X n ) dXr ¯ ≤ 2 E[ T ] ¯ b n ϕ(Tb k/2 ) ¯ k=0 ϕ(T k/2 )

11 L. Coutin and A. Lejay / Semi-martingales and rough paths theory which is ﬁnite from (12). From (11) and Lemma 1 with Y = 0, there exists some constant cp,q,T such that h i h i q/2 q/2 E Varq/2,[0,T ] Θ(X) = E Varq/2,[0,Tb] Θ(Xϕ) bq/p ≤ cp,q,T E[ T ]. Now, to consider the general case, ﬁx K and let τ = ϕ(K). Then τ τ is a stopping time and for a F·-adapted process Y , set Y = Y·∧τ . The τ continuous process X is a F·-semi-martingales with canonical decomposition τ τ τ b X = X0 + M + V . Remark that τ > T is equivalent to T ≤ K. Hence, it follows from (7) that for any L > 0, h i P Varq/2,[0,T ] Θ(X) > C h i τ ≤ P Varq/2,[0,T ] Θ(X ) > C + P [ τ < T ] 1 h i h i ≤ E Var Θ(Xτ )q/2 + P Tb > K Cq/2 q/2,[0,T ] √ XN h i h i cq,p,T q/p CK,p i 1/p + − 1/p ≤ q/2 K + L+ P hM iT > L +P Var1,[0,T ](V ,V ) > L . C K i=1

± 1 Now, remark that Vt = 2 (Vt ± Var1,[0,t](V )) (See for example Section I.6 in ± [24]). Hence, Var1,[0,t](V ) ≤ Var1,[0,t](V ). Thus, h i h i + − 1/p 1/p P Var1,[0,T ](V ,V ) > L ≤ P Var1,[0,T ](V ) > L . For any ε > 0, choosing ﬁrst L, then K and C, it follows that h i P Varq/2,[0,T ] Θ(X) > C < ε. (16)

Hence, Θ(X) is of ﬁnite q/2-variation almost surely. n Now, let (X )n∈N be a sequence of semi-martingales satisfying the condition UCV. In (16), when ε > 0 is ﬁxed, the choice of C is uniform in n, since with (16), if L is large enough such that

XN h i h i i,n 1/p n 1/p sup P hM iT > L + sup P Var1,[0,T ](V ) > L < 2ε, n∈N i=1 n∈N then there exists C large enough such that h i n sup P Varq/2,[0,T ] Θ(X ) > C < 3ε. (17) n∈N

n If (X )n∈N converges in distribution to X, it follows from the condition UT n N that Θ0,·(X ) converges in distribution to Θ0,·(X) in C([0,T ]; R ). On the

12 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

n n other hand, it follows from (9) and (17) that (X , Θ(X ))n∈N is tight in MVp([0,T ]; RN ), thanks to Corollary 6.1 in [16]. Thus, (Xn, Θ(Xn)) converges to (X, Θ(X)) in MVp([0,T ]; RN ). i j i j Concerning Θ(M), note that hMϕ,Mϕit = hM ,M iϕ(t) for all t ≥ 0. i,j 1 i j i j Thus, if Ξs,t(M) = 2 (hM ,M it − hM ,M is), then i,j i,j Varp/2,[0,T ] Ξ (M) = Varp/2,[0,Tb] Ξ (Mϕ). Besides, one knows that h i h i h i i,j q/2 i,i q 1/2 j,j q 1/2 E |Ξs,t(Mϕ)| ≤ E |Ξs,t(Mϕ)| + E |Ξs,t(Mϕ)| . Thus, one has only to act as previously.

