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El e c t ro n ic

Jo ur

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b a b i_{l i t y}

Vol. 13 (2008), Paper no. 67, pages 2031–2068. Journal URL

http://www.math.washington.edu/~ejpecp/

Delay equations driven by rough paths

A. Neuenkirch

∗

, I. Nourdin

†

and S. Tindel

‡

Abstract

In this article, we illustrate the flexibility of the algebraic integration formalism introduced in

M. Gubinelli (2004), Controlling Rough Paths, J. Funct. Anal. 216, 86-140, by establishing an existence and uniqueness result for delay equations driven by rough paths. We then apply our results to the case where the driving path is a fractional Brownian motion with Hurst parameter

H>1 3.

Key words:rough paths theory; delay equation; fractional Brownian motion; Malliavin calculus. AMS 2000 Subject Classification:Primary 60H05, 60H07, 60G15.

Submitted to EJP on November 16, 2007, final version accepted November 5, 2008.

∗_{Johann Wolfgang Goethe-Universität Frankfurt, FB 12 Institut für Mathematik, Robert-Mayer-Strasse 10, 60325} Frankfurt am Main, Germany,neuenkir@math.uni-frankfurt.de. Supported by the DFG-project "Pathwise Numerics and Dynamics for Stochastic Evolution Equations"

†_{Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Boîte courrier 188, 4 Place Jussieu,} 75252 Paris Cedex 5, France,ivan.nourdin@upmc.fr

‡_{Institut Élie Cartan Nancy (IECN), Boîte postale 239, 54506 Vandœuvre-lès-Nancy Cedex, France,}

tindel@iecn.u-nancy.fr

1

Introduction

In the last years, great efforts have been made to develop a stochastic calculus for fractional Brow-nian motion. The first results gave a rigorous theory for the stochastic integration with respect to fractional Brownian motion and established a corresponding Itô formula, see e.g. [1; 2; 3; 6; 18]. Thereafter, stochastic differential equations driven by fractional Brownian motion have been con-sidered. Here different approaches can be used depending on the dimension of the equation and the Hurst parameter of the driving fractional Brownian motion. In the one-dimensional case[17], existence and uniqueness of the solution can be derived by a regularization procedure introduced in

[21]. The case of a multi-dimensional driving fractional Brownian motion can be treated by means of fractional calculus tools, see e.g. [19; 23] or by means of the Young integral [13], when the Hurst coefficient satisfies H > 1₂. However, only the rough paths theory [13; 12]and its applica-tion to fracapplica-tional Brownian moapplica-tion[5]allow to solve fractionalSDEs in any dimension for a Hurst parameterH > 1₄. The original rough paths theory developed by T. Lyons relies on deeply involved algebraical and analytical tools. Therefore some alternative methods [8; 9] have been developed recently, trying to catch the essential results of[12]with less theoretical apparatus.

Since it is based on some rather simple algebraic considerations and an extension of Young’s integral, the method given in [9], which we call algebraic integration in the sequel, has been especially attractive to us. Indeed, we think that the basic properties of fractional differential systems can be studied in a natural and nice way using algebraic integration. (See also[16], where this approach is used to study the law of the solution of a fractionalSDE.) In the present article, we will illustrate the flexibility of the algebraic integration formalism by studying fractional equationswith delay. More specifically, we will consider the following equation:







X_t=ξ0+

Rt

0 b(Xs,Xs−r1, . . . ,Xs−rk)ds +R₀tσ(X_s,X_s−r

1, . . . ,Xs−rk)d Bs, t∈[0,T],

Xt=ξt, t∈[−rk, 0].

(1)

Here the discrete delays satisfy 0 < r1 < . . . < rk < ∞, the functions σ : Rn(k+1) → Rn,d, b : Rn(k+1) _→Rn _{are regular, B is a d-dimensional fractional Brownian motion with Hurst parameter}

H> 1₃ and the initial conditionξis aRn_{-valued weakly controlled process based onB, see Definition} 2.5. (In particularξcan be any smooth deterministic function from[₋rk, 0]toRn.) We also use, here and in the sequel, the following notation: we write Rn(k+1) _{for the set} Rn_{× · · · ×}Rn _(k+1 times), whileRn,d _{stands for the set ofn×d matrices with real entries.}

The stochastic integral in equation (1) is a generalized Stratonovich integral, which will be explained in detail in Section 2. Actually, in equations like (1), the drift term R₀tb(Xs,Xs−r1, . . . ,Xs−rk)ds is usually harmless, but causes some cumbersome notations. Thus, for sake of simplicity, we will rather deal in the sequel with delay equations of the type

¨

Xt=ξ0+

Rt

0σ(Xs,Xs−r1, . . . ,Xs−rk)d Bs, t∈[0,T],

X_t=ξt, t ∈[−rk, 0].

(2) Our main result will be as follows:

Theorem 1.1. Letσ∈C3_b(Rn(k+1)_;Rn,d_{), let B be a d-dimensional fractional Brownian motion with}

Hurst parameter H> 1₃ and letξbe aRn_{-valued weakly controlled process based on B. Then equation}

(2) admits a unique solution on[0,T]in the class of weakly controlled processes (see Definition 2.5).

Stochastic delay equations driven by standard Brownian motion have been studied extensively (see e.g. [15]and[14]for an overview) and are used in many applications. However, delay equations driven by fractional Brownian motion have been only considered so far in [7], where the one-dimensionalequation

¨

Xt =ξ0+

Rt

0 σ(Xs−r)d Bs+

Rt

0 b(Xs)ds, t ∈[0,T],

X_t =ξt, t∈[−r, 0],

(3)

is studied forH> 1₂. Observe that (3) is a particular case of equation (2).

To solve equation (2), one requires two main ingredients in the algebraic integration setting. First of all, a natural class of paths, in which the equation can be solved. Here, this will be the paths whose increments are controlled by the increments ofB. Namely, writing(δz)_st =zt−zs for the increments of an arbitrary functionz, a stochastic differential equation driven byBshould be solved in the class of paths, whose increments can be decomposed into

z_t−z_s=ζs(Bt−Bs) +ρst for 0≤s<t≤T, withζbelonging toC1γ andρbelonging toC

2γ

2 for a given 1

3 < γ <H. (Here,C µ

i denotes a space ofµ-Hölder continuous functions ofivariables, see Section 2.) This class of functions will be called the class ofweakly controlled pathsin the sequel.

To solve fractional differential equationswithoutdelay, the second main tool would be to define the integral of a weakly controlled path with respect to fractional Brownian motion and to show that the resulting process is still a weakly controlled path. To define the integral of a weakly controlled path, a double iterated integral of fractional Brownian motion, called theLévy area, will be required. Once the stability of the class of weakly controlled paths under integration is established, the differential equation is solved by an appropriate fixed point argument.

To solve fractional delay equations, we will have to modify this procedure. More specifically, we need a second class of paths, the class ofdelayed controlled paths, whose increments can be written as

zt−zs=ζs(0)(Bt−Bs) + k

X

i=1

ζ(_si)(Bt−ri−Bs−ri) +ρst for 0≤s<t≤T, where, as above, ζ(i) belongs to _C1γ for i = 0, . . . ,k, and ρ belongs to _C22γ for a given 1

3 < γ <

H. (Note that a classical weakly controlled path is a delayed controlled path with ζ(i) = 0 for

i = 1, . . . ,k.) For such a delayed controlled path we will then define its integral with respect to fractional Brownian motion. We emphasize the fact that the integral of a delayed controlled path is actually a classical weakly controlled path and satisfies a stability property.

