Perturbed linear rough differential equations

BLAISE PASCAL

Laure Coutin & Antoine Lejay

Perturbed linear rough differential equations Volume 21, n^o1 (2014), p. 103-150.

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Perturbed linear rough differential equations

Laure Coutin Antoine Lejay

Abstract

We study linear rough differential equations and we solve perturbed linear rough differential equations using the Duhamel principle. These results provide us with a key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebras more general than tensor algebras and to consider extensions of classical results such as the Magnus and the Chen-Strichartz formula.

Équations différentielles linéaires rugueuses perturbées

Résumé

Nous étudions les équations différentielles linéaires rugueuses et résolvons des équations linéaires rugueuses perturbées à l’aide du principe de Duhamel. Ces résultats donnent un argument technique pour étudier la différentiabilité de l’ap- plication d’Itô. La notion d’équation différentielle rugueuses nous condition à consi- dérer des fonctionnelles multiplicatives à valeurs dans des algèbres de Banach plus générales que celle des algèbres tensorielles, ainsi que des extensions de résultats classiques tels que les formules de Magnus et Chen-Strichartz.

1. Introduction

Linear Rough Differential Equations (RDE) have been considered by sev- eral authors since they are an essential tool for studying the derivative of the Itô map and its flow properties (See the bibliography in [19]). How- ever, with the exception of the works of D. Feyel, A. de la Pradelle and G. Mokobodzki [27] and S. Aida [1], linear RDE have hardly been consid- ered as objects as such with their own properties, excepted to control the growth of the solution as in [28, 36]. Instead, they are generally presented

Keywords:Rough paths, Rough differential equations, Banach algebra, Magnus formula Chen-Strichartz formula, perturbation formula, Duhamel’s principle.

Math. classification:34A25, 60H10.

as a special case of RDE. However, the Baker-Campbell-Hausdorff-Dynkin formula was among the inspirations of the theory of rough paths [3, 9].

The algebraic view of solution of controlled differential equations through for example Chen-Fliess series [17] as well as some numerical simulation algorithms stem directly from the theory of linear controlled ordinary dif- ferential equations (See [7] for an overview and [28, 29, 43, 42, 18, 44] for the relationship between algebra and RDE). Linear equations are among the first examples given in the article [43] and the book [42] as motivation for developing rough paths theory.

The main goal of this article is then to study linear RDE in a general setting. For this, we define the notion of p-rough resolvent, which is an extension of the notion of multiplicative functionals [43, 42] taking their values in a Banach algebra. Chen series, and their rough paths extensions which are the core of the theory, are solutions to linear RDE taking their values in tensor spaces, for which more precise results could be given.

Our initial motivation for this article was to consider the perturbation of the Itô map. We then deal with a variation of constant/Duhamel prin- ciple for perturbed linear RDE. Yet we also extend the Chen-Strichartz formula, and the Magnus formula providing exponential representations of solutions.

The content of this article may be applied to bounded linear oper- ators on an infinite dimensional Banach space. Of course, dealing with unbounded family of operators, for example for solving Stochastic Partial Differential Equations, is much more intricate and needs specific treat- ments. The reader is referred to the quickly growing literature on these subjects [30, 29, 21, 10, 11], ... The variation of constant principle is also linked to Volterra equations which have been studied in the rough path context by A. Deya [22, 23].

In Section 3, we define for anyp≥1 the notion ofrough resolvent which is a family of linear operators giving the solutions to the linear RDE. The idea is to solve

Yt=Id+ Z t

0

YsdAs

when (A_t)_t∈[0,T] is an operator values path of finite p-variation through the constitutive relation Yr,t = Yr,sYs,t for any 0 ≤ s≤ r ≤t ≤ T with Ys,t =Y⁻¹_s Yt. For this, we find an approximation Bs,t ofYs,twhen t−sis small enough. Thesewing lemma, which is the technical core of the theory of rough paths, is then extended to transform Bs,t intoYs,t.

In Section 4, we provide a series expansion for such solutions when the solutions are represented in the formal algebra of power series. As a byproduct, we obtain an exponential representation of the p-rough re- solvents, at least for a small time. This provides us with extensions of the Magnus [7] and Chen-Strichartz formula [53]. Although these formu- lae were already known for the (fractional) Brownian motion (See e.g.

[4, 5, 12, 6]), it was not clear whether or not it could be extended easily to the case of drivers of finite p-variation as soon as p <1.

In Sections 5 and 6.1, we consider perturbed linear RDE of type Y_t=Id+

Z t 0

Y_sdA_s+B_t

and we show a variation of constant/Duhamel principle. Of course, when p∈[2,3), we need to know some appropriate extensions of bothA•andB•. In Section 6.2, we show that our notion is well adapted for the kind of linear equations which arise when one differentiates the Itô map with respect to its starting point. In a subsequent article [19], we use these properties to show that the Itô map is differentiable with a Lipschitz or Hölder continuous Fréchet derivative.

Acknowledgements. This work has been supported by the ANR Ecru (ANR-09-BLAN-0114-01/02). A. Lejay also wants to thank the Centre In- traFacultaire Bernoulli (CIB) and the NSF (National Science Foundation) for its kind hospitality during the SPDE Semester in 2012.

