Large deviation principle for enhanced gaussian processes

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Large deviation principle for enhanced Gaussian processes

Peter Friz

^∗

, Nicolas Victoir

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK Received 10 January 2006; received in revised form 2 November 2006; accepted 22 November 2006

Available online 27 February 2007

Abstract

We study large deviation principles for Gaussian processes lifted to the free nilpotent group of stepN. We apply this to a large class of Gaussian processes lifted to geometric rough paths. A large deviation principle for enhanced (fractional) Brownian motion, in Hölder- or modulus topology, appears as special case.

Résumé

Nous etudions les principes de grandes deviations pour les « rough paths Gaussiens », pour une large classe de processus Gaus- siens. Cette classe contient le mouvement brownien fractionnaire.

MSC:60F10; 60G15; 60H99

Keywords:Rough paths; Large deviation principle; Gaussian processes; Fractional Brownian motion

1. Introduction

We prove large deviation principles for d-dimensional Gaussian processes lifted to the free nilpotent group of stepN. The example we have in mind is a class of zero mean Gaussian processes subject to certain conditions on the covariance, [5,17].

After recalls on nilpotent groups and Wiener Itô chaos, we give simple conditions under which a large deviation principle holds. To illustrate the method, we quickly check these conditions for enhanced Brownian motion (see [15,10] for earlier approaches). We then apply our methodology to the class of enhanced Gaussian processes considered in [5,17]. Enhanced fractional Brownian motion appears as a special case and we strengthen the results in [18,4].

* Corresponding author.

E-mail address:P.K.Friz@statslab.cam.ac.uk (P. Friz).

doi:10.1016/j.anihpb.2006.11.002

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1.1. The free nilpotent group and rough paths

LetN0. The truncated tensor algebra of degreeN is given by the direct sum T^N

R^d

=R⊕R^d⊕ · · · ⊕ R^d_⊗N

.

With the usual scalar product, vector addition, and tensor product⊗the spaceT^N(R^d)is an algebra. Letπ_i denote the canonical projection fromT^N(R^d)onto(R^d)^⊗ⁱ.

Letq∈ [1,2)andx∈C₀^q-var([0,1],R^d), the space of continuousR^d-valued paths of boundedq-variation started at the origin. The liftS_N(x):[0,1] →T^N(R^d)is defined as

S_N(x)_t=1+ N

i=1

0<s₁<···<s_i<t

dxs₁⊗ · · · ⊗dxs_i,

the iterated integrals are Young integrals. Then G^N

R^d

=

g∈T^N R^d

: ∃x∈C¹-var 0

[0,1],R^d

: g=SN(x)1

is a submanifold ofT^N(R^d)and, in fact, a Lie group with product⊗, called the free nilpotent group of stepN. The dilation operatorδ:R×G^N(R^d)→G^N(R^d)is defined by

π_i δ_λ(g)

=λⁱπ_i(g), i=0, . . . , N,

and a continuous norm onG^N(R^d), homogeneous with respect toδ, is given g =inf

length(x): x∈C¹-^var

0

[0,1],R^d

, S_N(x)₁=g . By equivalence of such norms there exists a constantK_Nsuch that

1 KN

i=max1,...,N

π_i(g)^{1/ i}gK_N max

i=1,...,N

π_i(g)^{1/ i}. (1.1)

The norm · induces a metric onG^N(R^d)known as Carnot–Caratheodory metric, d(g, h)= g⁻¹⊗h .

Letx, y∈C₀([0,1], G^N(R^d)), the space of continuousG^N(R^d)-valued paths started at the neutral element of (G^N(R^d),⊗). We define a supremum distance,

d_∞(x, y)= sup

t∈[0,1]d(x_t, y_t), a Hölder distance,α∈ [0,1],

d_α-Hölder(x, y)= sup

0s<t1

d(x_s⁻¹⊗x_t, y_s⁻¹⊗y_t)

|t−s|^α , write

xα-Hölder= sup

0s<t1

x_s⁻¹⊗xt

|t−s|^α = sup

0s<t1

d(xs, xt)

|t−s|^α ,

andLipinstead of 1-Hölder. Similarly, one definesp-variation regularity and distance forG^N(R^d)-valued paths. We refer to [11] for a more detailed discussion of these topics.