To define this integral we have to introduce a delayed Lévy areaB2(v) of B for v _∈[₋r_k, 0]. This process with values in the space of matricesRd,d will also be defined as an iterated integral: for

1_≤i,j_≤d and 0_≤s<t_≤T we set

B2_st(v)(i,j) = Z t

s

d B_ui

Z u+v

s+v

d B_wj = Z t

s

(B_uj_+v−B_sj_+v)d◦B_ui, (4) where the integral on the right hand side is a Russo-Vallois integral[21]. Finally, the fractional delay equation (2) will be solved by a fixed point argument.

Here, we would like to stress that we prefer to use the regularization procedure introduced by Russo and Vallois[21]instead of the direct limits of sums approach for the definition of the delayed Lévy area. Malliavin calculus allows us then to verify without too much effort that (4) properly defines a delayed Lévy area forB, see Section 5, while working with limits of sums as e.g. in Coutin and Qian

[5]would certainly require more work. Moreover, we would like to mention that we restrict here to a Hurst parameterH>1/3 for sake of conciseness and simplicity. Indeed, the structures needed in order to solve the delay equation (2) for 1/4<H <1/3 are more cumbersome, and include for instance the notions of doubly delayed controlled paths and of volume elements based onB. These problems will be addressed in the forthcoming paper[22].

This article is structured as follows: Throughout the remainder of this article, we consider the general delay equation

¨

d y_t = σ(y_t,y_t−r1, . . . ,y_t−rk)d x_t, t∈[0,T],

y_t = ξt, t∈[−rk, 0],

(5)

where x isγ-Hölder continuous function withγ > 1₃ andξis a weakly controlled path based on x. In Section 2 we recall some basic facts of the algebraic integration and in particular the definition of a classical weakly controlled path, while in Section 3 we introduce the class of delayed controlled paths and the integral of a delayed controlled path with respect to its controlling rough path. Using the stability of the integral, we show the existence of a unique solution of equation (5) in the class of classical weakly controlled paths under the assumption of the existence of a delayed Lévy area. Finally, in Section 4 we specialize our results to delay equations driven by a fractional Brownian motion with Hurst parameterH> 1₃.

2

Algebraic integration and rough paths equations

Before we consider equation (5), we recall the strategy introduced in [9] in order to solve an equation without delay, i.e.,

d y_t=σ(y_t)d x_t, t_∈[0,T], y₀=α∈Rn_{, (6)} where x is aRd_{-valuedγ-Hölder continuous function withγ >}1

3.

2.1

Increments

Here we present the basic algebraic structures, which will allow us to define a pathwise integral with respect to irregular functions. For real numbers 0≤a≤ b≤T <∞, a vector spaceV and an

integer k_≥1 we denote by_Ck([a,b];V)the set of functions g:[a,b]k_→V such that g_t

1···tk =0 wheneverti= ti+1 for some 1≤i≤k−1. Such a function will be called a(k−1)-increment, and we will setC∗([a,b];V) =∪k≥1Ck([a,b];V). An important operator for our purposes is given by

δ:_Ck([a,b];V)_{→ Ck+1}([a,b];V), (δg)_t

1···tk+1= k+1

X

i=1

(₋1)k−igt1···ˆti···tk+1, (7)

whereˆt_imeans that this argument is omitted. A fundamental property ofδis thatδδ=0, whereδδ

is considered as an operator from_Ck([a,b];V)to_Ck+2([a,b];V). We will denote_{Z Ck}([a,b];V) = Ck([a,b];V)∩KerδandB Ck([a,b];V) =Ck([a,b];V)∩Imδ.

Some simple examples of actions ofδare as follows: For g _{∈ C1}([a,b];V),h_{∈ C2}([a,b];V)and

f ∈ C3([a,b];V)we have

(δg)_st =g_t−g_s, (δh)_sut =h_st−h_su−h_ut and (δf)_{suv t}= f_{uv t−}f_{sv t}+f_sut−f_suv

for anys,u,v,t_∈[a,b]. Furthermore, it is easily checked that_{Z Ck}+1([a,b];V) =B Ck([a,b];V) for anyk_≥1. In particular, the following property holds:

Lemma 2.1. Let k_≥1and h_{∈ Z Ck+1}([a,b];V). Then there exists a (non unique) f _{∈ Ck}([a,b];V)

such that h=δf .

Observe that Lemma 2.1 implies in particular that all elementsh_{∈ C2}([a,b];V)withδh=0 can be written ash=δf for some f _{∈ C1}([a,b];V). Thus we have a heuristic interpretation ofδ|C2([a,b];V): it measures how much a given 1-increment differs from being an exact increment of a function, i.e., a finite difference.

Our further discussion will mainly rely onk-increments withk≤2. For simplicity of the exposition, we will assume that V =Rm _{in what follows, althoughV could be in fact any Banach space. We} measure the size of the increments by Hölder norms, which are defined in the following way: for

f ∈ C2([a,b];V)let

kfkµ= sup s,t∈[a,b]

|f_st|

|t₋s_|µ and

C2µ([a,b];V) =

¦

f ∈ C2([a,b];V);kfkµ<∞

©

.

Obviously, the usual Hölder spacesC1µ([a,b];V)are determined in the following way: for a contin-uous functiong_{∈ C1}([a,b];V)set

kgkµ=kδgkµ,

and we will say that g ∈ C1µ([a,b];V) iff kgkµ is finite. Note that k · kµ is only a semi-norm on C1([a,b];V), but we will work in general on spaces of the type

C1,αµ ([a,b];V) =

¦

g:[a,b]_→V; ga=α,kgkµ<∞

©

Forh_{∈ C3}([a,b];V)we define in the same way kh_kγ,ρ = sup

s,u,t∈[a,b]

|h_sut|

|u−s|γ_|t−u|ρ, (8)

khkµ = inf

( X

i

khikρi,µ−ρi;(ρi,hi)i∈Nwithhi ∈ C3([a,b];V), X

i

hi=h, 0< ρi< µ

)

and

C3µ([a,b];V) =

¦

h∈ C3([a,b];V);khkµ<∞

©

.

Eventually, let_C31+([a,b];V) =_∪µ>1C3µ([a,b];V) and note that the same kind of norms can be considered on the spaces _{Z C3}([a;b];V), leading to the definition of the spaces_{Z C3}µ([a;b];V) andZ C31+([a,b];V).

The crucial point in this algebraic approach to the integration of irregular paths is that the operator

δcan be inverted under mild smoothness assumptions. This inverse is called Λ. The proof of the following proposition may be found in[9], and in a simpler form in[10].

Proposition 2.2. There exists a unique linear mapΛ:_{Z C3}1+([a,b];V)_{→ C2}1+([a,b];V)such that

δΛ =Id_{Z C}1+

3 ([a,b];V) and Λδ=IdC 1+

2 ([a,b];V).

In other words, for any h _{∈ C3}1+([a,b];V) such that δh = 0, there exists a unique g = Λ(h) _∈ C21+([a,b];V) such that δg = h. Furthermore, for any µ > 1, the map Λ is continuous from Z C3µ([a,b];V)toC

µ

2([a,b];V)and we have kΛh_kµ≤ 1

2µ₋2khkµ, h∈ Z C µ

3([a,b];V). (9) This mappingΛallows to construct a generalized Young integral:

Corollary 2.3. For any 1-increment g _{∈ C2}([a,b];V)such thatδg_{∈ C3}1+([a,b];V)setδf = (Id₋

Λδ)g. Then

(δf)_st= lim

|Πst|→0 n

X

i=0

gtiti+1

for a_≤s<t_≤ b, where the limit is taken over any partitionΠ_st=_{t₀=s, . . . ,t_n=t_}of[s,t], whose mesh tends to zero. Thus, the 1-incrementδf is the indefinite integral of the 1-increment g.

