large deviations for a stochastic volterra-type equation in the besov–orlicz space

Stochastic Processes and their Applications 81 (1999) 39–72

Large deviations for a stochastic Volterra-type equation inthe Besov–Orlicz space

Boualem Djehichea ; ∗, M’hamed EddahbibaDepartment of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, SwedenbD�epartement de Math�ematiques et d’Informatique Facult�e des Sciences et Techniques,

Universit�e Cadi Ayyad, B.P. 618, Marrakech, Maroc, Morocco

Received 14 January 1998; received in revised form 30 September 1998; accepted 13 November 1998

Abstract

In this paper, we investigate the regularity of the solutions of a class of two-parameter Stochas-tic Volterra-type equations in the anisotropic Besov–Orlicz space B0

�; ! modulated by the Youngfunction �(t) = exp(t2)− 1 and the modulus of continuity !(t) = (t(1 + log(1=t)))1=2. Moreover,we derive in the Besov–Orlicz norm a large deviation estimate of Freidlin–Wentzell type for thesolution. c© 1999 Elsevier Science B.V. All rights reserved.

1991 Mathematics Subject Classi�cation: Primary 92D30; secondary 60J27; 60F17

Keywords: Brownian sheet; Besov–Orlicz norm; Hyperbolic stochastic partial di�erentialequation; Large deviations; Volterra equation

1. Introduction and results

Recently, there has been a growing interest in the study of stochastic partial di�er-ential equations (SPDEs) of hyperbolic type due to their importance in many appliedareas. Our primary ambition is to study statistical inference for the solutions of thistype of equations, taking advantage of wavelet bases which are known to enjoy goodproperties for statistical estimation – especially when the underlying function spacesallow less regular functions – combined with the Laplace–Varadhan Principle whichyields bounds on the rate of performance of any consistent parameter estimator. Itis then natural to seek a class of function spaces that produces a stronger topologythan the ones induced by, e.g. the sup-norm and the H�older norm, thus gives more(semi-)continuous functions which yields of course sharper large deviations estimatesand gives a broader scope to the Laplace–Varadhan Principle. Since the driving noisein such equations is the white noise, it is natural to consider a class of anisotropicBesov–Orlicz spaces of Lp-smooth functions that enjoys the above-mentioned proper-ties and includes (almost surely) the driving white noise. To this end, the purpose of

∗ Corresponding author.

0304-4149/99/$ – see front matter c© 1999 Elsevier Science B.V. All rights reserved.PII: S0304 -4149(98)00096 -9

40 B. Djehiche, M. Eddahbi / Stochastic Processes and their Applications 81 (1999) 39–72

this paper, as a �rst step, is to consider a particular class of SPDEs connected withthe stochastic Volterra equation in the plane and establish a large deviations principlein the limit of small perturbations of the driving noise in an appropriate anisotropicBesov–Orlicz space.Rovira and Sanz-Sol�e (1997) (referred to as [RS] in the sequel) consider the fol-

lowing stochastic Volterra-type equation on the plane.

Xz = xz +∫Rz

[�(z; �; X�)W (d�) + b(z; �; X�) d�]; (1)

where z ∈ I 2 :=[0; 1]2 and W = {Wz; z ∈ I 2} is a Brownian sheet.An important special case of (1) that appears in Norris (1995) and Rovira and

Sanz-Sol�e (1996) is the following SPDE:

LXs; t = a3((s; t); Xs; t)W s; t + a4((s; t); Xs; t); (2)

where W s; t is the white noise on I 2 and

L=@2

@s@t− a1(s; t)

@@s

− a2(s; t)@@t

:

If ( s; t(�); �∈ [0; s] × [0; t]) denotes the Green function associated to L, Rovira andSanz-Sol�e (1996) show that the solution of Eq. (2) admits the following integral form:

Xs; t = X0 +∫Rs; t

s; t(�)[a3(�; X�)W (d�) + a4(�; X�)d�]:

A large deviation principle (LDP) for the solutions of Eq. (1), in the limit of smallperturbations of the noise, has been established in [RS] in the uniform norm, generalis-ing their previous result for Eq. (2), (see Rovira and Sanz-Sol�e (1996)) and the one byDoss and Dozzi (1987) for the Brownian sheet. In the H�older space C� with exponent�¡ 1=2, Eddahbi (1997) proved an LDP result for a class of nonlinear SPDEs similarto Eq. (2).As mentioned above, in this paper, we go one step further and study small perturba-

tions of Eq. (1) in the stronger topology induced by the separable Besov–Orlicz spaceB0

�;! modulated by the Young function �(t)=exp(t2)−1 and the modulus of continuity!(t) = (t(1 + log(1=t)))1=2 (see the de�nition below).In Section 2, we collect some facts about the Besov–Orlicz space we are concerned

with and state the main results. A result of the regularity of the solutions to Eq. (1) inthe Besov–Orlicz space is given in Section 3. In Section 4, we prove a large deviationprinciple in the Besov–Orlicz norm, for the solution of Eq. (1), in the limit of smallperturbations of the noise.