2. Notations and standard results

By C, we denote a constant whose value may vary from line to line.

We set K = R or C and V denotes a Banach space over K with a norm | · | and its dual V^∗.

When V = U ⊕W for two Banach spaces U and W, we denote by πU: V→U the projection onto U orthogonal to W.

We also denote byLa Banach algebra with a normk · k(See Section 2.2 below for a definition). The unit element ofLis constantly denoted byId.

2.1. Times, multiplicative properties and control Fix 0≤S < T. Let

∆⁺₂(S, T) :={(s, t)|S≤s≤t≤T},

∆⁻₂(S, T) :={(s, t)|S≤t≤s≤T},

∆⁺₃(S, T) :={(s, r, t)|S ≤s≤r ≤t≤T} and ∆⁻₃(S, T) :={(s, r, t)|S ≤t≤r≤s≤T}.

For i= 2,3, we write ∆^±_i (S, T) to denote either ∆⁺_i (S, T) or ∆⁻_i (S, T) depending on the context. WhenS = 0, we set ∆^±_i (T) := ∆^±_i (S, T).

For the sake of simplicity, a family (At)_t∈T is also denoted byA• when this notation introduces no ambiguity.

For a family (At)_t∈T of elements ofLindexed by a set T, we write A^]_• := sup

t∈T

kA_tk.

Definition 2.1. A family (A_s,t)_(s,t)∈∆±

2(T) is said to satisfy the right/left multiplicative property if

A_s,rA_r,t=A_s,t for (s, r, t)∈∆^±₃(T). (2.1) By this, we mean that a family (A_s,t)_(s,t)∈∆+

2(T) satisfies the right mul- tiplicative property if As,rAr,t = As,t for (s, r, t) ∈ ∆⁺₃(T), and a family (A_t,s)_(s,t)∈∆+

2(T) satisfies the left multiplicative property if A_t,rA_r,s =A_t,s for (s, r, t)∈∆⁺₃(T). This notational trick will be widely used below.

Definition 2.2(Control). Acontrol is a functionω: ∆⁺₂(T)→R+which is continuous close to its diagonal and such thatω(s, r) +ω(r, t)≤ω(s, t) for all (s, r, t)∈∆⁺₃(T).

We extend a control ω on [0, T]² by settingω(t, s) =ω(s, t) for (s, t)∈

∆⁺₂(T).

Given a controlω, a constant C and p≥1, we write

A• ≺Cω^1/p to mean thatkA_s,tk ≺Cω(s, t)^1/p for (s, t)∈∆^±₂(T) for a family (As,t)_(s,t)∈∆^±

2(T) of elements ofL.

Remark 2.3. Of course, sinceω(t, t) = 0 fort∈[0, T],A• ≺Cω^1/pimplies that At,t= 0 for any t∈[0, T].

A family indexed byt∈[0, T] may be transformed into a family indexed by (s, t)∈∆^±(T) satisfying the left/right multiplicative property.

Lemma 2.4. Given a family(A_t)_t∈[0,T]of invertible elements inL, we set As,t :=A⁻¹_s At for (s, t) ∈[0, T]². Thus (As,t)_(s,t)∈∆+

2(T) satisfies the right multiplicative property and (A_t,s)_(s,t)∈∆+

2(T) satisfies the left multiplicative property.

2.2. Associative algebras and Banach algebras

A Banach algebra L is a unital algebra (L,+,·) over K with a unit Id which is a Banach space with a norm k · k such that kabk ≤ kak · kbk for any a,b∈L,kλak=|λ| · kak forλ∈Kand kIdk= 1 (See [24]).

2.2.1. Inverse, exponential and logarithm.

Several operations on L may be defined using series representations. For example, when converging (for example whenka−Idk<1),

a⁻¹:=

+∞

X

i=0

(−1)^k(a−Id)^k (2.2)

is a left and right inverse of a. The exponential and logarithm maps are defined by

exp(a) =Id+

+∞

X

k=1

1

k!a^k and log(a) =

+∞

X

k=1

(−1)^k−1

k (a−Id)^k when a∈exp(L).

(2.3) In particular, a condition for the existence of the logarithm iska−Idk<1.

In this case, klog(a)k ≤ ka−Idk/(1− ka−Idk).

Lemma 2.5. For a,b∈L,

kexp(a)−exp(b)k ≤ ka−bkexp(kak+kbk), klog(a−Id)−log(b−Id)k ≤ ka−bk

kak+kbklog 1 1− kak − kbk for kak+kbk<1,

ka⁻¹−b⁻¹k ≤ kb−ak

1− ka−Idk − kb−Idk for ka−Idk+kb−Idk<1.

Proof. All these inequalities stem from the following result easily shown by recurrence kaⁿ−bⁿk ≤ ka−bk(kak+kbk)ⁿ⁻¹ for anyn≥1.

There are several kind of associative algebras and Banach algebra we consider in this article.

2.2.2. Space of operators

LetL=L(V,V) be the space of linear bounded operators on V. The norm of Lis kAk= sup_{u∈V, |u|=1}|Au|.