We also need some product rules for the operatorδ. For this recall the following convention: for

g_{∈ Cn}([a,b];Rl,d_{)andh∈ Cm([a,b];}Rd,p_{)let ghbe the element ofCn+m}

−1([a,b];Rl,p)defined by

(gh)_t1,...,tm+n

−1=gt1,...,tnhtn,...,tm+n−1 (10) fort₁, . . . ,t_m+n−1∈[a,b].

(i) Let g_{∈ C1}([a,b];Rl,d_{)and h∈ C1([a,b];}Rd_{). Then gh∈ C1([a,b];}Rl_)and

δ(gh) =δg h+gδh.

(ii) Let g∈ C1([a,b];Rl,d)and h∈ C2([a,b];Rd). Then gh∈ C2([a,b];Rl)and

δ(gh) =₋δg h+gδh.

(iii) Let g∈ C2([a,b];Rl,d)and h∈ C1([a,b];Rd). Then gh∈ C2([a,b];Rl)and

δ(gh) =δg h+gδh.

(iv) Let g_{∈ C2}([a,b];Rl,d_{)and h∈ C2([a,b];}Rd,p_{). Then gh∈ C3([a,b];}Rl,p_)and

δ(gh) =₋δg h+gδh.

2.2

Classical weakly controlled paths (CCP)

In the remainder of this article, we will use both the notations R_st f d g or Jst(f d g) for the inte-gral of a function f with respect to a given function g on the interval [s,t]. Moreover, we set kf_k∞ =sup_x∈Rd,l|f(x)| for a function f :R

d,l

→ Rm,n and also_kg_k∞= sup_t∈[a,b]|g_t| for a path

g_{∈ C1}κ([a,b];V). To simplify the notation we will write_Ckγ instead of_Ckγ([a,b];V), if[a,b]and

V are obvious from the context.

Before we consider the technical details, we will make some heuristic considerations about the properties that the solution of equation (6) should have. Setσˆt =σ yt

, and suppose that y is a solution of (6), which satisfies y_{∈ C1}κfor a given 1₃ < κ < γ. Then the integral form of our equation can be written as

y_t=α+ Z t

0 ˆ

σ_ud x_u, t∈[0,T]. (11)

Our approach to generalized integrals induces us to work with increments of the form (δy)_st =

y_t− y_s instead of (11). It is immediate that one can decompose the increments of (11) into

(δy)_st= Z t

s ˆ

σud xu= ˆσs(δx)st+ρst with ρst=

Z t

s

( ˆσu−σˆs)d xu.

We thus have obtained a decomposition of yof the formδy = ˆσδx+ρ. Let us see, still at a heuristic level, which regularity we can expect forσˆ andρ: Ifσis bounded and continuously differentiable, we have thatσˆ is bounded and

|σˆt−σˆs| ≤ kσ′k∞kykκ|t−s|κ,

where kykκ denotes the κ-Hölder norm of y. Hence σˆ belongs to C1κ and is bounded. As far as ρ is concerned, it should inherit both the regularities of δσˆ and x, provided that the integral

Rt

s( ˆσu−σˆs)d xu=

Rt

s(δσˆ)sud xuis well defined. Thus, one should expect thatρ∈ C 2κ

2 . In summary, we have found that a solutionδy of equation (11) should be decomposable into

δy= ˆσδx+ρ with σˆ∈ C1κ bounded and ρ∈ C 2κ

2 . (12)

This is precisely the structure we will demand for a possible solution of equation (6) respectively its integral form (11):

Definition 2.5. Let a_≤ b_≤T and let z be a path in_C1κ([a,b];Rn_{)withκ≤γand2κ+γ >1. We}

say that z is a classical weakly controlled path based on x, if z_a=α∈Rn_{andδz∈ C}κ

2([a,b];R n_)can

be decomposed into

δz=ζδx+r, i.e. (δz)_st =ζs(δx)st+ρst, s,t∈[a,b], (13)

withζ∈ C1κ([a,b];Rn,d)andρ∈ C 2κ

2 ([a,b];Rn).

The space of classical weakly controlled paths on[a,b]will be denoted by_Qκ,α([a,b];Rn_{), and a path}

z_{∈ Qκ,α}([a,b];Rn)should be considered in fact as a couple(z,ζ). The norm onQκ,α([a,b];Rn)is given by

N[z;Qκ,α([a,b];Rn)] =kδzkκ+kρk2κ+kζk∞+kδζkκ.

Note that in the above definitionαcorresponds to a given initial condition andρcan be understood as a regular part. Moreover, observe thatacan be negative.

Now we can sketch the strategy used in[9], in order to solve equation (6): (a) Verify the stability of_Qκ,α([a,b];Rn_{)under a smooth mapϕ:}Rn_→Rn,d_.

(b) Define rigorously the integralRz_ud x_u=_J(zd x)for a classical weakly controlled pathzand compute its decomposition (13).

(c) Solve equation (6) in the space_Qκ,α([a,b];Rn)by a fixed point argument.

Actually, for the second point one has to impose a priori the following hypothesis on the driving rough path, which is a standard assumption in the rough paths theory:

Hypothesis 2.6. TheRd_{-valuedγ-Hölder path x admits a Lévy area, i.e. a processx}2=_J(d x d x)_∈ C22γ([0,T];R

d,d_{), which satisfiesδx}2_=δx

⊗δx,that is

”

(δx2)_sut—(i,j) = (δxi)_su(δxj)_ut, for all s,u,t∈[0,T],i,j∈ {1, . . . ,d}. Then, using the strategy sketched above, the following result is obtained in[9]:

Theorem 2.7. Let x be a process satisfying Hypothesis 2.6 and let σ ∈ C2(Rn_;Rn,d_{) be bounded}

together with its derivatives. Then we have:

1. Equation (6) admits a unique solution y in_Qκ,α([0,T];Rn_{)for anyκ < γsuch that2κ+γ >1.}

2. The mapping (α,x,x2) _7→ y is continuous from Rn _{× C}γ

1([0,T];Rd)× C 2γ

2 ([0,T];Rd,d) to Qκ,α([0,T];Rn), in a sense which is detailed in[9, Proposition 8].

3

The delay equation

In this section, we make a first step towards the solution of the delay equation

(

d y_t=σ(y_t,y_t−r

1, . . . ,yt−rk)d xt, t∈[0,T],

yt =ξt, t∈[−rk, 0],

(14)

wherex is aRd_{-valuedγ-Hölder continuous function withγ >}1

3, the functionσ∈C

3_(Rn(k+1)_;Rn,d₎ is bounded together with its derivatives, ξis aRn_{-valued weakly controlled path based on x, and} 0<r₁<. . .<r_k<∞. For convenience, we setr₀=0 and, moreover, we will use the notation

s(y)_t= (y_t−r

1, . . . ,yt−rk), t∈[0,T]. (15)

3.1

Delayed controlled paths

As in the previous section, we will first make some heuristic considerations about the properties of a solution: setσˆt =σ(yt,s(y)_t)and suppose that y is a solution of (14) with y ∈ Cκ

1 for a given 1

3< κ < γ. Then we can write the integral form of our equation as (δy)_st=

Z t

s ˆ

σud xu= ˆσs(δx)st+ρst with ρst=

Z t

s

( ˆσu−σˆs)d xu.

Thus, we have again obtained a decomposition of yof the formδy = ˆσδx+ρ. Moreover, it follows (still at a heuristic level) thatσˆ is bounded and satisfies

|σˆt−σˆs| ≤ kσ′k∞

k

X

i=0 |y_t−r

i −ys−ri| ≤(k+1)kσ

′_k

∞kykκ|t−s|κ.