2. Preliminaries and main results

2.1. Main results

On the �ltered probability space (; F; (Fz)z∈ I2 ; P), let W = {Wz; z=(s; t)∈ I 2}be a one-dimensional Brownian sheet, where (Fz)z∈ I2 is the natural �ltration associated

B. Djehiche, M. Eddahbi / Stochastic Processes and their Applications 81 (1999) 39–72 41

with W . Let Rz denote the rectangle [0; z], Rz1 ;z2 the rectangle [z1; z2] and �(·) theLebesgue measure on R2.This paper is devoted to the study of the equation:

Xz = xz +∫Rz

[�(z; �; X�)W (d�) + b(z; �; X�) d�];

under the following conditions:(H1) b; � : I 2 × I 2 × R → R are bounded. � is uniformly Lipschitz in the second

variable.(H2) �, b, @�=@s; @�=@t and @2�=@s@t are continuous and satisfy the following con-

ditions: There exist positive increasing functions ’ and ’ on [0; 1] (typically ’(r) =r�; �¿ 1

2 ) such that

’(r) = o(!(r)); r → 0

and ’ is continuous, such that(i) For every z1; z2 ∈ I 2, x; y∈R and �1; �2 ∈ I 2,

sup�∈ I2

|f(Rz1 ; z2 ; �; x)− f(Rz1 ; z2 ; �; y)|6L’(�(Rz1 ; z2 ))|x − y|;

supx∈R

|f(Rz1 ; z2 ; �1; x)− f(Rz1 ; z2 ; �2; x)|6L’(�(Rz1 ; z2 ))’(|�2 − �1|):

(ii) For every s1; s2 ∈ I , x; y∈R and �1; �2 ∈ I 2,

supt ∈ I; �∈ I2

|f((s2; t); �; x)− f((s2; t); �; y)− f((s1; t); �; x) + f((s1; t); �; y)|

6L’(|s2 − s1|)|x − y|;

supt ∈ I; x∈R

|f((s2; t); �1; x)− f((s2; t); �2; x)− f((s1; t); �1; x) + f((s1; t); �2; x)|

6L’(|s2 − s1|)’(|�2 − �1|):(iii) For every t1; t2 ∈ I , x; y∈R and �1; �2 ∈ I 2,

sups∈ I; �∈ I 2

|f((s; t2); �; x)− f((s; t2); �; y)− f((s; t1); �; x) + f((s; t1); �; y)|

6L’(|t2 − t1|)|x − y|:

sups∈ I; x∈R

|f((s; t2); �1; x)− f((s; t2); �2; x)− f((s; t1); �1; x) + f((s; t1); �2; x)|

6L’(|t2 − t1|)’(|�2 − �1|):Here, f denotes any of the above continuous functions, f(Rz1 ; z2 )=f(s2; t2)−f(s1; t2)−f(s2; t1) +f(s1; t1), for z1 = (s1; t1) and z2 = (s2; t2) and L denotes a constant that maydi�er from line to line.(H3) The function x : I 2 → R is continuous.We note that Condition (H2) implies that b and � are globally Lipschitz in the last

argument. Under these conditions, it can be shown (see [RS]) that Eq. (1) admits aunique solution X = {Xz; z ∈ I 2} that is (Fz)z∈ I2 -adapted with a.s. continuous samplepaths.


Let B0�;! denote the separable anisotropic Besov–Orlicz space on I 2, modulated by

the Young function �(·) and the function !(·), with norm || · ||�;! (see the de�nitionbelow). The following theorem, to be proved in Section 3, is a regularity result forthe solution of Eq. (1).

Theorem 1. Assume that Conditions (H1) and (H2) are satis�ed. If the function xis in B0

�;!; then the solution X of Eq. (1) is almost surely in B0�;!.

The next theorem to be proved in Section 4, is a large deviation principle on B0�;!

for the family {X �; �¿ 0} de�ned by

X �z = xz +

√�∫Rz

�(z; �; X �� )W (d�) +

∫Rz

b(z; �; X �� ) d�; (3)

provided that the function x is in B0�;!. Let H be the Cameron–Martin space associated

with the Brownian sheet W .

H :={h∈C(I 2;R); there exists h∈L2(I 2;R) s:t: hz =

∫Rz

h� d�; z ∈ I 2}

:

Denote

||h||2H =∫I2|h�|2 d�:

The rate function associated with the LDP for the Brownian sheet in the uniform normis then

�(h) ={ 1

2 ||h||2H; if h∈H;�(h) = +∞; otherwise:

Recall that, for every a¿0, the level set of �

L(a) = {h∈H; �(h)6a}is a compact subset of H. For every h∈H and every ∈C(I 2;R), let S (h) be thesolution of the deterministic di�erential equation, called the skeleton of Eq. (3):

S (h)z = z +∫Rz

�(z; �; S (h)�)h� d�+∫Rz

b(z; �; S (h)�) d� (4)

and set

�(f) ={inf{�(h); S (h) = f}; if (S )−1(f) 6= ∅;+∞; otherwise:

Our objective is to prove the following result:

Theorem 2. Assume (H1) and (H2) and let the function x be in B0�;!. Then the family

{X �; �¿ 0} of solutions of Eq. (3) satis�es a large deviation principle on B0�;! with

rate function

�(f) = inf{ 12 ||h||2H; Sx(h) = f; h∈H};


with Sx(h) given by Eq. (4). This means that the probabilities P�(A) :=P(X � ∈A);for measurable sets A in B0

�;!; satisfy(i) for each open subset G of B0

�;! lim inf �→0 � logP�(G)¿− inff∈G �(f);(ii) for each closed subset F of B0

�;! lim sup�→0 � logP�(F)6− inff∈ F �(f).

(iii) The level sets {f∈B0�;!; �(f)6N}; N ¿ 0; are compact.

To prove the LDP, we apply the transfer principle deviced by Azencott (1980) (seealso [RS]).Let us remark that these results extend easily to the more general situations where

the coe�cients may depend on � provided some suitable convergence assumptions.