An operator inL(V,V) will be seen as an operator acting on the right, will an operator in L(V^∗,V^∗) is seen as an operator acting on the left.

Then V =K^d, we identifyL(V,V) with the space of matricesM_d×d(K).

2.2.3. Space of sequences

An element aof L^Z+ will be written a= (a0,a1, . . . ,). Letπk :L^Z+ → L and π≤k:L^Z+ →L^Z+ be defined

π_k(a) =a_k and π≤k(a) = (a₀, . . . ,a_k,0, . . .).

Let 1 := (Id,0, . . .). We also consider L={a∈L^Z+;π0(a) =αId, α∈ K} and we identify (α,a1,a2, . . .) with (αId,a1,a2, . . .) for a∈L.

For a subspace Xof L^Z+, we set π≤kX={π_≤k(a);a∈X}.

The space (L,+,) is an algebra with the convolution product de- fined by

π_k(ab) =

k

X

i=0

a_ibn−i fork= 0,1,2, . . . . For a∈L, setkak_sum:=^P_i≥0kπ_i(a)k.

Lemma 2.6. S={a∈L;kak_sum<+∞}, equipped with addition + and product , is a Banach algebra with unit 1.

2.2.4. Graded algebra

An associative algebra L is a graded algebra if it may be decomposed as L = K⊕L₁⊕L₂ ⊕ · · · where the L_i’s are vector spaces and ab ∈ L_i+j when a∈L_i and b∈L_j for i, j≥1. We callL_i the subspace of elements homogeneous of degree j.

When L is a graded algebra, there exists an isomorphism φ between (L,+) and (L,+) by settingπ_kφ(a) =a_k fora=a₀Id+a₁+· · · witha_k ∈ L_k. We then define a norm k · k_sum on L by setting kak_sum :=kφ(a)k_sum. Note thatkak ≤ kak_sum. We then extend naturallyπ_kandπ≤kto a graded algebra L.

On the graded algebra, we define kak_hom := sup

i≥1

{kπ_i(a)k^1/i}. (2.4)

2.2.5. Tensor algebra

Given a vector space U, we construct a tensor algebra by T(U) =K⊕U⊕(U⊗U)⊕(U⊗U⊗U)⊕ · · · with the tensor product ⊗. This is a graded algebra.

The tensor space (U)^⊗` is endowed with a norm | · |such that|a⊗b| ≤

|a| · |b|for anya∈(U)<sup>⊗`0</sup> andb∈(U)<sup>⊗(`−`0)</sup>,`⁰= 1, . . . , `−1 [24, Ex. 1.36 and 2.31]. This norm is then extended to T(U).

For the sake of notational simplicity, for k= 1, . . . ,∞, we write T_k(U) for the subset of elementsa ofπ≤kT(U) with kak_sum<+∞.

With the tensor product⊗_kdefined bya⊗_kb=π≤k(a⊗b), (T_k(U),+,⊗_k) is a Banach algebra with k · k_sum. The space T_k(U) is the quotient space T_∞(U)/∼_k witha∼_k b when π≤k(a−b) = 0. To simplify the notations, when there is no ambiguity we simply writek · kfork · k_sumand⊗for⊗_k. Remark 2.7. With φ:x ∈V7→ 1 +x∈T₁(V), we embed V into T₁(V).

Since exp(V) ={1+x∈T₁(V);x∈V}forms a group, (V,+) is isomorphic to (exp(V),⊗₁). Besides, |x| ≤ kφ(x)k_sum and kφ(x)−1k_sum= 0 implies that x= 0.

2.2.6. Algebra of words

Fix d∈N, let us consider the algebra of words Wwhose basis is {∅} ∪ {I = (i₁, . . . , i_k) with (i₁, . . . , i_k)∈ {1, . . . , d}^k, k∈N}.

The multiplication on Wis defined by the concatenation

IJ = (i1, . . . , ik, j1, . . . , j`) for I = (i1, . . . , ik) and J = (j1, . . . , jk),

and ∅I = I∅ = I. The word ∅ is the null word. For I = (i1, . . . , i_k), we write|I|=k, thelength ofI, and|∅|= 0. The algebra of words is agraded algebra K⊕W₁⊕ · · · where W_k= Span{I;|I|=k}.

When U is a finite dimensional vector space with basis {e₁, . . . , e_d}, then Wis homomorphic to T(U) throughφby setting

φ(I) =e_I:=e_i1 ⊗ · · · ⊗e_ik when I = (i₁, . . . , i_k).

More generally, for a finite familyX:={a¹, . . . ,a^d}ofdelements, called the alphabet, we write a^I := aⁱ¹· · ·a^ik forI = (i₁, . . . , i_k). This way, to X is associated T(X) =K⊕L₁⊕ · · · which is isomorphic to T(R^d) through the isomorphism defined by φ(a^I) =eI for any word I.

2.2.7. Free Lie algebra and groups

A Lie algebra l is a vector space over K with a bilinear mapping [·,·]

satisfying [a,b] = −[b,a] and the Jacobi identity [a,[b,c]] + [c,[a,b]] + [b,[c,a]] = 0 for any a,b,c∈l. For a,b∈L, [a,b] =ab−ba defines a Lie bracket [33].