The interest for all this comes from T. Lyons’ rough path theory, [16,17], a deterministic theory of control differential equations (“Rough Differential Equation”), driven by a continuousG^[^p^](R^d)-valued path of finitep-variation.

Brownian motion and Lévy’s area can be viewed as aG^[^p^](R^d)-valued path of finitep-variation withp >2 and the corresponding RDE solution turns out to be a classical Stratonovich SDE solution. This gives a general recipe how to make sense of differential equations driven by anarbitrarystochastic processXwith continuous sample path of finite p-variation: find an appropriate lift ofXto aG^[^p^](R^d)-valued pathXwith finitep-variation use Lyons’ machinery.

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1.2. Gaussian process

The Wiener spaceC₀([0,1],R^d)is the space of continuous function started at 0; the sup norm induces a topology and hence a Borelσ-algebra. We assume there is a probability measurePsuch that the coordinate processX, defined by X(ω)_t =ω(t ), is a centered Gaussian process with covariancec(s, t ). We also defineHto be the reproducing kernel Hilbert space associated toP, i.e. the closure of the set of the linear span of the functionss→c(ti, s), under the Hilbert product

c(t_i,·), c(t_j,·)

=c(t_i, t_j).

Letidenote the canonical injection from the reproducing kernel Hilbert space to the Wiener space. We defineCn(B) the B-valued homogeneous Wiener chaos of degreen, where(B,| · |B)is a given Banach space. We also define C_n(B)the sum of the firstnhomogeneous Wiener chaos. We refer to [14,1–3] for more details on Gaussian spaces and Wiener chaos.

1.3. Enhanced Gaussian process

We make the assumption that there exists a lift of the Gaussian processXto a processXwith values inG^N(R^d) for some fixedN 1. Such lifts have been constructed first by Coutin and Qian [5] as a.s. limits inp-variation of (canonically lifted) dyadic piecewise linear approximations ofX. We shall be less specific here and extract those properties of the approximations that we need in the sequel. We make the following

Assumption.There exists a sequence of continuous linear maps{Φ_m}fromC([0,1],R^d)(with uniform topology) onto the space of bounded variation path from[0,1]intoR^d(with 1-variation topology) such that

(1) limm→∞Φ_m(x)=xfor all continuous pathsxwith respect to uniform topology;

(2) the uniform distance betweenSN◦Φm(X)andXconverges to 0 in probability.

In fact, Condition (2) is equivalent to the seemingly stronger condition ofL^p-convergence.

Lemma 1.Convergence of d_∞(S_N◦Φ_m(X),X)to zero in probability is equivalent to convergence inL^p for all p∈ [1,∞).

Proof. Theorem III.2 in [20] and Eq. (1.1). 2

Remark 1.Assumption 2 is always satisfied whenN=1. In the applications discussed in later sections,N will be related to the regularity ofX.

Example 1.Let{Φ_m}be the piecewise linear approximations based on a sequence of dissections{D_m}with mesh

|D_m| →0. IfXdenotes fractional Brownian motion of parameterH >¹₄, then the liftXforN= [1/H]and was first constructed in [5], see also [4].

Remark 2.As is well known (e.g. Theorem 1 in [13]) any continuous Gaussian processX on[0,1]is the uniform limit of

X^(m)_t ≡ m

j=1

ξ_j(ω)ψ_j(t )

where{ψ_j}is an orthonormal basis for the reproducing Kernel Hilbert spaceHand the{ξ_j}are i.i.d. standard Gaussian given by the image ofψ_j under the isometric isomorphismθ:H→L2(X_t: t∈ [0,1]), the closure of{X_t: [0,1]}in L²(P). The Hilbert structure ofHimplies that for allh∈H,

h^(m)≡ m

j=1

ψj, hψj→h inH

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and hence uniformly. The construction ofXcan be based on such (or similar) approximations,¹see [6,11,19,8], but they do not satisfy our assumption. It may be possible to adapt the subsequent proofs, aimed to establish a large deviation principle forX, to such approximations. As this requires further work without improving the results we shall not pursue this further here.