ε Counter-example 2 (Counter-example 1 (continuation)). In the case of (X )ε>0 ε ε deﬁned by (5), (X , Θ(X ))ε>0 converges in p-variation to (σβt, Θs,t(σβ) + c(t − s))0≤s≤t≤T , where c is an antisymmetric matrix (See [15, 17]). This im- R · ε ε R · plies that the limit of the stochastic integral 0 f(Xs )◦dXs is not 0 f(σβs)σ◦ dβs, but Ã ! Z · Z · 1 ∂fi ∂fj f(σβs)σ ◦ dβs + ci,j − (σβs) ds. 0 2 0 ∂xj ∂xi 5 An almost sure Wong-Zakai theorem

N Let X be a F·-semi-martingale with values in R . We prove a result of type Wong-Zakai with respect piecewise-linear approximation of X converging to X in p-variation, together with their L´evy area. We have seen in the previous section that (X, Θ(X)) is a multiplicative functional in MVp([0,T ]; RN ) for any p > 2. From now, we set

1 2 Xs,t = Xt − Xs and Xs,t = Θs,t(X). Notation and hypothesis. Let p > 2 and K be a positive random variable. Let ϕ be a compatible random time-change (that is ϕ is the right-continuous inverse of a non-decreasing, continuous, F·-adapted function and X is constant on the intervals [ϕ(t−), ϕ(t)]: see [26] for example) such that

p + + p − − p |Mϕ(t) − Mϕ(s)| + |Vϕ(t) − Vϕ(s)| + |Vϕ(t) − Vϕ(s)| ≤ K|t − s|. (18) Let us denote by ϕ−1 the right-continuous inverse of ϕ. Remark 2. We have seen in the proof of Proposition 1 how to construct a random time change such that (18) is satisﬁed with K = 1.

13 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

The properties of ϕ and the inequality (18) imply that

1/p −1 1/p Varp,[0,T ](X) = Varp,[0,ϕ−1(T )] Xϕ(·) ≤ K ϕ (T ) and Z t 2 Xϕ(s),ϕ(t) = (Xϕ(r) − Xϕ(s)) ⊗ ◦dXϕ(r). s n n n For any n ∈ N, let 0 ≤ s1 ≤ s2 ≤ ... ≤ s2n ≤ T be a partition of [0,T ]. n n For any n ∈ N and any k = 1, 2,..., 2 , we set ∆ X = X n − X n and k sk sk−1

n t − sk−1 n n n X(n) = X n + ∆ X for t ∈ [s , s ]. (19) t sk−1 n n k k−1 k sk − sk−1

5.1 A Wong-Zakai theorem for the Stratonovich SDEs The path X(n) is the piecewise linear approximation of X that coincides n with X at the points sk . We could then construct a geometric rough path (X(n)1,X(n)2) from X(n) by setting

1 X(n)s,t = X(n)t − X(n)s Z t 2 and X(n)s,t = (X(n)r − X(n)s) ⊗ dX(n)r. s

n n n n Let 0 ≤ s ≤ t ≤ T , and k and ` such that sk−1 < s ≤ sk ≤ s` ≤ t < s`+1,

Z sn Z t 2 k X(n)s,t = (X(n)r − X(n)s) ⊗ dX(n)r + (X(n)r − X(n)sn ) ⊗ dX(n)r s sn `  `  X`−1 X(n) n + X(n) n sj+1 sj +  − X(n)  ⊗ (X(n) n − X(n) n ). s sj+1 sj j=k 2

2 With this expression, it is easily seen that X(n)s,t converges in probability 2 R t to the Stratonovich integral Xs,t = s (Xr − Xs) ⊗ ◦dXr for any s ≤ t (See for example [26, exercise 2.18, p. 136]). The following proposition was already known for semi-martingales, but for the convergence in probability with respect to the uniform norm: See [23]. Yet, combining the next proposition and the usual Wong-Zakai result for semi-martingales implies that the integral deﬁned by the rough-path theory agrees with the Stratonovich integral. Besides, it is important here that the approximation is piecewise linear, since it is known that other approximations could lead to diﬀerent integrals: See for example [14, Chapter 5.7], [10, Section VI-7], [23, 4] and more specif- ically [17] regarding the rough paths theory.