Let X = {a¹, . . . ,a^d} be a finite set of d elements. A free Lie algebra on X is a Lie algebra Lie(X) and a map ı : X → Lie(X) such that for any Lie algebra g and any  : X → g there exists a unique Lie algebra morphism k:Lie(X)→g such thatk◦ı =  (Seee.g. [9, 51]). In addition, the Lie algebra Lie(X) is isomorphic to the Lie algebraLie(R^d) where for a d-dimensional vector space V with basis{e_i}_i=1,...,d,

Lie(V) :=^M

k≥1

Lie_k(V), Lie₁(V) := V and Lie_k+1(V) := Span{[a,b];a∈V,b∈Lie_k(V)}

with [ei, ej] =ei⊗e_j−e_j⊗e_i. Indeed,Lie(V) is the smallest Lie subalgebra of T(V) containing V.

Basically, in the theory of rough paths, there are three kinds of free Lie algebras that are under consideration:

• Tensor algebras and tensor Lie algebras, which allows one to de- fine rough paths, following the work of K.T. Chen on iterated integrals [14].

• Lie groups of matrices, in which live the flows of some linear dif- ferential equations [3].

• Left-invariant vector fields [8].

2.2.8. Baker-Campbell-Hausdorff-Dynkin formula

Foraandbin a Banach algebraL, the Baker-Campbell-Hausdorff-Dynkin (BCHD) formula states that under appropriate conditions, there exists c such that exp(a) expb= exp(c) with

c:=

+∞

X

j=1 j

X

n=1

X

(H,K)∈Wn

|H|+|K|=j

1

H!K!(h1+· · ·+hn+k1+· · ·+kn)Dh,k(a,b) (2.5) with H! =h₁!· · ·h_n!,

DH,K(a,b) =

h1times

z }| { [a,· · ·[a,

k1times

z }| { [b,· · ·[b,· · ·

hntimes

z }| { [a,· · ·[a,

kntimes

z }| {

[b,[· · · ,b]]]· · ·]· · ·]· · ·]]· · ·], and

Wn=ⁿ(I, J)∈(Z⁺)ⁿ; (i₁, j₁), . . . ,(i_n, j_n)6= (0,0)^o.

When this formula holds, we write a?b := c. If L is a nilpotent matrix group, then a?b is always defined. Otherwise, for a simply connected group, it holds for elements with norms small enough. A detailed discus- sion in given in [9, Sect. 5.5]. The BCHD formula stems from combinatorial considerations, so that (2.5) is formally valid in associative algebras.

Theorem 2.8 (Convergence of the BCHD formula [9, Theorem 5.54, p. 341]). For a,b ∈ L with max{kak,kbk} ≤ ¹₄log(2), the series in (2.5) is absolutely convergent. Besides, the series in (2.5) is totally convergent on any set {(a,b),kak+kbk ≤δ}, 0< δ < ¹₂log 2.

There exist several alternative representations for the series givinga?b.

We refer to [9] for a detailed account on this rich topic and various proofs.

2.2.9. Shuffle algebra and Lie elements

The shuffle algebra is the main tool to relate Lie elements and elements in tensor algebras.

Definition 2.9 (Shuffle product and shuffle algebra). For words I = (i1, . . . , i|I|) andJ = (j1, . . . , j|J|), let Sh(I, J) be the set of words of type K = (k1, . . . , k|I|+|J|) such that each letter of K corresponds exactly to one letter of I orJ and the order of the letters of I and the order of the letters in J is preserved.

Hence the shuffle product on two words is defined by linearity from I ~J = ^P_K∈Sh(I,J)K for words I and J in W, and (W,+,~) is the shuffle algebra.

Definition 2.10 (Lie element). An element x = ^P_Ie_Ix^I of T(R^d) is called a Lie element if x∅ = 0 and for each k, the homogeneous term P

I;|I|=ke_Ix^I of lengthk belongs to Lie_k(R^d).

Since T(R^d) is a unital algebra, we define inverse, exponential and log- arithm by power series using formal series given by (2.2) and (2.3). We set

gr(R^d) = exp(Lie(R^d))⊂T(R^d).

Hence, ifx is a Lie element, then exp(x)∈gr(R^d) and such an element is calledgroup like, and (gr(R^d),⊗) is a group, as exp(x)⊗exp(y) = exp(x?y) with x ? y given formally by (2.5).

Remark 2.11. Whenlis a matrix Lie algebra, then exp(l) is a matrix Lie group with respect to the product of matrices [3].

Following the work of K.T. Chen [15], R. Ree has proved the following result.

Theorem 2.12 (R. Ree [50, Theorem 2.5]). For elements x ∈ T(R^d) of type

x= 1 + ^X

I;|I|≥1

e_Iα^I, α^I∈K,

then log(x) is a Lie element if and only if the linear map φ : W → K defined by φ(∅) = 1 and φ(I) = α^I is an algebra homomorphism between (T(R^d),+,⊗)and (W,+,~), that is if and only ifα^Iα^J =^P_K∈Sh(I,J)α^K for any words I andJ.