2.2. Besov–Orlicz space

In this subsection, we give a short review of the Besov–Orlicz space we considerin this paper. For more details on these spaces, see Rao and Ren (1991), Peetre(1976), Jonsson and Wallin (1984), Ciesielski et al. (1993) and Ropela (1976) for theone-parameter case and Ciesielski and Domsta (1972), Ciesielski and Kamont (1995)and Kamont (1994, 1996) for the multiparameter case. This review follows closelyCiesielski and Kamont (1995) and Kamont (1994).The Orlicz space L(�; I 2) on I 2 associated to � is the space of measurable functions

f : I 2 → R for which

||f||∗� := inf�¿0

1�

[1 +

∫I2

�(�|f(t)|) dt]¡∞:

For the purpose of the paper it is more convenient to use an equivalent norm to || · ||∗�(see Fernique (1971) or Ciesielski (1993) for a proof):

||f||� := supp¿1

||f||p√p

; (5)

where, ||f||p is the usual Lp(I 2)-norm. For r ∈R and ei=(�1; i ; �2; i)∈R2; i=1; 2, unitvectors, let

�r; if(z) ={

f(z + rei)− f(z); if z and z + rei ∈ I 2;0; otherwise;

denote the progressive di�erence of a measurable function f on I 2 in the eith direction.Set

�(r1 ; r2)f = �r1 ;1 ◦ �r2 ;2f; (r1; r2)∈R2

and de�ne the moduli of smoothness of f in Lp(I 2) by

!p;i(f; t) := sup|r|6t

||�r; if||p; i = 1; 2; t ∈ (0; 1]

and

!p(f; (t1; t2)) := sup|r1|6t1 ;|r2|6t2

||�(r1 ; r2)f||p; t1; t2 ∈ (0; 1]:


Using Eq. (5), the moduli of smoothness of f in L(�; I 2) are de�ned by the usualformulas:

!�; i(f; t) := supp¿1

!p;i(f; t)√p

(6)

and

!�(f; (t1; t2)) := supp¿1

!p(f; (t1; t2))√p

: (7)

Consider the function !, sometimes called (0; 1)-modulus, de�ned on I by

!(t) =

√t(1 + log

1t

):

An anisotropic Besov–Orlicz space B�;! is a Besov space modulated with the Orlicznorm described in terms of !(·) and the moduli of smoothness de�ned in Eqs. (6)and (7):

B�;! :={f∈L(�; I 2); ||f||�;! ¡+∞};where

||f||�;! := sup0¡t61

!�;1(f; t)!(t)

+ sup0¡t61

!�;2(f; t)!(t)

+ sup0¡t1 ;t261

!�(f; (t1; t2))!(t1)!(t2)

+ ||f||�:

B�;! endowed with the norm || · ||�;! is a nonseparable Banach space. We are goingto consider a separable Banach subspace of B�;! de�ned as follows:

B0�;! :=

{f∈B�;!; ||f||p = o(√p) as p → +∞;

!p(f; (t1; t2)) = o(√p !(t1)!(t2)) as

1p

∧ t1 ∧ t2 → 0;

!p; i(f; t) = o(√p !(t)) as

1p

∧ t → 0 for i = 1; 2}

:

The fact that these spaces are isomorphic to some sequence spaces makes them rela-tively easy to manipulate. The isomorphism is given by the coe�cients of a functionin the tensor product Schauder system (see Theorem A in Kamont (1994)). Indeed, let{’n; n¿0} be the family of Schauder functions on I de�ned by

’0(s) = 1; ’1(s) = s;

’n(s) =√2−j’(2 j+1s− k) for n= 2 j + k; j∈N and k = 1; : : : ; 2 j:

With ’(u) = max(1− |u|; 0). We know that for each continuous function f on I , wehave

f(s) =+∞∑n=0

Cn(f)’n(s);

where the coe�cients are given by

C0(f) =f(0); C1(f) = f(1)− f(0);

Cn(f) = 2√2 j

[f(2k − 12 j+1

)− 12

(f(2k2 j+1

)+ f

(2k − 22 j+1

))]for n= 2 j + k:


Now, if f is continuous in I 2 we have the decomposition

f =+∞∑m=0

+∞∑n∨n′=m

Cn;n′(f)’n ⊗ ’n′

and the coe�cients are given by

Cn;n′(f) = C1n (f) ◦ C2n′(f);

with

C1n (f)(x2) = Cn(f(·; x2)) and C2n (f)(x1) = Cn(f(x1; ·)):The main tools to prove our results rely on the following characterisation of the spaceB0

�;! (see Ciesielski and Kamont, 1995):

Theorem 3. f∈B0�;! if and only if conditions (A1), (A2) and (A3) below are satis-

�ed.

(A1) limj′∨p→+∞

2−j′=p√p(1 + j′)

2 j′+1∑

n′=2 j′+1

|Cl;n′(f)|p1=p

= 0; l= 0; 1:

(A2) limj∨p→+∞

2−j=p√p(1 + j)

2 j+1∑

n=2 j+1

|Cn;l′(f)|p1=p

= 0; l′ = 0; 1:

(A3) limj∨j′∨p→+∞

2−(j+j′)=p√p(1 + j)(1 + j′)

2 j+1∑

n=2 j+1

2 j′+1∑n′=2 j′+1

|Cn;n′(f)|p1=p

= 0:

Two other norms || · ||∗ and || · ||∗∗ (inducing nonseparable Banach spaces) that playa crucial role in the proof of the LDP results are de�ned as follows. For f : I 2 → R,vanishing on the axes, set

||f||∗ =max(|f(1; 1)|; ||f||∗1; ||f||∗2; ||f||∗3)with

||f||∗1 = supj¿0

sup2 j+16n62 j+1

|Cn(f(:; 1))|√1 + j

;

||f||∗2 = supj¿0

sup2 j+16n62 j+1

|Cn(f(1; :))|√1 + j

;

||f||∗3 = supj; j′¿0

sup(n;n′)∈Kj; j′

|Cn;n′(f)|√(1 + j)(1 + j′)

;

where, Kj; j′ = {(n; n′); 2 j + 16n62 j+1; 2 j′ + 16n′62 j′+1}; and||f||∗∗ = ||f||∗∗1 + ||f||∗∗