2. Wiener chaos andG^N(R^d)-valued paths

The hypercontractivity of the Ornstein–Uhlenbeck semigroup leads to a useful Lemma 2.LetZ_nbe a random variable inCn(B). Assume1< p < q <∞. Then

Z_nL^q(P;B) q−1

p−1 n/2

Z_nL^p(P;B).

Proof. LetP_t denote the Ornstein–Uhlenbeck semigroup. Whenever 1< p < q <∞andt >0 satisfies e^t

q−1 p−1

1/2

, (2.1)

then, for allf ∈L^p(P;B), P_tfL^q fL^p. We also recall that

PtZn=e⁻^ntZn.

See [14] for this two results. We now chooset such that equality holds in (2.1) and find q−1

p−1 ₋n/2

Z_nL^q =e⁻^ntZ_nL^q = P_tZ_nL^q Z_nL^p. 2

It is well known thatL^p(P;B)- andL^q(P;B)-norms are equivalent on Cn(B)=C1(B)+ · · · +Cn(B).

The next lemma quantifies this equivalence.

Lemma 3.Letn∈NandZbe a random variable inC_n(B). Assume2pq <∞. Then there exists a constant Mnsuch that

ZL^p(P;B)ZL^q(P;B)M_n(q−1)^n/2ZL^p(P;B). Proof. Only the second inequality requires a proof. WriteZ=_n

i=0Z_i withZ_i∈Ci. From Lemma 2, for allin, Z_iL^q(P;B)

q−1 p−1

i/2

Z_iL^p(P;B). (2.2)

The sumCn(B)=C1(B)+ · · · +Cn(B)is topological direct inL⁰(P;B), see [3, p. 6] for instance, which implies that the projectionZ→Z_i is continuous inL²(P;B). It follows thatZ_iL²(P;B)cZL²(P;B)for some constant c=c(n). Then, forp2,

Z_iL^p(P;B)(p−1)î/2Z_iL²(P;B)c(p−1)î/2ZL²(P;B)c(p−1)î/2ZL^p(P;B). (2.3) By Hölder’s inequality for finite sums,

|Z|^q_Bn^q⁻¹ n

i=0

|Zi|^q_B,

1 At least for,N2 it is shown in [6] that reproducing kernel – and piecewise linear approximations yield the same lifted process.

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and after taking expectations, ZL^q(P;B)n¹⁻^1/q

_n

i=0

Z_i^q_L^q₍_P;_B) 1/q

. Hence, from (2.2) and (2.3),

ZL^q(P;B)cn¹⁻^1/q _n

i=0

q−1 p−1

qi/2

Z_i^q_L^p₍_P;_B) 1/q

cn¹⁻^1/q _n

i=0

(q−1)^qn/2Z^q_Lp(P;B)

1/q

cn(q−1)^n/2ZL^p(P;B). 2

Corollary 1.LetX, Y be two continuousG^N(R^d)-valued processes such that, for eachi=1, . . . , N, the projection t→πi(Xt), πi(Yt)∈

R^d_⊗i

belongs toC_i(B_i)=C0(B_i)⊕C1(B_i)⊕ · · · ⊕Cn(B_i), where B_i=C₀

[0,1], R^d_⊗i

andCj(B_i)is theB_i-valued homogeneous Wiener chaos of degreej. Assume2< q <∞. Then there exists a constant M_Nsuch that

d_∞(X, Y )

L^2N(P) d_∞(X, Y )

L^qN(P)√

q M_N d_∞(X, Y )

L^2N(P). Proof. Again, only the second inequality requires a proof. Fori=1, . . . , Ndefine

Z_i:t→π_i

X_t⁻¹⊗Y_t

∈ R^d_⊗i

.