14 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

Proposition 2. Let p, K and ϕ as above. Let X(n) be the piecewise lin- −1 n ear approximation of X along the partition (ϕ (ϕ(T )k/2 ))k=0,...,2n . Then, 1 2 the geometric multiplicative functional (X(n) ,X(n) )n∈N converges almost surely in MVp([0,T ]; RN ) to (X1,X2) as n → ∞, and (X1,X2) is a geometric multiplicative functional.

Remark 3. If for all β > 0, X is a semi-martingale satisfying for some Cβ > 0 the condition h i β β/2 E |Xt − Xs| ≤ Cβ|t − s| , for any 0 ≤ s, t ≤ T, then one knows from the Kolmogorov lemma, that there exists a α-H¨older continuous version of X for any α < 1/2. Moreover, there exists a random 1/α variable K such that |Xt(ω) − Xs(ω)| ≤ K(ω)|t − s|. Thus, Proposition 2 may be applied with the time change ϕ(t) = t. In particular, this proposition is true for the Brownian motion with ϕ(t) = t. Hence, we recover a result from [27]. −1 n Remark 4. The partition (ϕ (ϕ(T )k/2 ))k=0,...,2n is a random partition un- less the semi-martingale X has H¨older continuous paths. We have to note that ϕ−1 is not necessarily continuous, but the discontinuities of ϕ corre- sponds to the intervals on which X is constant, and are not taken into account in the integrals. Remark 5. If the partition is deterministic, but no longer related to dyadics, then computations similar to the ones used in the proof of Proposition 2 1 2 1 2 prove that the convergence in probability of (X(n) ,X(n) )n∈N to (X ,X ) holds in p-variation.

1 N m N Hypothesis 1. (i) The function g = (g1, . . . , gN ) is C (R , R ) with a derivative which is also α-Hölder continuous with α > p − 2. 2 m m N (ii) The function f = (f1, . . . , fN ) is C (R , R ) with a second-order derivative which is also α-Hölder continuous with α > p − 2. Corollary 1. Under the hypotheses of Proposition 2 on X and Hypothesis 1 on g and f, then the ordinary integral Z(n) and the solution of the ordinary differential equation Y (n) defined by

Z t Z t Z(n)t = z + g(X(n)s) dX(n)s and Y (n)t = y + f(Y (n)s) dX(n)s 0 0 converge respectively almost surely in p-variation to

Z t Z t Zt = z + g(Xs) ◦ dXs and Yt = y + f(Ys) ◦ dXs, 0 0 where the integrals are understood as Stratonovich integrals.

15 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

Proof. Denote by K (resp. I) the map that gives the rough path obtained PN i by integrating the diﬀerential form g = i=1 gi(x)dx (resp. the vector ﬁeld f) against a rough path. Let also X(n) be the canonical rough path con- structed above X(n). Let us set Z(n) = K(X(n)) and Y(n) = I(X(n)). From the continuity of K and I from MVp([0,T ]; RN ) to MVp([0,T ]; Rm), (Z(n))n∈N and (Y(n))n∈N converge almost surely respectively to the multiplicative functionals Z = K(X) and Y = I(X), that are also geometric. But 1 1 Y (n)t − y = Y(n)0,t and Z(n)t − y = Z(n)0,t. Besides, by the Wong-Zakai theorem (See for example [23]), one knows that (Y (n))n∈N and (Z(n))n∈N 1 converge in probability to Y and Z. It is now clear that Yt = y + Y0,t and 1 Zt = z + Z0,t for any t ∈ [0,T ]. First part of the proof of Proposition 2. Since X is continuous, it is clear that (X(n))n∈N converges to X almost surely in the space of continuous functions with the uniform norm. Let 2 < q < p. Let Π = { ui 0 ≤ u1 ≤ · · · ≤ uk ≤ T } be a partition of n [0,T ]. We introduce two sets of index (with the convention that s0 = u0 = 0 n n and uk+1 = s2n+1 = T ): For j = 0,..., 2 , we set n o n n Πj = i ui ∈ [sj , sj+1] , n o 0 n n Πleft = i ∃j, j s.t. ui ≤ sj ≤ sj0 < ui+1 .