2+ ||f||∗∗3 ;


with

||f||∗∗1= sup

06s1¡s261

|f(s1; 1)− f(s2; 1)|!(|s1 − s2|) ;

||f||∗∗2 = sup06t1¡t261

|f(1; t2)− f(1; t1)|!(|t1 − t2|) ;

and

||f||∗∗3= sup

06t1¡t26106s1¡s261

|f(Rz1 ; z2 )|!(|s1 − s2|)!(|t1 − t2|) :

In particular, for rectangles Rz; z = (s; t)∈ I 2, and functions f null on the axes, wehave f(s; t) = f(Rz) and therefore

||f||6||f||∗∗; (8)

where, ||·|| denotes the usual uniform norm for the corresponding space. The importanceof these norms relies on the fact that

||f||6C0||f||�;!6C1||f||∗∗6C2||f||∗;for some positive constants C0, C1 and C2, where the last inequality holds only forfunctions f being null on the axes.

2.3. Exponential inequalities

In this subsection, we list the exponential inequalities for continuous strong mar-tingales and stochastic integrals (that are not martingales) in the plane we will fre-quently use in the sequel. The �rst inequality appears in Proposition 7 in Dozzi (1989,p. 114) (cf. also Theorem 2.4 in Rovira and Sanz-Sol�e (1997) and Mishura (1987)).The proofs of the remaining propositions can be found in Appendix A.

Proposition 4. Let M = {Mz; z ∈ I 2} be a continuous strong Fz-martingale null onthe axes. Assume that there exists a function a: R2 → (0;+∞) such that P(〈M 〉z ¿a(z)) = 0 for all z ∈ I 2; where {〈M 〉z ; z ∈ I 2} is the quadratic variation of M . Then,for any u¿ 0

P

(sup�∈ Rz

|M�|¿u

)64 exp

(− u2

18a(z)

):

Proposition 5. For every u¿ 12√log 2 and every R-valued continuous and bounded

process H = {H (�); �∈ I 2}

P[∣∣∣∣∣∣∣∣∫R·

H (�)W (d�)∣∣∣∣∣∣∣∣∗¿u; ||H ||61

]6100 exp

(− u2

144

);

and therefore the inequality holds for the norms || · ||�;! and || · ||∗∗ instead of || · ||∗.

The next two propositions are a B0�;!-version of Theorems 2.3 and 2.4 in [RS].


Proposition 6. Let H : I 2× I 2× → R be a B(I 2)⊗B(I 2)⊗F-measurable processsatisfying:(a) For each z; H (z; �) is F�-adapted for which E[

∫I2 H 2(z; �) d�]¡∞:

(b) The map z → H (z; ·) satis�es a.s. Assumption (H2).Then, there exist positive constants C1 and C2; independent of L; such that(i)

P[∣∣∣∣∣∣∣∣∫R·

H (·; �)W (d�)∣∣∣∣∣∣∣∣∗¿�; ||H ||6�

]6exp−

(�2C2

12L2 + 4�2

)(9)

for all � for which �¿2C1√3L2 + �2.

(ii)

P

[∣∣∣∣∣∣∣∣∣∣∫R·

∫R�;·

H (�; �) d�W (d�)

∣∣∣∣∣∣∣∣∣∣∗¿�; ||H ||6�

]6exp−

(�2C2

12L2 + 4�2

): (10)

Consequently, inequalities (9) and (10) hold for the norms || · ||�;! and || · ||∗∗instead of || · ||∗.

Denote for a B(I 2)⊗B(I)⊗F-measurable process H

(H ·Ws)(s; t) :=∫ s

0

∫Rs; t

H (�; r)W (d�)dr

and for a B(I)⊗B(I 2)⊗F-measurable process H

(H · tW )(s; t) :=∫ t

0

∫Rs; t

H (r; �)W (d�)dr

Proposition 7. (i) Let H : I 2× I× → R be a B(I 2)⊗B(I)⊗F-measurable processsatisfying

E

[∫ 1

0

∫I2

H 2(�; r) d� dr

]¡∞:

If for each � = (u; v) and z = (s; t); H ((u; v); s) is F(s; v)-measurable and H ((u; v); s)vanishes for 06s¡u; then there exists a positive constant C such that

P[||(H ·Ws)(·; ·)||∗∗ ¿�; ||H ||6�

]6Cexp−

(�2

C(�2 + L2)

): (11)

(ii) Let H : I × I 2 × → R be a B(I)⊗B(I 2)⊗F-measurable process satisfying

E

[∫ 1

0

∫I2H2(r; �) d� dr

]¡∞:

If for each � = (u; v) and z = (s; t); H (t; (u; v)) is F(u; t)-measurable and H (t; (u; v))vanishes for 06t ¡ v; then there exists a positive constant C such that

P[||(H · tW )(·; ·)||∗∗ ¿�; ||H ||6�]6C exp−(

�2

4C(�2 + 3L2)

): (12)

Consequently, inequalities (11) and (12) hold for the norm || · ||�;! instead of || · ||∗∗.