Besides, if the coefficients satisfies the shuffle relation,

x⁻¹ = 1 +^X

k≥0

X

I;|I|=k

(−1)^kα

←− Ie_I

with ←−

I = (i_k, . . . , i₁) when I = (i₁, . . . , i_k).

Example: The Heisenberg group. The Heisenberg group provides us with a simple non trivial example of a non-commutative matrix Lie group, which is nilpotent of step 2. The Heisenberg algebra is

h:=











0 a c 0 0 b 0 0 0



 (a, b, c)∈R³





 .

The space his the Lie algebra of the Heisenberg group H:=











1 a c 0 1 b 0 0 1



 (a, b, c)∈R³







= exp(h).

The product of 3 matrices inhequals 0. Indeed,his a simply connected nil- potent Lie algebra of step 2. LetA=^{h0 1 0}0 0 0

0 0 0

i,B =^{h0 0 0}0 1 0 0 0 0

iandC=^{h0 0 1}0 0 0 0 0 0

i. Then [A, B] := AB−BA=C so that the Lie algebra his generated by the two matrices A and B, althoughhis a vector space of dimension 3.

The BCHD formula is always valid, with

exp(A) exp(B) = exp(A ? B) withA ? B=A+B+1

2[A, B] forA, B∈h.

2.3. Spaces of functions

2.3.1. Chen series and Chen-Strichartz formula

The notion of Chen series, initiated by K.T. Chen in the 50’s, provides us with a way to transform a regular path x : [0, T]→R^d into an algebraic object taking its values inT(R^d). This is one of the core ideas of the theory of rough paths to extend this theory to paths of irregular variations.

In the next statement, formula (2.7) has been given by R. Strichartz in [53] and relies on the Baker-Campbell-Hausdorff-Dynkin formula.

Theorem 2.13 (K.T. Chen [16, 14, 15], R. Strichartz [53]). Let x be a path of bounded variation with values in R^d. Then the solutions, called Chen series, to

x_t= 1 + Z t

0

x_s⊗ dx_s and y_t= 1− Z t

0

dx_s⊗y_s (2.6)

are such that for any t∈[0, T], x_t=y⁻¹_t , x_t= 1 + ^X

I;|I|≥1

e_Ix^I_0,t and y_t= 1 + ^X

I;|I|≥1

e_Ix

←− I

0,t withx^Ij_s,t= Z t

s

x_s,r^I dx^j_r. Besides, log(x_s,t) and log(y_s,t) are Lie elements in T(R^d), and for e_[I] = [e_i1,[e_i2, . . . ,[e_ik−1,eik]· · ·]],

log(x_s,t) =^X

k≥1

X

I;|I|=k

γ_I(s, t)e_[I] withγ_I(•) := ^X

σ∈Perm_k

(−1)^e(σ) k^{2 k−1}_e(σ)x^σ(I)_•

(2.7) where Perm_k is the set of permutations of {1, . . . , k},

σ(I) := (σ(i₁), . . . , σ(i_k))for I = (i₁, . . . , i_k) ande(σ) := #{j∈ {1, . . . , k−1};σ(j)> σ(j+ 1)}.

Let us consider a family A¹, . . . ,A^d of elements in B as well as a path x:R+ →R^dof bounded variation. SetAt=^Pd_i=1Aⁱxⁱ_t fort∈[0, T]. The linear equation

Yt=Id+ Z t

0

YrdAr=Id+ Z t

0

Yr d

X

i=1

Aⁱdxⁱ_r (2.8) is easily solved by considering the algebra homomorphismψ:T(R^d)→B byψ(e_I) =A^Iwith the conventions of Section 2.2.6. Indeed, forxsolution to (2.6) with x_t=^Pd_i=1e_ixⁱ_t,Y_t=ψ(x_t), t∈[0, T].

The proof of eq. 2.7 relies on the Baker-Campbell-Hausdorff-Dynkin formula, and contains it. For this, simply set x¹_t =t1_t∈[0,1]+ 11_t∈(1,2] and x²_t = (t−1)1_t∈[1,2] for t ∈ [0,2]. Then log(Y₂) = log(exp(A¹) exp(A²)) with Y• solution to (2.8) by applying the homomorphism φto (2.7).

For an operators-valued path (At)_t∈[0,T] of bounded variation and a partition {tⁿ_i}ⁿ_i=0 of [0, T], we may consider solving

Yⁿ_t =Id+ Z t

0

Y_rⁿ

n−1

X

i=0

1_r∈[tn

i,tⁿ_i+1)A_tⁿ

i,tⁿ_i+1. (2.9) This means that (2.9) may be seen as a special case of (2.8) with

d=n, ei=Atⁿ_i,tⁿ_i+1 andxⁱ_t= (t−tⁿ_i)1_t∈[tⁿ

i,tⁿ_i+1)+ (tⁿ_i+1−tⁿ_i)1_t∈[tⁿ

i,tⁿ_i+1].

With Theorem 2.12,Yt may be expressed as the exponential of operators defined as series defined in the Lie algebra containing the A_t,t ∈ [0, T].

This remains true as n→ ∞ [53].

The Magnus formula, and its variant, gives an explicit expression for this formula, and could be thought as a “continuous analogue of the Baker- Campbell-Hausdorff-Dynkin formula”. We refer to [47, 7, 48] on this topic.