For each j, we remark that (with the convention that a sum over the empty set is 0),

Thus,

2Xn−1 X 2Xn−1 q q q |X(n) − X(n) | ≤ |X n − X n | ≤ Var (X) . ui+1 ui sj+1 sj q j=0 i∈Πj , i6=max Πj j=0

16 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

n n |X n − X n | for any s ∈ [s , s ], sj+1 sj j j+1

Hence, X q q−1 q |X(n)ui+1 − X(n)ui | ≤ 3 Varq,[0,T ](X) . i∈Πleft Let us remark that

q−1 1/q Varq,[0,T ](X(n)) ≤ (3 + 1) Varq,[0,T ](X). (20) Finally, let us remark that for any p > q,

p Varp,[0,T ](X − X(n)) ³ ´ q−1 p−q q q ≤ 2 sup |Xt − X(n)t| Varq,[0,T ](X) + Varq,[0,T ](X(n)) . t∈[0,T ]

Thus, one obtains the almost sure convergence of (X(n))n∈N to X in p- variation.

ϕ Let K0 be ﬁxed, and let T0 be the F -stopping time such that ( ) + − + − p |(M,V ,V )ϕ(t) − (M,V ,V )ϕ(s)| T0 = inf r > 0 sup ≥ K0 . 0≤s

b Let also T be a ﬁxed real number. Let us set Yt = X , which is a ϕ(·)∧T0∧Tb F ϕ-semi-martingale. Let Y (n) denotes the piecewise linear approximation of Y along the dyadics at level n, i.e., X and X(n) are replaced by Y and Y (n) in (19), b n 2 but with the partition (T k/2 )k=0,...,2n . Similarly, one can deﬁne Y (n)s,t. n b n n From now, we use the notation tk = T k/2 for k = 0, 1,..., 2 . The proof relies on the following lemma.

17 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

Lemma 3. For all n ∈ N and j, ` ∈ { 0,..., 2n } such that ` > j, we have

2 2 n a Y (n) n n = Y n n + [θ n n ] , tj ,t` tj ,t` tj ,t`

n a a 1 t where [θs,t] denotes the anti-symmetric part (i.e., [A] = 2 (A − A ) for a d × d-matrix A) of

n Z t X2 n n n θ = δ Y ⊗ dY with δ Y = (Y − Y n )1 n n (u). s,t u u u u tk−1 [tk−1,tk ) s k=1

n n Proof. We set s = tj and t = tk . Then, a direct computation shows that

Xk µ ¶ 2 n 1 n n Y (n) = (Y n − Y ) ⊗ ∆ Y + ∆ Y ⊗ ∆ Y . s,t t`−1 s ` ` ` `=j+1 2

As the random variable Y n − Y is F n -measurable, t`−1 s ϕ(t`−1)

Z tn n ` (Ytn − Ys) ⊗ ∆ Y = (Ytn − Ys) ⊗ dYu `−1 ` n `−1 t`−1 Z tn Z tn ` ` n = (Yu − Ys) ⊗ dYu − δ Y ⊗ dYu. n n u t`−1 t`−1 On the other hand, the Itˆoformula implies that " # Z tn s ³ ´ 1 n n ` n 1 ∆ Y ⊗ ∆ Y = δ Y ⊗ dYu + hY itn − hY itn , ` ` n u ` `−1 2 t`−1 2

s 1 t where [A] = 2 (A + A ) denotes by symmetric part of a matrix A. Since

Z t ³ ´ 2 1 Ys,t = (Yu − Ys) ⊗ dYu + hY it − hY is , s 2 the result is now clear. Final part of the proof of Proposition 2. Using the H¨older continuity of Y and (20), one get easily that for any q ≥ p and any m ∈ N,

1 q 1 q q q/p |Ys,t| + |Y (m)s,t| ≤ cqK0 |t − s| , where cq is a constant depending only on q. Besides, as Y (m)s,t = Ys,t if s = Tb i/2m, t = Tb j/2m for any i, j ∈ { 0,..., 2m },   1 1 q 0 if m ≥ n, |Y n n − Y (m) n n | ≤ ti ,ti+1 ti ,ti+1  bq/p q/p −nq/p cqT K0 2 if m < n.