Thus, with the notation of Section 2.1, we have that σˆ belongs to_C1κ and is bounded. The term

ρshould again inherit both the regularities ofδσˆ and x. Thus, one should have thatρ∈ C22κ. In conclusion, the incrementδy should be decomposable into

δy= ˆσδx+ρ with σˆ∈ C1κ bounded and ρ∈ C 2κ

2 . (16)

This is again the structure we will ask for a possible solution to (14). However, this decomposition does not take into account that equation (14) is actually a delayequation. To define the integral

Rt

s σˆud xu, we have to enlarge the class of functions we will work with, and hence we will define a

delayed controlled path(hereafterDCPin short).

Definition 3.1. Let0≤a≤b≤T and z∈ C1κ([a,b];Rn)with 1

3< κ≤γ. We say that z is a delayed

controlled path based on x, if z_a=αbelongs toRn_{and ifδz∈ C}κ

2([a,b];Rn)can be decomposed into (δz)_st=

k

X

i=0

ζ(_si)(δx)_s−r

i,t−ri +ρst for s,t∈[a,b], (17)

whereρ∈ C22κ([a,b];R

n_)andζ(i)

∈ C1κ([a,b];R

n,d_{)for i=0, . . . ,k.}

Dκ,α([a,b];Rn)should be considered in fact as a(k+2)-tuple(z,ζ(0), . . . ,ζ(k)).

The norm onDκ,α([a,b];Rn)is given by

N[z;Dκ,α([a,b];Rn)] =kδzkκ+kρk2κ+ k

X

i=0

kζ(i)k∞+

k

X

i=0

kδζ(i)kκ.

Now we can sketch our strategy to solve the delay equation:

1. Consider the map Tσdefined onQκ,α([a,b];Rn)× Qκ, ˜α([a−rk,b−r1];Rn)by

(T_σ(z, ˜z))_t=σ(z_t,s(z˜)_t), t∈[a,b], (18)

where we recall that the notation s(˜z) has been introduced at (15). We will show that

Tσ maps Qκ,α([a,b];Rn)× Qκ, ˜α([a −rk,b− r1];Rn) smoothly onto a space of the form Dκ,αˆ([a,b];Rn,d).

2. Define rigorously the integral Rzud xu = J(zd x) for a delayed controlled path z ∈ Dκ,αˆ([a,b];Rn,d), show that J(zd x) belongs to Qκ,α([a,b];Rd), and compute its decom-position (13). Let us point out the following important fact: Tσ creates “delay”, that is

Tσ(z, ˜z)∈ Dκ,αˆ([a,b];Rn,d), whileJ creates “advance”, that isJ(zd x)∈ Qκ,α([a,b];Rn). 3. By combining the first two points, we will solve equation (14) by a fixed point argument on

the intervals[0,r₁],[r₁, 2r₁], . . . .

3.2

Action of the map

T

on controlled paths

The major part of this section will be devoted to the following two stability results:

Proposition 3.2. Let 0 ≤ a ≤ b ≤ T , let α, ˜α be two initial conditions in Rn _{and let ϕ ∈}

C3(Rn(k+1)_;Rl_{) be bounded with bounded derivatives. Define Tϕ on Qκ,α([a;b];}Rn_{)× Qκ, ˜α([a−}

rk;b−r1];Rn)by Tϕ(z, ˜z) = ˆz, with ˆ

z_t=ϕ(z_t,s(z˜)_t), t∈[a,b].

Then, settingαˆ=ϕ(α,s(˜z_a)) =ϕ(α, ˜z_a

−r1, . . . , ˜za−rk−1, ˜α), we have Tϕ(z, ˜z)∈ Dκ,αˆ([a;b];R

l_{)and it}

admits a decomposition of the form

(δzˆ)_st = ˆζs(δx)st+ k

X

i=1 ˆ

ζ(_si)(δx)_{s−ri,t−ri}+ ˆρst, s,t∈[a,b], (19)

whereζˆ,ζˆ(i)are theRl,d_{-valued paths defined by} ˆ

ζs=

‚ ∂ ϕ ∂x1,0

(z_s,s(˜z)_s), . . . , ∂ ϕ ∂xn,0

(z_s,s(˜z)_s) Œ

ζs, s∈[a,b],

and

ˆ ζ(_si)=

‚ ∂ ϕ ∂x1,i

(z_s,s(z˜)_s), . . . , ∂ ϕ ∂xn,i

(z_s,s(˜z)_s) Œ

˜

for i=1, . . . ,k.Moreover, the following estimate holds:

N[ˆz;_Dκ,aˆ([a;b];Rl)] (20)

≤cϕ,T

€

1+_N2[z;Qκ,α([a,b];Rn)] +N2[˜z;Qκ, ˜α([a−rk,b−r1];Rn)]

Š

,

where the constant cϕ,T depends onlyϕand T .

Proof. Fixs,t_∈[a,b]and set

ψ(_si)= ‚

∂ ϕ

∂x_1,i(zs,s(˜z)s), . . . ,

∂ ϕ

∂x_n,i(zs,s(˜z)s)

Œ

. fori=0, . . . ,k. It is readily checked that

(δˆz)_st = ϕ(z_t−r0, ˜z_t−r1, . . . , ˜z_t−rk)₋ϕ(z_s−r0, ˜z_s−r1, . . . , ˜z_s−rk) = ψ(_s0)ζs(δx)st+

k

X

i=1

ψ(_si)ζ˜s−ri(δx)s−ri,t−ri + ˆρ

(1) st + ˆρ

(2) st ,

where ˆ

ρ(st1) = ψ(s0)ρst+ k

X

i=1

ψ(_si)ρ˜s−ri,t−ri, ˆ

ρ(_st2) = ϕ(z_t−r

0, ˜zt−r1, . . . , ˜zt−rk)−ϕ(zs−r0, ˜zs−r1, . . . , ˜zs−rk) −ψ(_s0)(δz)_st−

k

X

i=1

ψ(_si)(δ˜z)_s−r i,t−ri.

(i)We first have to show thatρˆ(1),ρˆ(2)∈ C22κ([a,b];R

l_{). For the second remainder term Taylor’s} formula yields

|ρˆ(_st2)| ≤ 1 2kϕ

′′_k ∞

|(δz)_st|2+ k

X

i=1

|(δ˜z)_s−r i,t−ri|

2_,

and hence clearly, thanks to some straightforward bounds in the spaces_Qκ,α, we have |ρˆ_st(2)|

|t−s|2κ ≤ 1 2kϕ

′′_k ∞

N2[z;Qκ,α([a,b];Rn)] + k

X

i=1

N2[˜z;Qκ,α([a−ri,b−ri];Rn)]

. (21) The first term can also be bounded easily: it can be checked that

|ρˆ(st1)|

|t−s|2κ ≤ kϕ′k∞

N ”ρ;_C22κ([a,b];Rn)—+ k

X

i=1

N ”ρ˜,_C22κ([a−ri,b−ri];Rn)

—

. (22) Putting together the last two inequalities, we have shown that decomposition (19) holds, that is

(δzˆ)_st =ψ_s(0)ζs(δx)st+ k

X

i=1

ψ(_si)ζ˜(_si₋)_r

withρˆ_st= ˆρ(_st1)+ ˆρ_st(2)∈ C22κ([a,b];Rl).

(ii)Now we have to consider the “density” functions ˆ

ζ_s=ψ(_s0)ζ_s, ζˆ(_si)=ψ_s(i)ζ˜_s−ri, s∈[a,b].