3. Regularity of the solutions in the Besov–Orlicz space

This section is devoted to the proof of Theorem 1 on the regularity of the processX , solution of Eq. (1), in B0

�;!. Recall

Xz = xz +∫Rz

[�(z; �; X�)W (d�) + b(z; �; X�) d�];

where � and b satisfy Conditions (H1) and (H2) above and the function x : I 2 → Ris in B0

�;!.For the sequel, we denote L=Lip(�)∨Lip(b), the largest of the Lipschitz constants

of the coe�cients and M = ||�|| ∨ ||b||, where || · || denotes the usual uniform norm forthe corresponding spaces.The proof of Theorem 1 consists in checking Conditions (A1)–(A3) in Theorem

2.1 above. First we need an Lp-estimate for the stochastic integral

Yz :=∫Rz

�(z; �; X�)W (d�):

We note that for � :=(u; v)6z :=(s; t) and � :=(r; w), our stochastic integral can bewritten as a representable semimartingale (cf. Eq. (1:5) in [RS]):

Yz = J 1z + J 2z + J 3z + J 4z ;

where

J 1z =∫Rz

�(�; �; X�)W (d�);

J 2z =∫ s

0

∫Rz

@�@s((r; v); �; X�)W (d�) dr;

J 3z =∫ t

0

∫Rz

@�@t((u; w); �; X�)W (d�) dw

and

J 4z =∫Rz

∫R�; z

@2�@s@t

(�; �; X�) d�W (d�):

Lemma 8. For every integer p¿2;

E|Yz|p6Cp pp=2 �(Rz)p=2; (13)

where C is a positive constant.

Proof (Sketch). The idea of the proof uses arguments applied to the uniform casein [RS], the two-parameters Ito’s formula and Burkholder–Davis–Gundy inequality toeach of the (J i

z)p’s above.

Proof of Theorem 1 (Sketch). First we remark from the de�nition of B0�;! and (H2)

that z → ∫Rz

b(z; �; X�)d� belongs a.s to B0�;!. Indeed it is not hard to show that this

map belongs a.s. to B�;!. The fact that it belongs to B0�;! follows from the assumption


that ’(r)= o(!(r)) as r → 0. It remains to show that the process (Yz; z ∈ I 2) satis�es(A1)–(A3) of Theorem 3. We will only prove (A3). (A1) and (A2) can be derivedin the same fashion and therefore we omit the details.To prove (A3) for J i

z ; i = 1; : : : ; 4 we shall show that for all 0¡�¡ 12 ,

(B1) supj; j′¿0

supp¿1

2−(j+j′)=p

√p(1 + j)�(1 + j′)�

2 j+1∑

n=2 j+1

2 j′+1∑n′=2 j′+1

|Cn;n′(J i· )|p

1=p

¡+∞ a:s:

and

(B2) limp→+∞ sup

j; j′¿0

2−(j+j′)=p

√p√(1 + j)(1 + j′)

2 j+1∑

n=2 j+1

2 j′+1∑n′=2 j′+1

|Cn;n′(J i· )|p

1=p

= 0 a:s:

To check relation (B1), let A¿ 0. Using Chebyshev’s inequality, we get

P

2−(j+j′)=p

√p(1 + j)�(1 + j′)�

2 j+1∑

n=2 j+1

2 j′+1∑n′=2 j′+1

|Cn;n′(J i· )|p

1=p

¿A

62−(j+j′)

Ap(√p)p(1 + j)�p(1 + j′)�p

2 j+1∑

n=2 j+1

2 j′+1∑n′=2 j′+1

E|Cn;n′(J i· )|p

:

Let us �rst estimate the expectation of the pth power of Cn;n′(J 4· ). Recall that

J 4z =∫Rz

∫R�; z

@2�@s@t

(�; �; X�)d�W (d�) :=∫Rz

H (z; �)W (d�)

with

H (z; �) :=∫R�; z

@2�@s@t

(�; �; X�)d�:

By de�nition, |Cn;n′(J 4· )| is less than four terms of the form2(j+j′)=2(D1 + D2 + D3 + D4);

where

D1 =

∣∣∣∣∣∫ (k−1)=2 j

0

∫ (k′−1)=2 j′

0H (R((k−1)=2 j ; k=2 j);((k′−1)=2 j′ ; k′=2 j′ ); �)W (d�)

∣∣∣∣∣ ;

D2 =

∣∣∣∣∣∫ (k−1)=2 j

0

∫ k′=2 j′

(k′−1)=2 j′

(H((

k2 j ;

k ′

2 j′

); �)− H

((k − 12 j ;

k ′

2 j′

); �))

W (d�)

∣∣∣∣∣ ;

D3 =

∣∣∣∣∣∫ k=2 j

(k−1)=2 j

∫ k′=2 j′

(k′−1)=2 j′H((

k2 j ;

k ′

2 j′

); �)

W (d�)

∣∣∣∣∣ ;

D4 =

∣∣∣∣∣∫ k=2 j

(k−1)=2 j

∫ (k′−1)=2 j′

0

(H((

k2 j ;

k ′

2 j′

); �)− H

((k2 j ;

k ′ − 12 j′

); �))

W (d�)

∣∣∣∣∣ :


Hence,

E|Cn;n′(J 4· )|p

64p 2(j+j′)p=2 sup|t′−s′|62−j′−1

|t−s|62−j−1

E

∣∣∣∣∣∫ s

0

∫ s′

0H (R(s;t);(s′ ;t′); �)W (d�)

∣∣∣∣∣p

+4p2(j+j′)p=2 sup|t′−s′|62−j′−1

|t−s|62−j−1

E

∣∣∣∣∣∫ t

s

∫ t′

s′H ((t; t′); �)W (d�)

∣∣∣∣∣p

+4p2(j+j′)p=2 sup|t′−s′|62−j′−1

|t−s|62−j−1

E

∣∣∣∣∣∫ s

0

∫ t′

s′(H ((t; t′); �)− H ((s; t′); �))W (d�)

∣∣∣∣∣p

+4p2(j+j′)p=2 sup|t′−s′|62−j′−1

|t−s|62−j−1

E

∣∣∣∣∣∫ t

s

∫ s′

0(H ((t; t′); �)− H ((t; s′); �))W (d�)

∣∣∣∣∣p

:

Therefore by Eq. (13), for integers p¿2,

E|Cn;n′(J 4· )|p64(2M)ppp=2:

Using the multiparameter Fubini theorem, H�older’s inequality and Lemma 8, estimatesof the expectation of each of the Cn;n′(J i

· )’s can be checked in a similar way. Therefore,

maxi=1;2;3;4

E|Cn;n′(J i· )|p6CMppp=2:

Hence,

P

2−(j+j′)=p

√p(1 + j)�(1 + j′)�

2 j+1∑

n=2 j+1

2 j′+1∑n′=2 j′+1

|Cn;n′(J i· )|p

1=p

¿A

64(

M2A

)p 1(1 + j)�p(1 + j′)�p

:

Choosing p0¿ 1=� and A large enough, the series

∑j; j′¿0

∑p¿p0

(M2A

)p 1(1 + j)�p(1 + j′)�p

converges. Relation (B1) is then a consequence of Borel–Cantelli lemma.To check relation (B2), let

Kj; j′ = {(n; n′); 2 j + 16n62 j+1; 2 j′ + 16n′62 j′+1}and note that

2−(j+j′)=p

2 j+1∑

n=2 j+1

2 j′+1∑n′=2 j′+1

|Cn;n′(J i· )|p

1=p

6 sup(n;n′)∈Kj; j′

|Cn;n′(J i· )|:


Now, using the fact that each of the Cn;n′(J i· )’s is dominated by terms of the same

form as D1; : : : ; D4, the exponential inequalities of Propositions 4–7 yield that thereexist positive constants C1 and C2 such that for all A¿ 0 large enough

P

[1√

(1 + j)(1 + j′)sup

(n;n′)∈Kj; j′|Cn;n′(J i

· )|¿A

]

6C1exp(−(

A2

C2M 2

)(1 + j)(1 + j′)

):

Borel–Cantelli lemma yields then

supj; j′

1√(1 + j)(1 + j′)


|Cn;n′(J i· )|¡+∞ a:s:

But

supj; j′

2−(j+j′)=p

√p√(1 + j)(1 + j′)

2 j+1∑

n=2 j+1

2 j′+1∑n′=2 j′+1

|Cn;n′(J i· )|p

1=p

61√psupj; j′

1√(1 + j)(1 + j′)


|Cn;n′(J i· )|;

which completes the proof of relation (A3).

4. Large deviations in the Besov–Orlicz norm

In this section, we prove Theorem 2. Following the transfer principle deviced byAzencott (see also Rovira and Sanz-Sol�e (1996, 1997)), the proof of the theorem is adirect consequence of the next two propositions:

Proposition 9. For every a¿0 and every x∈B0�;!; the map

S :H → B0�;!;

h → Sx(h)

is continuous in L(a) endowed with the Besov–Orlicz norm || · ||�;!. In particular, thelevel sets {f∈B0

�;!; �(f)6N}; N ¿ 0; are compact.

Proposition 10. For every R¿ 0; a¿ 0 and �¿ 0 large enough, there exist �¿ 0and �0¿ 0 such that for every h∈L(a) and every �∈ (0; �0] we have

P(||X � − Sx(h)||�;!¿�; ||√�W − h||6�)6exp(−R

�

):

Proof of Proposition 9. Let h; k ∈H such that ||h||H∨ ||k||H6a. Lemma 3.2 in [RS]tells us that the map S : h → Sx(h)· is continuous in the set L(a) with respect to the


uniform norm. On the other hand

||Sx(h)· − Sx(k)·||�;!6∣∣∣∣∣∣∣∣∫R·(b(·; �; Sx(h)�)− b(·; �; Sx(k)�))d�

∣∣∣∣∣∣∣∣�;!

+∣∣∣∣∣∣∣∣∫R·(�(·; �; Sx(h)�)− �(·; �; Sx(k)�))k� d�

∣∣∣∣∣∣∣∣�;!

;

+∣∣∣∣∣∣∣∣∫R·

�(·; �; Sx(h)�)(h� − k�) d�∣∣∣∣∣∣∣∣�;!

= I1 + I2 + I3:

Since I1 is a particular case of I2 if we replace, in I2; � by b and k� by 1, it su�cesto prove that I2 and I3 tend to zero as ||h− k||�;! tends to zero.For ease of notation we set �(z; �; x; y)=b(z; �; x)−b(z; �; y); (z; �; x; y)=�(z; �; x)−

�(z; �; y) and

(z) =∫Rz

(z; �; Sx(h)�; Sx(k)�)k� d�:

Hence,

I2 = ||||� + sup0¡t61

!�;1(; t)!(t)

+ sup0¡t61

!�;2(; t)!(t)

+ sup0¡t1 ;t261

!�(; (t1; t2))!(t1)!(t2)

:

Using Eq. (5) and the Lipschitz property of � we get

||||�6L||Sx(h)− Sx(k)|| ||k||H: (14)

To estimate the remaining terms of I2, recall that

!p;i(; t) = sup|r|6t

||�r; i||p; i = 1; 2; 0¡t61;

!p(; (t1; t2)) = sup|r2|6t2|r1|6t1

||�(r1 ; r2)||p; 0¡t1; t261;

!�; i(; t) = supp¿1

!p;i(; t)√p

; 0¡t61; i = 1; 2

and

!�(; (t1; t2)) = supp¿1

!p(; (t1; t2))√p

; 0¡t1; t261:

We have

||�r; i||p =(∫

I2r; i

|(z + rei)−(z)|p dz)1=p

;

where I 2r; i = {z ∈ I 2 : z + rei ∈ I 2}. For i = 1 and z = (s; t),

|(z + re1)−(z)|6∣∣∣∣∫ s+r

s

∫ t

0 ((s+ r; t); �; Sx(h)�; Sx(k)�)k� d�

∣∣∣∣+∣∣∣∣∫ s

0

∫ t

0( ((s+ r; t); �; Sx(h)�; Sx(k)�)− ((s; t); �; Sx(h)�; Sx(k)�))k� d�

∣∣∣∣ :