18 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

ϕ We decompose Y as Y = Y0 +N +W , where N is a F -martingale and W a term of ﬁnite variation. Lemma 1 asserts that one can evaluate the p/2-variation of Y 2 −Y (m)2 if 2 2 2 2 one knows Y n n −Y (m) n n , that is the diﬀerence between Y and Y (m) tj ,tj+1 tj ,tj+1 only at the dyadic points of [0, Tb]. If m ≥ n, for any j = 0,..., 2n, there exists j0 and j00 in { 0,..., 2m } n m n m 2 2 such such t = t 0 and t = t 00 . With Lemma 3, Y n n − Y (m) n n = j j j+1 j tj ,tj+1 tj ,tj+1 m a m −[θtm,tm ] . We focus on θs,t. For m ≥ n, j0 j00  ¯ ¯  h i ¯Z tn ¯q/2 m q/2 q−1  ¯ j+1 m ¯  E |θ n n | ≤ 2 E ¯ δ Y ⊗ dNu¯ tj ,tj+1 ¯ n u ¯ tj  ¯ ¯  ¯Z tn ¯q/2 q−1 ¯ j+1 m ¯ + 2 E  ¯ δ Y ⊗ dWu¯  . ¯ n u ¯ tj We follow the arguments of the proof of Proposition 1. Using the both sides of the Burkholder-Davies-Gundy inequality and the Cauchy-Schwarz inequality, there exists a constant Cq depending only on q such that  ¯ ¯  ¯Z tn ¯q/2  ¯ j+1 m ¯  E ¯ δ Y ⊗ dNu¯ ¯ n u ¯ tj  1/2  1/2  m q   q  ≤ CqE sup |δt Y | E sup |Nt − Ntn | . n n n n j t∈[tj ,tj+1] t∈[tj ,tj+1]

Using the very deﬁnition of T0 and Y , b m p K0T sup |δt Y | ≤ m n n 2 t∈[tj ,tj+1] and that   Ã ! b q/p q K0T E  sup |N − N n |  ≤ C . t tj n n n 2 t∈[tj ,tj+1] p−1 + − p p−1 From (18), Varp,[s,t](W ) ≤ 2 Varp,[s,t](W ,W ) ≤ 2 K0|t − s| and then  ¯ ¯  Ã ! ¯Z n ¯q/2 b q/p tj+1 K T  ¯ m ¯  p−1 0 E ¯ δ Y ⊗ dWu¯ ≤ 2 . ¯ n u ¯ m/2 n/2 tj 2 2

n n m m If m < n, a direct computation shows that if [tj , tj+1] ⊂ [ti , ti+1], then n n 2 2m b 2/p 2 (tj+1 − tj ) 2 2 (K0T ) |Y (m) n n | = |Ytm − Ytm | ≤ . tj ,tj+1 m m 2 i+1 i 2n 2m/p (ti+1 − ti ) 2 2

19 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

Choose q > q0 > p. It follows that there exists a constant C depending only on p, q and q0 such that

X 2Xn−1 b q/p γ 2 q/2 (K0T ) m(1−q0+q−q/p) γ n(q0−q) n |Y (m) n n | ≤ C 0 2 sup n 2 . tj ,tj+1 1−q n>m j=0 1 − 2 n>m

As q/p > 1, if we choose q0 so that q − q0 < q/p − 1, then there exists a constant C depending only on q, q0, p and γ such that

X 2Xn−1 γ 2 q/2 q/p bq/p mδ n |Y (m) n n | ≤ CK T 2 tj ,tj+1 0 n>m j=0 with δ = 1 − q0 + q − q/p < 0. On the other hand, we have seen in the proof of Proposition 1 that