Observe thatZ_i∈C_i(B_i)⊂C_N(B_i).Let| · |_∞denote the supremum norm on(R^d)^⊗ⁱ-valued paths. From Eq. (1.1), 1

KN

i=max1,...,N|Z_i|^{1/ i}_∞ d_∞(X, Y )K_N max

i=1,...,N|Z_i|^{1/ i}_∞. (2.4)

Therefore, for allq2, 1

KN

i=max1,...,NZ_i^{1/ i}_LqN/ i(P) d_∞(X, Y )

L^qN(P)K_N max

i=1,...,NZ_i^{1/ i}_LqN/ i(P). (2.5)

From Lemma 3

Z_i^{1/ i}_LqN/ i(P)M_N^{1/ i}

qN/ i−1Z_i^{1/ i}_L2N/ i(P)

√

qM_N Z_i^{1/ i}_L2N/ i(P)

√

qM_N d_∞(X, Y )

L^2N(P)

where we used (2.5) withq=2 in the last line. Another look at (2.5) finishes the proof. 2 3. Large deviation results for(δεX)ε>0

From general principles,(εX)ε>0satisfies a large deviation principle with good rate functionIin uniform topology, whereI is given by (see [14] for example):

I (y)= 1

2x²_H wheny=i(x)for somex∈H, +∞ otherwise.

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It is clear that S_N◦Φ_m:

C

[0,1],R^d ,| · |_∞

→C

[0,1], G^N R^d

, d_∞

is continuous. By the contraction principle [7],S_N(εΦ_m(X))satisfies a large deviation principle with good rate function

J_m(y)=inf

I (x), xsuch thatS_N Φ_m(x)

=y ,

the infimum of the empty set being+∞. Essentially, a large deviation principle forδεXis obtained by sendingmto infinity. To this end we need

Lemma 4.Letδ >0fixed. Then

mlim→∞lim

ε→0ε²logP d_∞

S_N

Φ_m(εX) , δ_εX

> δ

= −∞.

Proof. First observe that d_∞

SN

Φm(εX) , δεX

=εd_∞ SN

Φm(X) ,X

. By standing assumption onXand Lemma 1 we have

α_m:= d_∞ S_N

Φ_m(X) ,X

L^2N −→

m→∞0.

Then, by Corollary 1 , d_∞

S_N Φ_m(X)

,X

L^qM_N√

q α_m ∀q2N. (3.1)

We then estimate P

d_∞ S_N

Φ_m(εX) , δ_εX

> δ

=P

d_∞ S_N

Φ_m(X) ,X

>δ ε

δ

ε ₋q√

q^qα^q_m

exp

qlog

ε δαm√

q

, and after choosingq=1/ε²we obtain, forεsmall enough,

ε²logP

t∈D

d S_N

Φ_m(εX)

t, δ_εXt

> δ

log α_m

δ

.

Now take the limits limε→0and limm→∞to finish the proof. 2

The following theorem is a straight forward application of the extended contraction principle [7, Theorem 4.2.23]

and Lemma 4. The point is that althoughS_N is a only a measurable map, defined as a.s. limit, it has exponentially good approximations given by{S_N◦Φ_m: m1}.

Theorem 1.Assume thatSN(h),defined as the pointwise limit of(SN◦Φm)(h).whenm→ ∞exists forhsuch that I (h) <∞(i.e. allhin the Cameron–Martin space), and that for allΛ >0,

mlim→∞ sup

{h:I (h)Λ}d_∞

(S_N◦Φ_m)(h), S_N(h)

=0. (3.2)

Then the family(δεX)ε>0satisfies a large deviation principle in uniform topology with good rate function J (y)=inf

I (x), xsuch thatS_N(x)=y .

The following corollary allows us to strengthen the topology.

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Corollary 2.Under the above assumptions and the additional condition

∃α>0, c >0:E

expcX²_α-Hölder

<∞,

the family (δ_εX)ε>0 satisfies a large deviation principle in α-Hölder topology for all α∈ [0, α)with good rate functionJ (y).