Clearlyζˆ,ζˆ(i)are bounded on[a,b], because the functionsψ(i)are bounded (due to the bounded-ness ofϕ′) and becauseζ, ˜ζ(i)_{are also bounded. In particular, it holds}

sup s∈[a,b]|

ˆ

ζs| ≤ kϕ′k∞ sup

s∈[a,b]|

ζs|, sup

s∈[a,b]|

ˆ

ζ(_si)| ≤ kϕ′k∞ sup

s∈[a,b]| ˜

ζs−ri| (23) fori=1, . . . ,k. Moreover, fori=1, . . . ,k, we have

|ζˆ(_si)

1 −

ˆ ζ(_si)

2|

≤ |(ψ(_si)

1 −ψ (i) s2)

˜

ζs1−ri|+|(ζ˜s1−ri−ζ˜s2−ri)ψ

(i) s2|

≤ kϕ′′k∞|zs1−zs2| sup s∈[a,b]

|ζ˜_s−ri|+_kϕ′′k∞

k

X

j=1

|z˜_s1−rj−˜z_s2−rj| sup s∈[a,b]

|ζ˜_s−ri| +_kψ(i)k∞|ζ˜s1−ri−ζ˜s2−ri|

≤ kϕ′′k∞N[z;C1κ([a,b];R

n_{)] sup} s∈[a,b]|

˜

ζs−ri| |s2−s1|

κ ₍₂₄₎

+_kϕ′′k∞

k

X

j=1

N[˜z;_C1κ([a₋r_j,b₋r_j];Rn_{)] sup} s∈[a,b]|

˜

ζs−ri| |s2−s1| κ

+_kψ(i)k∞N[ζ˜;C1κ([a−ri,b−ri];Rn,d)]|s2−s1|κ. Similarly, we obtain

|ζˆs1−

ˆ

ζs2| ≤ kϕ

′′_k

∞N[z;C1κ([a,b];R

n_{)] sup} s∈[a,b]|

ζs| |s2−s1|κ (25)

+_kϕ′′k∞

k

X

j=1

N[˜z;_C1κ([a₋r_j,b₋r_j];Rn_{)] sup} s∈[a,b]|

ζs| |s2−s1|κ +_kψ(0)k∞N[ζ;C1κ([a,b];R

n,d_)]

|s₂₋s_1|κ. Hence, the densities satisfy the conditions of Definition 3.1.

(iii)Finally, combining the estimates (21) – (25) yields the estimate (20), which ends the proof.

We thus have shown that the mapT_ϕ is quadratically bounded inzand ˜z. Moreover, for fixed ˜zthe mapTϕ(·, ˜z):Qκ,α([a;b];Rn)→ Dκ,αˆ([a;b];Rl)is locally Lipschitz continuous:

Proposition 3.3. Let the notations and assumptions of Proposition 3.2 prevail. Let0_≤a≤b≤T , let z(1),z(2)∈ Qκ,α([a,b];Rn)and let˜z∈ Qκ, ˜α([a−rk,b−r1];Rn). Then,

N[Tϕ(z(1), ˜z)−Tϕ(z(2), ˜z);Dκ,0([a;b];Rl)] (26) ≤cϕ,T 1+C(z(1),z(2), ˜z)

2

where

C(z(1),z(2), ˜z) =_N[˜z;_{Qκ, ˜α}([a₋r_k,b₋r₁];Rn_{)] (27)} +_N[z(1);_Qκ,α([a,b];Rn_{)] +N[z}(2)_{;Qκ,α([a,b];}Rn_)]

and the constant c_ϕ,T depends only onϕ and T .

Proof. Denoteˆz(j)=Tϕ(z(j), ˜z)for j=1, 2. By Proposition 3.2 we have

€ δzˆ(j)Š

st = ˆζ (j)

s (δx)st+ k

X

i=1 ˆ

ζ(_si,j)(δx)_s−r

i,t−ri+ ˆρ

(j)

st , s,t∈[a,b], with

ˆ

ζ_s(j)=ψ(_s0,j)ζ(_sj), ζˆ_s(i,j)=ψ(_si,j)ζ˜s−ri, s∈[a,b], where

ψ(_si,j) = ‚

∂ ϕ ∂x1,i

(z_s(j),s(˜z)_s), . . . , ∂ ϕ ∂xn,i

(z_s(j),s(˜z)_s) Œ

, s_∈[a,b],

fori=0, . . . ,k, j=1, 2. Furthermore, it holdsρˆ_st(j)= ˆρ_st(1,j)+ ˆρ_st(2,j), where

ˆ

ρ_st(1,j)=ψ_s(0,j)ρ_st(j)+ k

X

i=1

ψ(_si,j)ρ˜s−ri,t−ri, ˆ

ρ(_st2,j)=ϕ(z_t(₋j)_r

0, ˜zt−r1, . . . , ˜zt−rk)−ϕ(z

(j)

s−r0, ˜zs−r1, . . . , ˜zs−rk) −ψ(_s0,j)(δz(j))_st−

k

X

i=1

ψ(_si,j)(δ˜z)_s−r i,t−ri. Thus, we obtain forˆz= ˆz(1)₋ˆz(2)the decomposition

(δˆz)_st= k

X

i=0 ˆ

ζ(_si)(δx)_{s−ri,t−ri}+ ˆρst

with ζˆ(_s0) = ψ(_s0,1)ζ_s(1)−ψ(_s0,2)ζ(_s2), the paths ζˆ(i) are defined by ζˆ(_si) = (ψ_s(i,1)−ψ(_si,2))ζ˜s−ri for

i=1, . . . ,k, andρˆst = ˆρ (1) st −ρˆ

(2) st .

In the following we will denote constants (which depend only on T and ϕ) by c, regardless of their value. For convenience, we will also use the short notations N[z˜], N[z(1)], N[z(2)] and N[z(1)₋z(2)]instead of the corresponding quantities in (26) and (27).

(i)We first control the supremum of the density functionsζˆ(i),i=0, . . . ,k. Fori=0 we can write ˆ

ζ_s(0)=ψ_s(0,1)(ζ(_s1)−ζ_s(2)) + (ψ(_s0,1)−ψ_s(0,2))ζ(_s2)

and thus it follows

|ζˆ(_s0)| ≤ kϕ′k∞|ζ(s1)−ζ (2)

s |+kϕ′′k∞|zs(1)−z (2) s ||ζ

(2)

s | (28)

Similarly, we get

|ζˆ(_si)| ≤cN[˜z]_N[z(1)−z(2)]. (29)

(ii)Now, consider the increments of the density functions. Here, the key is to expand the expression

ψ_s(i,1)−ψ_s(i,2)fori=0, . . . ,k. For this define

u_s(r) =r(z_s(1)−z_s(2)) +z_s(2), r∈[0, 1], s∈[a,b]. We have

∂ ϕ ∂x_p,i(z

(1)

s ,s(z˜)s)−

∂ ϕ ∂x_p,i(z

(2)

s ,s(˜z)s) =

∂ ϕ

∂x_p,i(us(1),s(˜z)s)−

∂ ϕ

∂x_p,i(us(0),s(˜z)s)

= θ_s(p,i)(z_s(1)₋z_s(2)), where

θ_s(p,i)= Z 1

0

‚

∂2ϕ ∂x1,0∂xp,i

(u_s(r),s(z˜)_s), . . . ,

∂2ϕ ∂xn,0∂xp,i

(u_s(r),s(˜z)_s) Œ

d r. Hence it follows

ψ_s(i,1)−ψ(_si,2)=€θ_s(1,i)(z_s(1)₋z_s(2)), . . . ,θ_s(n,i)(z_s(1)₋z(_s2))Š. (30) Note thatθ(p,i)is clearly bounded and, under the assumptionϕ∈C_b3, it moreover satisfies

|θt(p,i)−θs(p,i)| ≤c

€

N[z(1)] +_N[z(2)] +_N[˜z]Š_|t₋s_|κ. (31) Fori=0 we can now write

ˆ

ζ(t0)−ζˆ(s0)= (ψ (0,1)

t −ψ(s0,1))(ζ (1) s −ζ

(2) s ) +ψ

(0,1) t ((ζ

(1) t −ζ

(2)

t )−(ζ(s1)−ζ (2) s )) + (ψ(_s0,1)−ψ(_s0,2))(ζ(t2)−ζ(s2)) +ζ

(2) t ((ψ

(0,1)

t −ψ

(0,2)

t )−(ψ(s0,1)−ψ (0,2) s )). It follows

|ζˆ(_t0)−ζˆ(_s0)| ≤c€N[z(1)] +_N[˜z]Š_|t−s|κN[z(1)−z(2)] +cN[z(1)−z(2)]_|t−s|κ +cN[z(1)−z(2)]_N[z(2)]_|t−s|κ

+_N[z(2)]_|(ψ(_t0,1)−ψ_t(0,2))₋(ψ(_s0,1)−ψ(_s0,2))_|. (32) Using (30) and (31) we obtain

|(ψ(_t0,1)−ψ(_t0,2))₋(ψ(_s0,1)−ψ(_s0,2))_| (33) ≤c€1+_N[z(1)] +_N[z(2)] +_N[z˜]Š_N[z(1)₋z(2)]_|t₋s_|κ.