2Xn−1 h i 2 q/2 n(1−q/p) bq/p E |Y n n | ≤ C2 T . tj ,tj+1 j=0 Hence, using all these estimates and Lemma 1, one gets for m large enough, h i 2 2 q/2 E Varq/2(Y − Y (m) ) Ã ! Xm γ X γ n 1/2 q/p bq/p mδ n n 2 ≤ CK0 T 2 + (m+n)q/2p + nq/p , n=1 2 n>m 2 where the constant C depends only on p, q and γ. The Bienaym´e-Tchebitchev inequality implies that for all αm > 0 and any δ > 0 Ã ! h i 0 γ+1 2 2 C mδ m P Varq/2,[0,Tb](Y − Y (m) ) ≥ αm ≤ 2 + mq/2p + Sm αm 2 R 0 q/p bq/p P γ n−nq/p +∞ γ with C = CK0 T and Sm = n>m n 2 . Clearly, Sm ≤ m t exp(−ζt) dt −3 for ζ = p(1 − β) ln 2. Hence, we deduce easily that Sm ≤ m /3 for m large enough. Setting

2 mδ γ+1 −mq/2p −3 αm = m (2 + m 2 + m /3), we remark that αm −−−→ 0 and that m→∞ X h i 2 2 P Varq/2,[0,Tb](Y − Y (m) ) ≥ αm < +∞. m≥1

20 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

The Borel-Cantelli Lemma implies that

Var (Y 2 − Y (m)2) −−−→ 0 q/2,[0,Tb] m→∞

P-almost surely. Now, for a given trajectory, let us choose K0 and T0 large b b enough such that ϕ(T ) ≤ T and T0 ≥ T . Since

2 2 2 2 Y (m)s,t = X(m)ϕ(s),ϕ(t) and Ys,t = Xϕ(s),ϕ(t), we deduce that

2 2 2 2 Varq/2,[0,Tb](Y − Y (m) ) = Varq/2,[0,T ](X − X(m) ).

Besides, since

2 2 2 2 q/2 sup |X(m)s,t − Xs,t| ≤ Varq/2,[0,T ](X(m) − X ) , 0≤s≤t≤T

2 2 one obtains the uniform convergence of X(m)s,t to Xs,t with respect to (s, t) such that 0 ≤ s ≤ t ≤ T .

5.2 A Wong-Zakai theorem for the ItôSDEs The difference between an Itôintegral and a Stratonovich integral depends of the cross-variation of the process that is integrated. Let us recall that we have denoted the family of cross-variations of X n n n by Ξ. For t ∈ [0,T ], we set j (t) = sup { i ∈ { 0,..., 2 } si ≤ t } and

1 t − sjn(t) Ξ(n)t = (Xsjn(t)+1 − Xsjn(t) ) ⊗ (Xsjn(t)+1 − Xsjn(t) ) 2 sjn(t)+1 − sjn(t) n 1 jX(t) + (X n − X n ) ⊗ (X n − X n ). si si−1 si si−1 2 i=1

We set Ξ(n)s,t = Ξ(n)t − Ξ(n)s. We consider now the (non-geometric) multiplicative functional (X(n)1,X(n)2 + Ξ(n)).

Proposition 3. Let p, K and ϕ as above. Let X(n) be the piecewise linear −1 n approximation of X along the partition (ϕ (ϕ(T )k/2 ))k=0,...,2n . Then, the multiplicative functional (X(n)1,X(n)2 + Ξ(n)) converges almost surely in MVp([0,T ]; RN ) to (X1,X2 + Ξ) as n → ∞.