Proof. By the inverse contraction principle [7, Theorem 4.2.4.] it suffices to show that the laws of(δεX)are exponentially tight inα-Hölder topology. But this follows from the compact embedding of

C^α^-Hölder

[0,1], G^N R^d

⊂⊂C^α-Hölder

[0,1], G^N R^d and Gauss tails ofXα-Hölder. Indeed, for everyM >0,

KM=

x: xα-HölderM

defines a precompact set w.r.t.α-Hölder topology. Then ε²log

(δεX)_∗P K_M^c

=ε²logP

δεXα-Hölder > M

=ε²logP

Xα-Hölder> M/ε

−M². 2

Remark 3.The same proof applies to the (weaker)p-variation topology and the (stronger) modulus topology consid- ered in [10].

We emphasise the generality of this approach: a large deviation result for an enhanced Gaussian process holds provided (a) the enhanced Gaussian process exists and (b) a condition on the reproducing Kernel of the Gaussian process.

As a warm-up we show how to apply this to enhanced Brownian motion. We will then discuss a wider class of Gaussian process, which contains fractional Brownian motion with Hurst parameterH >¹₄.

4. LDP for enhanced Brownian motion

LetX=Bis bed-dimensional Brownian motion. We takeΦmbe the piecewise linear approximation over at the dyadic dissection of[0,1]with mesh 2⁻^m. Then S2(Φm(B))converges uniformly on[0,1]and in probability to a processB, see [17,9,10]). Large deviations for(δ_εB)εwere first obtained inp-variation topology [15], then in Hölder and modulus topology [10]. The following proof is considerably quicker. There are only two things to check.

(1) Gauss tails ofBα-Hölder for anyα∈ [0,1/2). There are many different ways to see this. In [10] we use the scalingd(Bs,Bt)= |^D t−s|^1/2B0,1, noting thatB0,1has Gauss tails, followed by an application of the Garsia–

Rodemich–Rumsey lemma. One could prove the result directly by extending Lemma 1 toα-Hölder metric.

(2) Condition (3.2). But this is a simple consequence of the fact that the 1-variation norm ofhis bounded by√ 2I (h) and that the 1-variation ofΦ_m(h)is bounded by the 1-variation ofh. From Theorem 1 and Corollary 2 we deduce Theorem 2.LetB=B(ω)∈C([0,1], G²(R^d))denote enhanced Brownian Motion. Letα∈ [0,1/2). Then(δ_εB)ε>0

satisfies a large deviation principle inα-Hölder topology with good rate function J (y)=inf

I (x), xsuch thatS_N(x)=y .

When restrictingαtoα∈(1/3,1/2)the universal limit theorem gives the Freidlin–Wentzell theorem (inα-Hölder topology) as corollary. See [15,10,4] for details on this by now classical application of Lyons’ limit theorem.

Remark 4.The case of enhanced fractional Brownian motion,H > ¹₄, is similar. First, Gauss tails of theα-Hölder norm, for anyα∈ [0, H ), follow from scaling just as above. (This has already been used in [11,4].) Secondly, condition (3.2) follows from the fact that the Cameron–Martin space for fBM is embedded in the space of finiteq-variation for

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anyq less than 1/(1/2+H ), see [12]. In any case, enhanced fractional Brownian motion is a special case of the processes considered in the next section.

5. LDP for a class of Gaussian processes

We fix a parameterH >0. We now specialise to zero-meanR^d-valued Gaussian processes{X_t: t∈ [0,1]} with independent componentsx=Xⁱ, i=1, . . . , d, satisfying

E

|x_s,t|²

c|t−s|^2H, for alls < t, (5.1)

E(x_s,s₊_hx_t,t₊_h)c|t−s|^2H h

t−s

², for alls, t, hwithh < t−s (5.2)

for some constantc >0. Note that such a process has a.s. 1/p-Hölder continuous sample paths when p >1/H.

A particular example of such a Gaussian process is fractional Brownian motion of Hurst parameterH. This condition is the one which allowed Coutin and Qian [5] to construct the lift ofX to a rough path. As beforeHdenotes the Cameron–Martin space. We takeΦ_mto be the piecewise linear approximation over at the dyadic dissections of[0,1]

with mesh 2⁻^m.