Assumption (H2) implies that

|(z + re1)−(z)|6L||Sx(h)− Sx(k)||(∫ s+r

s

∫ t

0|k�|d�+ ’(r)

∫ s

0

∫ t

0|k�|d�

):

Using Cauchy–Schwarz inequality we get

|(z + re1)−(z)|6L||Sx(h)− Sx(k)|| ||k||H(√rt + ’(r)

√st):

Similarly,

|(z + re2)−(z)|6L||Sx(h)− Sx(k)|| ||k||H(√rs+ ’(r)

√st):

Hence,

||�r;1||p+ ||�r;2||p6 2L||Sx(h)−Sx(k)|| ||k||H(∫

I2r;1

(√rt + ’(r)

√st)p d s dt

)1=p

6 4√rL||Sx(h)− Sx(k)|| ||k||H

(∫I2r;1

tp=2 d s dt

)1=p

+4’(r)L||Sx(h)− Sx(k)|| ||k||H(∫

I2r; 2

(st)p=2 d s dt

)1=p

6 4(√r + ’(r))L||Sx(h)− Sx(k)|| ||k||H:

Therefore,

!p;i(; t) = sup|r|6t

||�r; i||p64L||Sx(h)− Sx(k)|| ||k||H(√t + ’(t))

and then

!�; i(; t) = supp¿1

!p;i(; t)√p

64L||Sx(h)− Sx(k)|| ||k||H(√t + ’(t));

which implies that

sup0¡t61

!�; i(; t)!(t)

64L||Sx(h)− Sx(k)|| ||k||H D1

with

D1 := sup0¡t61

√t + ’(t)!(t)

;

which is �nite by assumption on ’. Consequently,

sup0¡t61

!�;1(; t)!(t)

+ sup0¡t61

!�;2(; t)!(t)

68L||Sx(h)− Sx(k)|| ||k||H D1: (15)

Now, for r = (r1; r2)∈R2+ we have

|�(r1 ; r2)(z)|6∣∣∣∣∫Rz

(Rz; z+r ; �; Sx(h)�; Sx(k)�)k� d�∣∣∣∣

+

∣∣∣∣∣∫Rz; z+r

((z + r); �; Sx(h)�; Sx(k)�)k� d�

∣∣∣∣∣


+

∣∣∣∣∣∫ s

0

∫ t+r2

t( ((z + r); �; Sx(h)�; Sx(k)�)

− ((s; t + r2); �; Sx(h)�; Sx(k)�)k� d�

∣∣∣∣∣+

∣∣∣∣∣∫ s+r1

0

∫ t

0( ((z + r); �; Sx(h)�; Sx(k)�)

− ((s+ r1; t); �; Sx(h)�; Sx(k)�)k� d�

∣∣∣∣∣ :Assumption (H2) on � implies that

|�(r1 ; r2)(z)|6 L’(r1r2)||Sx(h)− Sx(k)||∫Rz

|k�| d�

+L||Sx(h)− Sx(k)||∫Rz; z+r

|k�| d�

+L’(r1)||Sx(h)− Sx(k)||∫ s

0

∫ t+r2

t|k�| d�

+L’(r2)||Sx(h)− Sx(k)||∫ s+r1

0

∫ t

0|k�| d�

since �(Rz; z+r) = r1r2. Using Cauchy–Schwarz inequality we get

|�(r1 ; r2)(z)|6L||Sx(h)− Sx(k)|| ||k||H �(r1; r2; s; t)

where

�(r1; r2; s; t) = ’(r1r2)√st +

√r1r2 + ’(r1)

√sr2 + ’(r2)

√tr1:

Therefore,

!p(; (t1; t2)) = sup|r2|6t2|r1|6t1

||�(r1 ; r2)||p

6 L||Sx(h)− Sx(k)|| ||k||H sup|r2|6t2|r1|6t1

(∫I2r

|�(r1; r2; s; t)|p d s dt)1=p

;

where I 2r = {z ∈ I 2; z + r ∈ I 2}. But,(∫I2r

|�(r1; r2; s; t)|p d s dt)1=p

64’(r1r2)

(∫I2r

√st

pd s dt

)1=p+ 4

√r1r2

+ 4’(r1)√r2

(∫I2r

√spd s dt

)1=p+ 4’(r2)

√r1

(∫I2r

√tpd s dt

)1=p:


Since ’ is an increasing function we get

sup|r2|6t2|r1|6t1

(∫I2r

|�(r1; r2; s; t)|p d s dt)1=p

64(’(t1t2) +√t1t2 + ’(t1)

√t2 + ’(t2)

√t1):

Hence,

!�(; (t1; t2)) = supp¿1

!p(; (t1; t2))√p

6 4L||Sx(h)− Sx(k)|| ||k||H(’(t1t2) +√t1t2)

+4L||Sx(h)− Sx(k)|| ||k||H(’(t1)√t2 + ’(t2)

√t1);

which implies that

sup0¡t1 ; t261

!�(; (t1; t2))!(t1t2)

64L||Sx(h)− Sx(k)|| ||k||H(D2 + D3) (16)

with

D2 := sup0¡t1 ; t261

’(t1t2) +√t1t2

!(t1)!(t2)

and

D3 := sup0¡t1 ; t261

’(t1)√t2 + ’(t2)

√t1

!(t1)!(t2)

which are �nite. Summing up, inequalities (14)–(16) imply that I2 tends to 0 as||h− k||�;! → 0.To complete the proof of the Proposition 9 it remains to prove that I3 tends to zero

when ||h− k||�;! tends to zero.De�ne the following approximation of Sx(h)

Sx(h)N� = Sx(h)( j

2N; k2N

) if �∈�Nj; k :=

[j2N

;j + 12N

[×[

k2N

;k + 12N

[and

�N =(

j2N

;k2N

)if �∈�N

j; k ;

for all j = 0; : : : ; 2N − 1 and k = 0; : : : ; 2N − 1. Now,

I3 =∣∣∣∣∣∣∣∣∫R·

�(·; �; Sx(h)�)(h� − k�) d�∣∣∣∣∣∣∣∣�;!