21 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

Proof. Clearly, we have only to prove that (Ξ(n))n∈N converges in q/2-variation to Ξ. The proof is similar to the one of Proposition 2. One has to work with Y instead of X. Let us define Ξ(n)Y and ΞY by a substitution of X to Y in the definition of Ξ(n) and Ξ. We use the same notations as in the proof of b Proposition 2 for K0, T , ... We assume in a first time that m ≥ n. Then, n m m n for i ∈ { 0,..., 2 } and for j < k in { 0,..., 2 } such that tj = ti and m m b −n tj − tk = T 2 ,

Y Y Y Y Ξ(m) n n − Ξ n n = Ξ(m) n n − Ξ n n ti ,ti ti ,ti+1 tj ,tk tj ,tk Xk Xk Z tn 1 ⊗2 Y 1 i = (Ytm − Ytn ) − Ξ m n = (Yr − Ytm ) ⊗ dYr. i−1 i ti−1,ti m i i=j+1 2 i=j+1 2 ti−1 And if n > m, for i ∈ { 0,..., 2n },

2m b 2/p Y Y ⊗2 2 (K0T ) |Ξ(m)tn,tn − Ξtn,tn | = |(Ytn − Ytn ) | ≤ . i i+1 i i+1 i−1 i 22n 22m/p Besides, we have already seen that

h i q/p bq/p Y q/2 K0 T E |Ξtn ,tn | ≤ . i−1 i 2nq/p We have already used these estimates in the proof of Proposition 2, and the proof of Proposition 3 is similar to the one of Proposition 2. Finally, let us remark that the relation between the Itôand the Stratonovich integrals and SDEs together with Corollary 1 allows to deduce that K((X1,X2+ Ξ)) and I((X1,X2 + Ξ)) are multiplicative functionals lying above the Itô R · integral z + 0 g(Xs) dXs and the solution Y to the Itô’s SDE Yt = y + R t 0 f(Ys) dXs, where K and I have been defined in the proof of Corollary 1. The proof of the following corollary is now clear. Corollary 2. Under the hypotheses of Proposition 2 and Hypothesis 1 on g and f, the ordinary integral

Z t XN Z t ∂gi i,j Z(n)t = z + g(X(n)s) dX(n)s + (X(n)s) dΞ (n)s 0 i,j=1 0 ∂xj and the solution of the ordinary diﬀerential equation deﬁned by

Z t Xm XN Z t j ∂fi i,k Y (n)t = y + f(Y (n)s) dX(n)s + fk (Y (n)s) dΞ (n)s 0 j=1 i,k=1 0 ∂xj converge almost surely to Z and Y deﬁned above.

22 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

References

[1] R.F. Bass, B. Hambly and T.J. Lyons. Extending the Wong-Zakai theorem to reversible Markov processes. J. Eur. Math. Soc., vol. 4, no3, pp. 237–269, 2002. . 2, 3, 5

[2] A. Bensoussan, J.L. Lions and G. Papanicolaou. Asymptotic Anal- ysis for Periodic Structures. North-Holland, 1978. 7

[3] M. Capitaine and C. Donati-Martin. The L´evy area process for the free Brownian motion. J. Funct. Anal., vol. 179, no1, pp. 153–169, 2001. . 2

[4] S. Cohen and A. Estrade. Non-symmetric approximations for manifold-valued semimartingales. Ann. Inst. H. Poincar´e Probab. Statist., vol. 36, no1, pp. 45–70, 2000. 14

[5] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields, vol. 122, no1, pp. 108–140, 2002. . 2

[6] H. Follmer¨ . Dirichlet processes. In Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980), vol. 851 of Lecture Notes in Math., pp. 476–478. Springer, Berlin, 1981.