Definition 1.ForH >1/4,we define, according to [5], the enhanced Gaussian processXto be the (a.s. and inL^qfor allq) limit in 1/p-Hölder norm ofS_[_p_](Φ_m(X)), where 1/p < H.²

To obtain the large deviation principle for(δεX)ε>0there are, as in the case of EBM, only two things to check.

(1) Gauss tails ofX1/p-Hölder forp >1/H,that is Eexp

αX²1/p-Hölder

<∞. (5.3)

First note that for allq1, sup

0s<t1

E

Xs,t

|t−s|^H q

<∞. Similar to Corollary 1, for alls, t,andk1,

E Xs,t

|t−s|^H

^2k1/2k

C√

k E Xs,t

|t−s|^H ^q1/q

,

whereq is any fixed real greater than 2[1/H]. Now expand the exponential in a power series to see that there existsα >0 s.t.Eexp(αXs,t²/|t−s|^2H) <∞for alls, t. The proof of 5.3 is then finished using the Garsia–

Rodemich–Rumsey lemma, just as in [10].

(2) Condition (3.2). We show (a) that for some q ∈ [1,2) the q-variation norm of a Cameron–Martin path is bounded by a constant times its Cameron–Martin norm and (b) a uniform modulus of continuity for elements in{h: I (h)Λ}. A soft argument, spelled out in Theorem 3 below, will then imply (3.2).

We start by proving that the quadratic variation of Cameron–Martin paths converges exponentially fast to 0.

Lemma 5.LetH∈(1/4,1). Then there exists a constantC_H,csuch that for alln0and allh∈H

2ⁿ−1 k=0

|hk

2n,^k₂⁺_n¹|²C_H,c|h|²_H2ⁿ⁽¹⁻^{4H )/2}.

2 The convergence in was proven inp-variation topology, but it is obvious when reading the proof that it also holds in Hölder norm.

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Proof. Without loss of generalities, we assumed=1. WriteR(s, t )=E(XsXt)for the covariance. The Cameron–

Martin space is the closure of functions of formh=

ia_iR(t_i,·)with respect to the norm

|h|_H=

i

a_iR(t_i,·) ²

H=

i,j

a_ia_jR(t_i, t_j)=E Z² whereZ is the zero-mean Gaussian r.v. given byZ=

iaiXt_i. Observe thaths,t=E(ZXs,t). We keepnfixed for the rest of the proof.

Q_n:=

2ⁿ−1 k=0

|hk

2n,^k₂⁺n¹|²=

2ⁿ−1 k=0

E(Xk

2n,^k₂⁺n¹Z)².

Note that ifU, V are two zero-mean Gaussian random variables, then E

U²V²

=2E(U V )²+E U²

E V²

or Cov U², V²

=2E(U V )². (5.4)

This allows us to write Q_n=1

2

2ⁿ−1 k=0

E Z²

X²_k

2n,^k₂⁺_n¹ −E X²_k

2n,^k₂⁺_n¹

=1 2E

Z²

2ⁿ−1 k=0

X²_k

2n,^k₂⁺_n¹ −E X²_k

2n,^k₂⁺_n¹

1

2E Z⁴1/2

₂n−1 k,l=0

Cov X²_k

2n,^k₂⁺_n¹, X²_l

2n,^l₂⁺_n¹

1/2

= 3

2E

Z²₂n−1 k,l=0

E(Xk 2n,^k₂⁺_n¹Xl

2n,^l₂⁺_n¹)² 1/2

where we used (5.4) again. The double sum in line just above splits naturally in a diagonal part, where k=l, and twice the sum overk < l. With (5.1) we control the diagonal part,

2ⁿ−1 k=0

E X²k

2n,^k⁺¹

2n

2

c²2ⁿ⁽¹⁻^{4H )}.