2.4. Functions of finite p-variation

Forp≥1, letR^p(V) be the set of continuous paths with values in V such that |x₀| < +∞ and x• ≺ Cω^1/p, which means with our conventions of Section 2.1 and Remark 2.7, that

kxk_p:= sup

(s,t)∈∆⁺₂(T)

|x_s,t|

ω(s, t)^1/p <+∞ withx_s,t:=x_t−x_s.

Note that k · k_p is just a semi-norm, but x7→ kxk_p+|x₀|defines a norm.

An element in R^p(V) is called a function of finite p-variation controlled by ω. Under the condition that ω(s, t) = t−s, then paths in R^p(V) are α-Hölder continuous with α= 1/p.

For a∈V, we setR^p_a(V) the subset of pathsx∈ R^p(V) with x0 =a.

For θ > 1, if |x_s,t| ≤ Cω(s, t)^θ for (s, t) ∈ ∆⁺₂(T), then x is constant since for any partition {t_i}_i=0,...,n of [s, t],

|x_s,t| ≤

n−1

X

i=0

|x_ti,ti+1| ≤ω(0, T) sup

i=0,...,n−1

ω(t_i, t_i+1)^θ−1.

2.5. Rough paths

Let us consider a control ω : ∆₂(T)→R+.

Definition 2.14 (p-rough path). A p-rough path x of order ` is a path taking its values in T<sub>`</sub>(V) with

(i) The order `satisfies`≥ bpc, wherebac is the integer part of a.

(ii) For anyt∈[0, T]π0(x_t) = 1 and x_t is invertible in T_`(V).

(iii) With (2.4), kx_•k_hom≺Cω^1/p. We write

x_s,t=^X

k≥0

x^(k)_s,t with x_s,t^(k) = ^X

wordsI,|I|=k

eIx^I_s,t

with e_∅ = 1 and x^∅_s,t= 1 for the null word ∅.

We denote byRP^p_`(R^d) the set ofp-rough paths. This space is equipped with the semi-norm

kxk_p := sup

(s,t)∈∆⁺₂(T)

sup

k=1,...,`

|x^(k)_s,t| ω(s, t)^k/p

and the norm kxk_∞,p := sup_t∈[0,T]|x_t|+kxk_p.

As we can see, a rough path is an extension of a function of finite p-variation andRP^p(R^d) =R^p(R^d) forp∈[1,2).

Let us present briefly some particular classes of rough paths (See [28, 42]

for a detailed account on these notions).

• A smooth rough path is a rough path x ∈ RP^p_`(R^d) whose projection on R^d is a smooth path from [0, T] toR^dand such that

x^I_s,t= Z Z

0≤s1<···<sk≤t

dxⁱ_s,t¹₁· · · dxⁱ_s,s^k_k

for any word I = i₁· · ·i_k, k ≤ `. Such a path takes its values in π≤`gr(R^d). It is the projection onto T_`(R^d) of the Chen series of Theo- rem 2.13 above a given path.

• A geometric rough path is the closure inRP^p_`(R^d) with respectk · k∞,p

of the set of smooth rough paths.

• Aweak geometric rough pathis a rough path taking its values inG_`(R^d).

The set of such paths is denoted byWGRP^p_`(R^d).

Indeed, any path in WGRP^p_{`</sub>(R<sup>d</sup>) is the limit of a sequence of smooth rough paths in RP<sup>q</sup><sub>`}(R^d) for q > p. Besides, for p∈[2,3), non-geometric rough paths may be interpreted as geometric ones lying above a path with values in R^d⊕(R^d⊗R^d) [40]. With the appropriate algebraic structure involving trees, a similar result also holds for any value of p[32].

We refer to [43, 42, 28, 35, 37] among others for the properties of rough paths.

2.6. Young integrals

The theory of rough paths is an extension of the theory of Young integrals [42, 54, 27].

Let X, Y and Z be Banach spaces with a product (x, y) ∈ X×Y 7→

xy ∈Z, such that |xy| ≤ |x| · |y|.

Letxandybe two paths of finitep-variation controlled byωwithp <2 respectively with values in X and Y.

In the sequel, we make use of the following inequalities:

kxyk_p ≤ kxk_py_•^]+kyk_px^]_• and x^]_•≤ |x₀|+ω(0, T)^1/pkxk_p. (2.10) For any (s, t) ∈ ∆⁺₂(T), ^R_s^tyrdxr may be defined as a Young inte- gral, that is as limit of Riemann sums^Pn−1_i=0 ytⁿ_i(xtⁿ_i+1−xtⁿ_i) for partitions {tⁿ_i}ⁿ_i=0 whose meshes decrease to 0.

Let us recall some standard results about Young integrals.

Proposition 2.15. With the above setting, (i) If u∈ R^p(Z)satisfies for C≥0 and θ >1,

u0 = 0 and (ut−us−ysxs,t)_(s,t)∈∆+

2(T)≺Cω^θ, thenut=^R₀^tysdxs for t∈[0, T].

(ii) For any (s, t)∈∆⁺₂(T),

Z t s

y_rdx_r−y_s(x_t−x_s)

≤ζ(2p)kxk_pkyk_pω(s, t)^2/p (2.11) withζ(q) :=^P_n≥11/n^q, q >1.