[7] P. Friz. Continuity of the Itô-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 Vol. Math. Appl., pp. 117–135, Springer, New York, 2005. . 4

[8] P. Friz and N. Victoir. Approximations of the Brownian Rough Path with Applications to Stochastic Analysis. Ann. Inst. H. Poincar´eProbab. Statist., vol. 41, no4, pp. 703–724, 2005. . 4

[9] B. M. Hambly and T. J. Lyons. Stochastic area for Brownian motion on the Sierpinski gasket. Ann. Probab., vol. 26, no1, pp. 132–148, 1998. 5

[10] N. Ikeda and S. Watanabe. Stochastic Diﬀerential Equations and Diﬀusion Processes. North-Holland, 2nd edition, 1989. 14

23 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

[11] A. Jalubowski, J. Memin´ and G. Pages` . Convergence en loi des suites d’int´egrales stochastiques sur l’espace D1 de Skorokhod. Probab. Theory Related Fields, 81, pp. 11–137, 1989. 3

[12] J. Jacod and A.N. Shiryaev. Limit Theorems for Stochastic Processes. Springer-Verlag, 1987.

[13] T.G. Kurtz and P. Protter. Weak Convergence of Stochastic In- tegrals and Diﬀerential Equations. In Probabilistic Models for Nonlin- ear Partial Diﬀerential Equations, Montecatini Terme, 1995, edited by D. Talay and L. Tubaro, vol. 1627 of Lecture Notes in Math., pp. 1–41. Springer-Verlag, 1996. 6, 7

[14] H. Kunita. Stochastic ﬂows and stochastic diﬀerential equations. Cam- bridge University Press, 1990. 14

[15] A. Lejay. On the convergence of stochastic integrals driven by processes converging on account of a homogenization property. Electron. J. Probab., vol. 7, no18, pp. 1–18, 2002. 7, 13

[16] A. Lejay. An introduction to rough paths. In S´eminaire de Probabilit´es XXXVII, vol. 1832 of Lecture Notes in Math., pp. 1–59. Springer-Verlag Heidelberg, 2003. 2, 4, 13

[17] A. Lejay and T.J. Lyons. On the Importance of the L´evy Area for Systems Controlled by Converging Stochastic Processes. Application to Homogenization. In Current Trends in Potential Theory Conference Proceedings, Bucharest, September 2002 and 2003 edited by D. Bakry, L. Beznea, Gh. Bucur, M. Rockner¨ . The Theta Foundation, 2005. 7, 13, 14

[18] M. Ledoux, T. Lyons and Z. Qian. L´evy area of Wiener processes in Banach spaces. Ann. Probab., vol. 30, no2, pp. 546–578, 2002. 2, 5

[19] T. Lyons and Z. Qian. System Control and Rough Paths. Oxford Mathematical Monographs. Oxford University Press, 2002. 2, 4

[20] M. Ledoux, Z. Qian and T. Zhang. Large deviations and support theorem for diﬀusions via rough paths. Stochastic Process. Appl., vol. 102, no2, pp. 265–283, 2002. . 3, 4

[21] T.J. Lyons. Diﬀerential equations driven by rough signals. Rev. Mat. Iberoamericana, vol. 14, no2, pp. 215–310, 1998. 2, 4

24 L. Coutin and A. Lejay / Semi-martingales and rough paths theory

[22] J. Memin´ and L. SÃlominski´ . Condition UT et stabilitéen loi des solutions d’équations différentielles stochastiques. In Séminaire de Proba- bilitésXXV, vol. 1485 of Lecture Notes in Math., pp. 162–177. Springer- Verlag Heidelberg, 1991. 3, 6

[23] P. Protter. Approximations of solutions of stochastic diﬀerential equations driven by semimartingales. Ann. Probab., vol. 13, no3, pp. 716–743, 1985. 14, 16

[24] P. Protter. Stochastic Integration and Diﬀerential Equation, A New Approach. Applications of Mathematics, Springer-Verlag, 1990. 12

[25] A. Rozkosz and L. Slominski´ . Extended Convergence of Dirichlet Pro- cesses. Stochastics Stochastics Rep., vol. 65, no1–2, pp. 79–109, 1998.

[26] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer-Verlag, 1990. 2, 9, 10, 13, 14

[27] E.-M. Sipilainen¨ . A pathwise view of solutions of stochastic diﬀerential equations. PhD thesis, University of Edinburgh, 1993. 3, 15