Combining (32) and (33) yields |ζˆ(t0)−ζˆ(s0)| ≤c

€

By similar calculations we also have

|ζˆ(_ti)−ζˆ(_si)| ≤c€1+_N[z(1)] +_N[z(2)] +_N[˜z]Š2_N[z(1)−z(2)]_|t−s|κ (35) fori=1, . . . ,k.

(iii)Now, we have to control the remainder termρˆ. For this we decomposeρˆas ˆ

ρst =ρ (1) st +ρ

(2) st , where

ρ_st(1)=ψ(_s0,1)ρ_st(1)−ψ(_s0,2)ρ(_st2)+ k

X

i=1

€

ψ(_si,1)−ψ(_si,2)Šρ˜s−ri,t−ri,

ρ(_st2)=

ϕ(z(_t1₋)_r

0, ˜zt−r1, . . . , ˜zt−rk)−ϕ(z

(1)

s−r0, ˜zs−r1, . . . , ˜zs−rk)

−

ϕ(z(_t2₋)_r

0, ˜zt−r1, . . . , ˜zt−rk)−ϕ(z

(2)

s−r0, ˜zs−r1, . . . , ˜zs−rk)

−€ψ(_s0,1)(δz(1))_st−ψ(_s0,2)(δz(2))_stŠ₋ k

X

i=1

€

ψ(_si,1)−ψ(_si,2)Š(δ˜z)_{s−ri,t−ri}.

We consider firstρ(1)_{: for this term, some straightforward calculations yield}

|ρ(_st1)| ≤c(1+_N[z(1)] +_N[z(2)] +_N[˜z])_N[z(1)−z(2)]_|t−s|2κ. (36) Now considerρ(2). The mean value theorem yields

ρ_st(2)=€ψ¯_s(0,1)−ψ(_s0,1)Š(δz(1))_st−€ψ¯(_s0,2)−ψ_s(0,2)Š(δz(2))_st +

k

X

i=1

€€

¯

ψ_s(i,1)−ψ¯_s(i,2)Š−€ψ(_si,1)−ψ(_si,2)ŠŠ(δz˜)_{s−ri,t−ri} =€ψ¯(_s0,1)−ψ(_s0,1)Š(δ(z(1)₋z(2)))_st

+€€ψ¯(_s0,1)−ψ¯_s(0,2)Š−€ψ_s(0,1)−ψ(_s0,2)ŠŠ(δz(2))_st +

k

X

i=1

€€

¯

ψ_s(i,1)−ψ¯(_si,2)Š−€ψ_s(i,1)−ψ(_si,2)ŠŠ(δ˜z)_s−r i,t−ri

¬_Q1+Q2+Q3, with

¯

ψ(_si,j)= Z 1

0

‚ ∂ ϕ ∂x1,i

(v_s(j)(r)), . . . , ∂ ϕ

∂xn,i

(v_s(j)(r)) Œ

d r,

v_s(j)(r) =

z(_sj)+r(z_t(j)₋z_s(j)), ˜z_s−r1+r(˜z_t−r1−˜z_s−r1), . . . , ˜z_s−r

k+r(˜zt−rk−˜zs−rk)

. We shall now boundQ1,Q2 andQ3 separately: it is readily checked that

and thus we obtain

Q1≤c

€

1+_N[z(1)] +_N[z(2)] +_N[˜z]Š_N[z(1)−z(2)]_|t−s|2κ. (37) In order to estimateQ₂andQ₃, recall that by (30) in part(ii)we have

ψ_s(i,1)−ψ(_si,2)=€θ_s(1,i)(z_s(1)−z_s(2)), . . . ,θ_s(n,i)(z_s(1)−z(_s2))Š, (38) where

θ_s(p,i)= Z 1

0

‚

∂2ϕ

∂x_1,0∂x_p,i(us(r

′_),s(z˜)

s), . . . ,

∂2ϕ

∂x_n,0∂x_p,i(us(r

′_),s(z˜)

s)

Œ

d r′,

u_s(r′) =z_s(2)+r′(z_s(1)₋z_s(2)). Similarly, we also obtain that

¯

ψ_s(i,1)−ψ¯(_si,2)=€θ¯_s(1,i)(z_s(1)−z_s(2)), . . . , ¯θ_s(n,i)(z_s(1)−z(_s2))Š+q_st(i) (39) with

¯

θ_s(p,i) = Z 1

0

Z 1

0

‚

∂2ϕ

∂x_1,0∂x_p,i(u¯s(r,r

′_{)), . . . ,} ∂

2_ϕ

∂x_n,0∂x_p,i(u¯s(r,r

′₎₎ Œ

d r d r′, ¯

u_s(r,r′) = v_s(2)(r) +r′€v_s(1)(r)₋v_s(2)(r)Š and

|q(_sti)_{| ≤}c_N[z(1)₋z(2)]_|t₋s_|κ. Now, using (38) and (39) we can write

€

¯

ψ(_si,1)−ψ¯(_si,2)Š−€ψ(_si,1)−ψ(_si,2)Š

=€(θ¯_s(1,i)−θ_s(1,i))(z(1)−z(2)), . . . ,(θ¯_s(n,i)−θ_s(n,i))(z(1)−z(2))Š+q_st(i)

for anyi=0, . . . ,k. Since moreover ¯

us(r,r′)−(us(r′),s(˜z)_s)

=r

(z(_t2)−z_s(2)) +r′(z(_t1)−z_s(1)−(z_t(2)−z_s(2))), ˜zt−r1−z˜s−r1, . . . , ˜zt−rk−˜zs−rk

, another Taylor expansion yields

|θ¯_s(p,i)−θ_s(p,i))_{| ≤}c€N[z(1)] +_N[z(2)] +_N[˜z]Š_|t−s|κ. Hence, we obtain

¯ ¯ ¯ €

¯

ψ(_si,1)−ψ¯(_si,2)Š−€ψ(_si,1)−ψ(_si,2)Š ¯ ¯ ¯

≤c€N[z(1)] +_N[z(2)] +_N[˜z]Š_N[z(1)−z(2)]_|t−s|κ, (40) from which suitable bounds forQ₂andQ₃ are easily deduced. Thus it follows by (37) and (40) that

|ρ(st2)| ≤c

€

1+_N[z(1)] +_N[z(2)] +_N[z˜]Š2 _N[z(1)₋z(2)]_|t₋s_|2κ. Combining this estimate with (36) we finally have

|ρst| ≤c

€

1+_N[z(1)] +_N[z(2)] +_N[z˜]Š2 _N[z(1)₋z(2)]_|t₋s_|2κ. (41)

3.3

Integration of delayed controlled paths (DCP)

The aim of this section is to define the integralJ(m∗d x), where m is a delayed controlled path

m_{∈ D}κ,αˆ([a,b];Rd). Here we denote byA∗the transposition of a vector or matrixAand byA1·A2 the inner product of two vectors or two matricesA₁ andA₂, namely, ifA₁,A_2∈Rn,d_{, thenA1·A2=} Tr A1A∗2

. We will also writeQκ,α. We will also writeQκ,α (resp. Dκ,α) instead ofQκ,α([a,b];V) (resp._Dκ,α([a,b];V)) if there is no risk of confusion about[a,b]andV.