(iii) The following control holds:

Z · 0

y_rdx_{r p}

≤(|y₀|+kyk_pω(0, T)^1/p)kxk_p+ζ(2p)kxk_pkyk_pω(0, T)^1/p. (2.12) This construction is very general. We will either use X = Y = Z = L for an algebra Ldefined as in Section 3.3, or X = Z = V and Y =L(V,V) for a Banach space V.

We present only Young integrals. From the results in Proposition 2.15, differential equations driven by finitep-variation paths withp <2 are easy to consider [38].

2.7. The Gamma function and the neo-classical inequality Let Γ(z) := ^R₀^+∞e^−tt^z−1dtbe the Gamma function [52, Chap. 6, p .76].

We use a majoration as in [1].

Lemma 2.16. Let x be a positive real and p≥1. Then for `≥ bpc, X

k≥`

x^k/p

Γ(k/p) ≤(1 +bpc) exp(x) x^b`/pc+1

Γ(b`/pc+ 1)(1∨x).

Proof. For an integer i, the set of integers k such that bk/pc = i is included in {dipe, . . . ,bp(i+ 1)c}. This set contains at most (bpc + 1) elements. Hence,

X

k≥`

x^k/p

Γ(k/p) ≤ ^X

i≥b`/pc+1

X

k∈Ns.t.bk/pc=i

xⁱ(x∨1) Γ(i+ (k/p−i)). The Γ function is increasing, so that

X

k≥`

x^k/p

Γ(k/p) ≤ ^X

i≥b`/pc

(bpc+ 1)xⁱ(x∨1)

Γ(i) ≤(bpc+ 1)(x∨1) ^X

i≥b`/pc

xⁱ Γ(i). Again with the properties of the Gamma function,

X

i≥k

xⁱ

Γ(i) ≤ x^k

Γ(k)exp(x).

since Γ(i+k)≥Γ(i)Γ(k) for i, k ≥1 [52]. Hence the result.

We give give now the so-called neo-classical inequality, proved first by T. Lyons [43] and then improved by K. Hara and H. Masanori [34].

Proposition 2.17. [Neo-classical inequality, [42, Theorem 3.1.1, p.35], [34]] For any p≥1, n∈Nand a, b≥0,

n

X

i=0

a^i/pb^(n−i)/p Γ_pⁱΓⁿ⁻ⁱ_p

≤p(a+b)^n/p Γⁿ_p

. (2.13)

3. Rough resolvent

Before introducing our main result, we present the features of linear RDE in the case p <2 which motivates our definition of a rough resolvent.

3.1. Linear RDE in the Young case

Let A : [0, T]→ Lbe a path of finite p-variation with 1 ≤p <2, that is A ∈ R^p(L).

Let us consider the equations Yt=Id+

Z t 0

YrdA_r (3.1)

and Zt=Id− Z t

dA_rZr fort∈[0, T]. (3.2) Proposition 3.1. The following properties hold:

(i) There exist unique solutionsY andZ to (3.1) and (3.2).

(ii) For some constant C and (s, t)∈∆⁺₂(T),

kY_t−Y_s−Y_sA_s,tk ≤Cω(s, t)^2/p and kZ_t−Z_s+A_s,tZ_sk ≤Cω(s, t)^2/p (3.3) and Y,Z∈ R^p_Id(L).

(iii) Any paths Y and Z in R^p_Id(L) satisfying (3.3) are solutions to equa- tions (3.1) and (3.2).

(iv) For any t∈[0, T], YtZt=ZtYt=Id.

(v) For A_s,t := Y_s⁻¹Y_t with (s, t) ∈ [0, T]², (A_s,t)_(s,t)∈∆±

2(T) satisfies the right/left multiplicative property.

(vi) For(s, t)∈∆^±₂(T),

kA_s,t−Id− A_s,tk ≤Cω(s, t)^2/p. (3.4) Proof. FixT₁≤T and consider onlyt∈[0, T₁]. SetY_t⁽⁰⁾=IdandY^(k+1)_t = Id+^R₀^tY^(k)s dA_s. From the properties of the Young integral, since p ≤ 2 and Y⁽⁰⁾ ∈ R^p_Id(L), an induction proves that Y^(k) ∈ R^p_Id(L). Besides, the Young integral is linear, so that

kY^(k+1)−Y^(k)k_p ≤ζ(2p)kY^(k)−Y^(k−1)k_pkAk_pω(0, T₁)^1/p. (3.5) For T₁ such that ζ(2p)kAk_pω(0, T₁)^1/p < 1, (Y^(k))_k∈N is a Cauchy se- quence inR^p(L) which converges inq-variation for anyq > ptoY∈ R^p(L) which is solution to (3.1).

With an inequality similar to (3.5), the solution Y to (3.1) is unique, and by linearity, the solution to y_t = a+^R₀^ty_sdA_s is given by y_t = aY_t, t∈[0, T1].