The other part is estimated by using (5.2). Indeed,

0k<l2ⁿ−1

E(Xk 2n,^k+1₂n X l

2n,^l+1₂n )²=

2ⁿ−1 k=0

2ⁿ−k−1 m=1

E(Xk

2n,^k+1₂n Xk+m

2n ,^k+m+1₂n )² c²

2ⁿ−1 k=0

2ⁿ−k−1 m=1

m 2ⁿ

^4H⁻⁴2⁻⁴ⁿ

_∞

m=1

m^4H⁻⁴

c²2ⁿ⁽¹⁻^{4H )}. Putting everything together yields a constantC=C(H, c)such that

Q_nC_HE Z²

2ⁿ⁽¹⁻^{4H )/2}=C_H,c|h|_H2ⁿ⁽¹⁻^{4H )/2}. This finishes the proof. 2

Corollary 3.LetH∈(1/4,1/2]andq∈(1∨1/(H+1/4),2). ThenHis continuously embedded inC^q-var₀ ([0,1],R^d).

More precisely, there exists a constantCH,q,csuch that for allh∈H,

|h|q-^varCH,q,c|h|_H.

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Proof. We use the inequality

|h|^qq-varCγ ,q

∞ n=1

n^γ

2ⁿ−1 k=0

|hk 2n,^k⁺¹

2n |^q,

see [17] for example. From the previous lemma and Hölder’s inequality,

2ⁿ−1 k=0

|hk

2n,^k₂⁺_n¹|^q ₂n−1

k=0

|hk 2n,^k₂⁺_n¹|²

q/2

2ⁿ⁽¹⁻^q/2) C|h|^q_H2ⁿ^[⁽¹⁻^{4H )}^q⁴⁺¹⁻^q²^]. Observe

(1−4H )q

4 +1−q

2<0⇔q

H+1 4

>1.

Hence, for everyq1 such thatq >1/(H+1/4), we have

|h|q-varC_H,q,c|h|_H. 2

This result is not optimal. Cameron–Martin paths associated to usual Brownian motion are of bounded variation (≡finite 1-variation) whereas the above corollary applied toH=1/2 only gives finiteq-variation forq >4/3. In [12], we prove that Cameron–Martin paths associated to fractional Brownian motion are of finiteq-variation for any qgreater than 1∨1/(H+1/2).

Lemma 6. Let H∈(0,1). Then His continuously embedded in C₀^H^-Hölder([0,1]). More precisely, there exists a constantC_H,csuch that for allh∈H,

|h|H-HölderCH,c|h|_H.

Proof. We use the notation of the above proofs. In particular,h_s,t=E(ZX_s,t). By Cauchy–Schwartz and (5.1),

|h_s,t| E

Z² E X²_s,t

C|h|_H|t−s|^H. 2 Theorem 3.For any fixedN1,for allΛ >0,

mlim→∞ sup

{h:I (h)Λ}d_∞

(S_N◦Φ_m)(h), S_N(h)

=0.

Proof. Observe first that forhsuch thatI (h)Λ, we have Φ_m(h)−h

∞ sup

|t−s|2^−m

|h_s,t| Λ^1/2C_H2⁻^mH. Observe also that forq1/(1/4+H ),

Φ_m(h)

q-var3|h|q-var3Cq,hΛ^1/2.

By interpolation, forq> q,we therefore obtain that

mlim→∞ sup

{h:I (h)Λ}

Φm(h)−h

q-var=0.

Then using Theorem 2.2.2 in [16] (or just the continuity of the Young integral), we obtain our theorem. 2 We have therefore the following:

(11)

Corollary 4.Assume that for someH∈(1/4,1), the Gaussian process satisfies condition(5.1)and(5.2)and letX be as in Definition1. Then,(δ_εX)satisfies a large deviation principle in1/p-Hölder topology wherep >1/H with good rate function

J (y)=inf 1

2|x|_H, xsuch thatS_[p](x)=y

.

Once again, applying the Itô map to δ_εX,we can obtain a generalisation of the Freidlin–Wentzell theorem, i.e.

a large deviation principle for solution of stochastic differential equation driven by the enhanced Gaussian processX.

Acknowledgements

The authors would like to thank Christer Borell, Ben Garling for helpful discussions and an anonymous referee for his/her careful reading. The first author acknowledges partial support from a Leverhulme Research Fellowship.

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Further reading

[1] L. Decreusefond, A.S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1997) 177–214.