Note that if the increments ofmcan be expressed like in (17),m∗admits the decomposition (δm∗)_st =

k

X

i=0

(δx)∗_s−r i,t−riζ

(i)∗

s +ρst∗, (42)

whereρ∗∈ C22κ([a,b];R

1,d_{)and the densitiesζ}(i)_{,i=0, . . . ,k, satisfy the conditions of Definition} 3.1.

To illustrate the structure of the integral of a DCP, we first assume that the paths x,ζ(i) andρare smooth, and we expressJ(m∗d x)in terms of the operatorsδandΛ. In this case,J(m∗d x)is well defined, and we have

Z t

s

m∗_ud x_u=m∗_s(x_t−x_s) + Z t

s

(m∗_u−m∗_s)d x_u

fora_≤s_≤t_≤b, or in other words

J(m∗d x) =m∗δx+_J(δm∗d x). (43) Now consider the termJ(δm∗d x): Using the decomposition (42) we obtain

J(δm∗d x) = Z t

s k

X

i=0

(δx)∗_s−r i,u−riζ

(i)∗

s +ρsu∗

!

d x_u=A_st+_Jst(ρ∗d x) (44) with

A_st= k

X

i=0

Z t

s

(δx)∗_s−r i,u−riζ

(i)∗

s d xu.

Since, for the moment, we are dealing with smooth paths, the densityζ(i) can be taken out of the integral above, and we have

Ast= k

X

i=0

ζ(_si)·x2_st(₋ri), with thed×d matrixx2_st(v)defined by

x2_st(v) = Z t

s

Z u+v

s+v

d x_w

!

d x_u(1), . . . ,

Z t

s

Z u+v

s+v

d x_w

!

d x_u(d)

!

, 0_≤s_≤t_≤T, forv_{∈ {−}r_k, . . . ,₋r_0}. Indeed, we can write

Z t

s

(δx)∗_s−r i,u−riζ

(i)∗

s d xu =

Z t

s

ζ(_si)·[(δx)_{s−ri,u−ri⊗}d x_u] = ζ(_si)·

Z t

s

(δx)_s−r

i,u−ri ⊗d xu=ζ

(i) s ·x

2

where B1, . . . ,Bd are d independent one-dimensional fBms, i.e. each Bi is a centered Gaussian process with continuous sample paths and covariance function

R_H(t,s) =1 2

€

|s_|2H+_|t_|2H_{− |}t₋s_|2HŠ (79) fori=1, . . . ,d. The fBm verifies the following two important properties:

(scaling) For anyc>0,B(c)=cHB_·/cis a fBm, (80) (stationarity) For anyh_∈R_{,B·+h−Bhis a fBm. (81)} Notice that, for Malliavin calculus purposes, we shall assume in the sequel that B is defined on a complete probability space(Ω,_F,P), and that_F =σ(B_s;s∈R).

5.2

Malliavin calculus with respect to fBm

Let us give a few facts about the Gaussian structure of fractional Brownian motion and its Malliavin derivative process, following Section 2 of[18]. Let_E be the set of step-functions onR _{with values} inRd. Consider the Hilbert space _H defined as the closure of_E with respect to the scalar product induced by

¬

(1[t1,t1_], . . . ,1_[t

d,td]),(1[s1,s1], . . . ,1[sd,sd])

¶

H =

d X

i=1

R_H(ti,si)₋R_H(ti,s_i)₋R_H(t_i,si) +R_H(t_i,s_i)

,

for any−∞<s_i <si < +_∞and −∞< t_i < ti <+_∞, and where R_H(t,s) is given by (79). The mapping

(1_[t

1,t1], . . . ,1[td,td])7→

d X

i=1 Bi

ti −B

i ti

can be extended to an isometry between _H and the Gaussian space H₁(B) associated with B = (B1, . . . ,Bd). We denote this isometry byϕ7→B(ϕ). Let_S be the set of smooth cylindrical random variables of the form

F= f(B(ϕ1), . . . ,B(ϕk)), ϕi∈ H, i=1, . . . ,k,

where f _∈C∞(Rd k_,R_{)is bounded with bounded derivatives. The derivative operatorDof a smooth} cylindrical random variable of the above form is defined as the_H-valued random variable

DF=

k X

i=1

∂f

∂xi

(B(ϕ1), . . . ,B(ϕk))ϕi.

This operator is closable fromLp(Ω)intoLp(Ω;_H). As usual,D1,2denotes the closure of the set of smooth random variables with respect to the norm

kFk21,2 = E|F| 2_+E

In particular, if DiF denotes the Malliavin derivative ofF _∈D1,2with respect toBi, we haveDiB_tj=

δi,j1[0,t]fori,j=1, . . . ,d.

The divergence operator I is the adjoint of the derivative operator. If a random variable u ∈ L2(Ω;_H) belongs to dom(I), the domain of the divergence operator, then I(u) is defined by the duality relationship

E(F I(u)) =E_〈DF,u_〉H, (82)

for everyF∈D1,2_{. Moreover, let us recall two useful properties verified byDandI:}

• Ifu∈dom(I)andF ∈D1,2 such thatF u∈L2(Ω;_H), then we have the following integration by parts formula:

I(F u) =F I(u)_{− 〈}DF,u_〉H. (83)

• If uverifies Ekuk2

H +EkDuk2H ⊗H < ∞, Dru∈ dom(I) for all r ∈R and (I(Dru))r∈R is an

element ofL2(Ω;_H), then

DrI(u) =ur+I(Dru). (84)

5.3

Delayed Lévy area and fractional Brownian motion

The stochastic integrals we shall use in order to construct our delayed Lévy area are defined, in a natural way, by Russo-Vallois symmetric approximations, that is, for a given processφwe have

Z t

s

φwd◦Bwi =L

2 −lim

ǫ→0(2ǫ) −1

Z t

s

φw Bw+i ǫ−B

i w−ǫ

d w,

provided the limit exists. This pathwise type notion of integral can be related to some stochastic analysis criterions in the following way (for a proof, see[1]):

Theorem 5.1. Fix t_≥0and letφ∈D1,2_{(H)be a process such that} Tr[0,t]DB

i

φ:=L2−lim ǫ→0(2ǫ)

−1

Z t

0

〈DBiφu,1[u−ǫ,u+ǫ]〉Hdu exists, and such that, settingℓ(t,u) =u2H−1+ (t₋u)2H−1 _for0≤u<t,

Z t

0

Eφ_u2ℓ(t,u)du+

Z t

0

Z t

0

E(DB_riφu)2ℓ(t,u)dud r<∞.

ThenRt 0φwd

◦_Bi

w exists and verifies Z t

0

φ_wd◦B_wi =IBi(φ1[0,t]) +Tr[0,t]DB

i

φ. With these notations in mind, the main result of this section is the following:

Proposition 5.2. Let B be a d-dimensional fractional Brownian motion and suppose H > 1₃. Then almost all sample paths of B satisfy Hypothesis 3.4.

Proof. A natural candidate to be the delayed Lévy area (provided it is well-defined) is given, for any v∈ {−rk, . . . ,−r0}, by:

B2_st(v) =

Z t

s

d◦Bu⊗ Z u+v

s+v

d◦Br, i.e.B2st(v)(i,j) = Z t

s

d◦B_ui

Z u+v

s+v

d◦B_rj, i,j∈ {1, . . . ,d}. (85)

WhenH= 1

2, the desired conclusion is easily obtained, because the Russo-Vallois symmetric integral coincides with the Stratonovich integral.