As the choice of the maximal time T1 depends only on ω, kAk_p and p, it is possible to extend the solution to [T₁, T₂] withζ(2p)kAk_pω(T₁, T₂)^1/p by solving y_t=Y_T1+^R₀^ty_sdA_T1+sfort∈[0, T₂−T₁], thenY_t=yt−T₁ for t ∈ [T1, T2], and so on... Since ω is continuous close to its diagonal, it is possible to find a finite family (T_i)_i≥0 such that ω(T_i, T_i+1) < δ for any δ >0 and Tj =T for somej≥1.

A similar reasoning applies toZ•.

Inequalities (3.3) in (ii) follow immediately from (2.11). It is then im- mediate that Y andZ belong toR^p_Id(L).

Conversely, that (3.3) characterizes uniquely (3.1) and (3.2) follows from Proposition 2.15(i).

For (s, t)∈∆⁺₂(T),

YtZt−Y_sZs = (Yt−Y_s−Y_sA_s,t)Zt−Ys(Zt−Z_s+A_s,tZs) +YsA_s,t(Zt−Z_s).

Thus t7→ Y_tZ_t is a continuous path of p/2-finite variation and then con- stant and equal to IdsinceY0 =Z0 =Id.

The multiplicative properties in (iv) are immediate from the construc- tion of As,t.

Finally, (3.4) in (vi) follows by multiplying Yt−Ys−YsA_s,t by YtY_s⁻¹

and using the fact that Y• is bounded.

3.2. Rough resolvent

Based on the previous computations, we define, using the vocabulary of differential equations, the notion of rough resolvent, which is amultiplica- tive functional [42, 43] taking its values in the Banach spaceL.

Definition 3.2 (Rough resolvent). A family (A_s,t)_(s,t)∈∆±

2(T) of elements in Lsatisfying the right/left multiplicative property (2.1) and

A•−Id≺Cω^1/p (3.6)

for some constant C is called aright/left p-rough resolvent.

Let us start with some simple properties.

Lemma 3.3 (Extension property). Assuming that on a partition 0 = T0 < T1 <· · ·< Tk =T of [0, T], we have a family of p-rough resolvents (Aⁱ_s,t)_(s,t)∈∆±

2(Ti,Ti+1). Then there is a right resolvent (A_s,t)_(s,t)∈∆±

2(T) such that A_s,t=Aⁱ_s,t for (s, t)∈∆^±₂(T_i, T_i+1), i= 0, . . . , k−1.

Proof. For s, t ∈ ∆^±₂(Ti, Ti+1), set As,t = Aⁱ_s,t. For s ∈ [Ti, Ti+1) and t∈[Tj, Tj+1) with i < j, set As,t=Aⁱ_s,Ti+1· · ·A^j_T

j,t. Then (As,t)_(s,t)∈∆+ 2(T)

satisfies the multiplicative property (2.1) and A_s,t−Id= (Aⁱ_s,Ti+1−Id)Aⁱ⁺¹_T

i+1,Ti+2· · ·A^j_T

j,t+Aⁱ⁺¹_T

i+1,Ti+2· · ·A^j_T

j,t. Iterating this procedure leads to (3.6). A similar construction holds for

left p-rough resolvent.

Lemma 3.4 (Existence of an Inverse). For any (A_s,t)_(s,t)∈∆±

2(T) that is a right/left p-rough resolvent, there exists a left/right p-rough resolvent (At,s)_(s,t)∈∆^±

2(T) such that As,tAt,s = At,sAs,t =Id, that is At,s = A⁻¹_s,t for (s, t)∈∆^±₂(T).

Proof. Using the extension property of Lemma 3.3, it is sufficient to prove the existence of an inverse ofAs,tforω(s, t) small enough, whose existence

follows from (2.2) and (3.6).

The next proposition is the converse of Lemma 2.4.

Proposition 3.5. A path (At)_t∈[0,T] ∈ R^p_Id(L) may be associated to a right/left p-rough resolvent A•.

Proof. For a right p-rough resolvent (A_s,t)_(s,t)∈∆+

2(T), A_t := A_0,t is such that A_s,t=A⁻¹_s A_t, sinceA_0,sA_s,t=A_0,t for (s, t)∈∆⁺₂(T). Besides,

kA_t−A_sk=kA_sA⁻¹_s (A_t−A_s)k ≤A^]_•kA_s,t−Idk ≤A^]_•Cω(s, t)^1/p. Thus,t7→At∈ R^p_Id(L). Similar results hold for leftp-rough multiplicative

resolvent with A_t=A_t,0,t∈[0, T].

Lemma 3.6 (Uniqueness of a p-rough resolvent). Let A• and B• be two left/right p-rough resolvents such that A•'B•. ThenA• =B•.

Proof. Let us consider that A• and B• are right p-rough resolvents. Then for f(t) :=A0,tB⁻¹_0,t,

f(t)−f(s) =A0,s(As,tB⁻¹_s,t −Id)C⁻¹_0,s and then

|f(t)−f(s)| ≤A^]_•(B⁻¹_• )^]kA_s,tB⁻¹_s,t −Idk.

Since

A_s,tB⁻¹_s,t −Id= (A_s,tB⁻¹_s,t −Id)B_s,tB⁻¹_s,t = (A_s,t−B_s,t)B⁻¹_s,t,