WhenH > 1₂, the Russo-Vallois symmetric integral coincides with the Young integral, which is well defined in this case, and the assertion still follows easily from the properties of Young integrals. When 1₃ < H < 1₂, by Theorem 5.1, we shall obtain (see below) the almost sure existence of the iterated integral (85) for fixed integration bounds, but unfortunately not as a process. In order to bypass this difficulty, and rather than trying to prove the uniform version of Theorem 5.1, our strategy actually consists in definingB2(v)by (85) only for rationaltimes and then to consider its continuous version (which well exists, sinceB2(v)will verify the Kolmogorov criterion).

Now, let us proceed with the proof in the case whereH _∈(1₃,1₂) is fixed. It is a classical fact that B∈ C1γ([0,T];Rd)for any

1

3 < γ <H. Due to the stationarity property (81) we will work without loss of generality on the interval[0,t₋s]instead of[s,t]in the sequel.

1) Case i=j. When v=0, it is readily checked that E|B2_st(0)(i,i)_|2=1

4E|Bt−Bs| 4₌3

4|t−s| 4H_.

Let us now consider the case where v < 0. For φ = (B_·i_{+v −} Bi_v)1_[0,t−s](_·), the conditions of Theorem 5.1 are easily verified, hence R₀t−sφ_ud◦B_ui exists. Notice moreover that we have DB_riφu = 1[v,u+v](r)1[0,t−s](u) and, for u ∈ [0,t−s] and ǫ ∈[0,−v] (which is always the case,

for a fixedv<0 andǫsmall enough) it holds 〈1[v,u+v],1[u−ǫ,u+ǫ]〉H =

1 2

€

|v+ǫ|2H− |v₋ǫ|2H+_|v₋u₋ǫ|2H− |v₋u+ǫ|2HŠ = 1

2

€

(₋v−ǫ)2H₋(₋v+ǫ)2H+ (₋v+u+ǫ)2H₋(₋v+u−ǫ)2HŠ. Thus, we obtain

Tr[0,t−s]DB

i

φ=₋H(₋v)2H−1(t₋s) +1 2

€

(t−s−v)2H₋(₋v)2HŠ.

Forx≥0, it is well-known that 0≤((₋v)+x)2H_−(−v)2H_≤2H(−v)2H−1_{x. Applying this inequality} to the second term of the right hand side of Tr[0,t−s]DB

i

φwe get

¯ ¯

¯Tr[0,t−s]D Bi_φ¯¯

¯≤H(−v)

2H−1_(t

−s). (86)

On the other hand, we have by (84) DBi

r I

Bi_{(φ) =φ}

r+IBi(DBriφ) = φr+IBi(1[r−v,+∞)∩[0,t−s])

When₋v_≥t₋s, then[r₋v,+_∞)_∩[0,t₋s] =_;for anyr_∈[0,t₋s]. By using (82) we deduce E|IBi(φ)_|2=Ekφk2_H =EkB_·i_+v−B_vik2_H([0,t−s])

=E_kBik2_H([0,t−s])= (t−s)

4H_E

kBik2_H([0,1]), (88)

where the two last equalities are due to the stationarity (81) and scaling (80) properties of fractional Brownian motion.

When−v<t−s, then IBi₍₁

[r−v,+∞)∩[0,t−s]) = (Bit−s−B i

r−v)1[0,t−s+v](r). (89)

We deduce

E_|IBi(φ)_|2

=E_〈DIBi(φ),φ〉H by (82) =E_kφk2_H([0,t−s])+E〈I

Bi₍₁

[r−v,∞)∩[0,t−s]),φ〉H([0,t−s]) by (87)

=Ekφk2_H([0,t−s])+E〈(B i t−s−B

i

·−v)1[0,t−s+v],φ〉H([0,t−s]) by (89)

≤E_kφk2_H([0,t−s])+E €

k(Bi_t−s−Bi_·−v)1_{[0,t−s+v]kH([0,t−s])k}φkH([0,t−s]) Š

≤3 2Ekφk

2

H([0,t−s])+

1 2Ek(B

i t−s−B

i

·−v)1[0,t−s+v]k2_H([0,t−s]) becausea b≤

1 2(a

2_+b2₎ =3

2(t−s) 4H_E

kBi_k2_H([0,1])+ 1 2Ek(B

i

t−s+v−B i₎₁

[0,t−s+v]k2_H([0,t−s]) by (80) and (81)

=3 2(t−s)

4H_E

kBi_k2_H([0,1])+ 1

2(t−s+v) 2H_E

k(B_ti_−s+v−Bi_(t+s−v)·)_k2_H([0,1]) =3

2(t−s) 4H_E

kBi_k2_H([0,1])+ 1

2(t−s+v) 4H_E

k(B₁i ₋Bi)_k2_H([0,1]) by (80) ≤1

2(t−s) 4H_3E

kBik2_H([0,1])+Ek(B i

1−B

i₎

k2_H([0,1])

. (90)

Finally, we can summarize (88) and (90) in

E_|IBi(φ)_|2_≤c_H|t₋s_|4H,

with a constantcH >0, which is in particular independent ofv. Putting together this last estimate

with inequality (86), we end up with

E|B2_st(v)(i,i)_|2_≤cH(1+|v|2H−1)|t−s|4H

for anyv_{∈ {−}r_k, . . . ,₋r_1}.

2) Case where i₆= j. By stationarity (81), we have for anyv_{∈ {−}r_k, . . . ,₋r_0}that B_u+vj −B_vj,B_ui

u∈[0,t−s]

L

= B_uj,Bi_u

u∈[0,t−s].

Thus, the delayed Lévy areaB2_0,t−s(v)(i,j) =R₀t−s(B_u+vj ₋Bvj)d◦Bui forv <0 behaves as in the case

where v =0. But it is a classical result that B2_0,t−s(0)is well-defined for H >1/3 (see e.g. [20]). Moreover, it follows again by the stationarity (81) and the scaling (80) properties that

Immediately, we deduce that

E_|B2_st(v)(i,j)_|2_≤cH|t−s|4H

for anyv∈ {−rk, . . . ,−r0}.

Both in the casesi= jandi₆= jthe substitution formula for Russo-Vallois integrals easily yields that

δB2(v) =δB(v)_⊗δB. Furthermore, sinceB2(v) is a process belonging to the second chaos of the fractional Brownian motionB, on which allLp norms are equivalent forp>1, we get that

E|B2_st(v)(i,j)_|p_≤cp|t−s|2pH (91)

fori6= jand

E|IBi_(φ)|p_≤c

p|t−s|2pH (92)

wheni= j. In order to conclude thatB2(v)_{∈ C2}2γ(Rd×d)for any1

3< γ <Handv∈ {−rk, . . . ,−r0}, let us recall the following inequality from[9]: Letg∈ C2(V)for a given Banach spaceV. Then, for anyκ >0 andp_≥1 we have

kg_kκ≤c€U_κ+2/p;p(g) +_kδg_kκŠ with U_γ;p(g) =

Z T

0

Z T

0

|gst|p

|t−s|γpdsd t !1/p

. (93)

By plugging inequality (91)-(92) into (93), by recalling thatδB2_{(v) =δB(v)⊗δB and (86) hold,}

we obtain thatB2(v)(i,j)_{∈ C2}2γ(Rd×d_{)for any}1

3 < γ <H andi,j=1, . . . ,d.

ƒ

Acknowledgements.The authors are grateful to the anonymous referees for a careful and thorough reading of this work and also for their valuable suggestions.

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