6∣∣∣∣∣∣∣∣∫R·(�(·; �N ; Sx(h)N� )− �(·; �; Sx(h)�))(h� − k�) d�

∣∣∣∣∣∣∣∣�;!

+∣∣∣∣∣∣∣∣∫R·

�(·; �N ; Sx(h)N� )(h� − k�) d�∣∣∣∣∣∣∣∣�;!

= I 13 + I 23 :

In view of the above calculations, I 13 can be estimated as I2 by replacing ||k||H by||h− k||H and Sx(k)� by Sx(h)N� in Eqs. (14)–(16). Hence, with D = D1 + D2 + D3,

I 136DL||Sx(k)− Sx(k)N ||||h− k||H:


Now, in Appendix B below we prove that

I 23622N M ||h− k||�;!: (17)

Therefore,

I362aDL||Sx(h)− Sx(h)N ||+ 22N M ||h− k||�;!:Combining this inequality and Eqs. (14)–(16) and letting ||h − k||�;! → 0 and thenN → ∞ we are done.

In the proof of Proposition 10, we closely follow the strategy used in Ben Arous andLedoux (1994) and Eddahbi (1997). In particular, we give an exponential upper boundof the probability that simultaneously the Besov–Orlicz norm of stochastic integralsw.r.t the Brownian sheet is large when the uniform norm of the Brownian sheet issmall. This is established in the following series of lemmas:

Lemma 11. For every u¿ 0 and v¿ 0 with u¿ 16v and u¿ 2√log 2

P[||W ||∗¿u; ||W ||6v]662g−1(u2=16v2)exp

(− u2

log 2log( u16v

));

where; g−1 is the inverse function of g(r) = 2r=(1 + r); r¿1.Consequently; the inequality holds with the norm || · ||∗∗ instead of || · ||∗.

Proof. Let u¿ 0 and v¿ 0 and recall that

||W ||∗ =max(|W (1; 1)|; ||W ||∗1; ||W ||∗2; ||W ||∗3)with

||W ||∗1 = supj¿0

sup2 j+16n62 j+1

|Cn(W (:; 1))|√1 + j

;

||W ||∗2 = supj¿0

sup2 j+16n62 j+1

|Cn(W (1; :))|√1 + j

;

||W ||∗3 = supj; j′¿0


|Cn;n′(W )|√(1 + j)(1 + j′)

:

Here, Kj; j′ = {(n; n′); 2 j + 16n62 j+1; 2 j′ + 16n′62 j′+1}. We haveP[||W ||∗¿u ; ||W ||6v]6P0 + P1 + P2 + P3;

where

P0 = P[|W (1; 1)|¿u; ||W ||6v];

P1 = P[||W ||∗1¿u; ||W ||6v];

P2 = P[||W ||∗2¿u; ||W ||6v];

P3 = P[||W ||∗3¿u; ||W ||6v]:


Now, since u¿v; P0 = 0. We are going to estimate P1, P2 and P3. We have

P1 = P

[supj¿0

sup2 j+16n62 j+1

|Cn(W (:; 1))|√1 + j

¿u; ||W ||6v

]

6∑j¿0

2 j+1∑n=2 j+1

P[|Cn(W (·; 1))|¿u√1 + j; ||W ||6v]:

Using the fact that, Cn(W (:; 1)) are one-dimensional centred Gaussian random vari-ables with variance equals to 1, and |Cn(W (:; 1))| is dominated by 4v

√2 j on the set

{||W ||6v}, it follows that, if

j0 = inf

{j¿1; (1 + j)2−j6

(4vu

)2};

i.e. we may choose j0 such that

j0 =[g−1

(u2

16v2

)]+ 1

with g−1 being the inverse function of g(r) = 2r=(1 + r); r¿1 and [r] the entire partof the positive number r, we have

P16∑j¿j0

2 jP[|N (0; 1)|¿u√1 + j ]

6 22 j0 exp(−u2j0=2):

Here, we used the exponential inequality for a one-dimensional centred Gaussian vari-able N (0; 1):

P[|N (0; 1)|¿a]62 exp(−a2=2):

Therefore,

P1622g−1(u2=16v2) exp

(− u2

log 2log( u4v

)):

P2 can be estimated by the same arguments and therefore we omit the details. Hence,

P1 + P2642g−1(u2=16v2) exp

(− u2

log 2log( u4v

)): (18)

It remains to estimate P3. We have

P3 = P

[sup

j; j′¿0sup

(n;n′)∈Kj; j′

|Cn;n′(W )|√(1 + j)(1 + j′)

¿u; ||W ||6v

]

6∑

j; j′¿0

∑(n;n′)∈Kj; j′

P[|Cn;n′(W )|¿u√(1 + j)(1 + j′); ||W ||6v]:

Since Cn;n′(W ) are one-dimensional centred Gaussian random variables with varianceequals to 1, and |Cn;n′(W )| is dominated by 16v

√2 j′+j on the set {||W ||6v}, it follows


that ∑j; j′¿0

∑(n;n′)∈Kj; j′

P[|Cn;n′(W )|¿u√(1 + j)(1 + j′); ||W ||6v]

6∑

(j; j′); j+j′¿k0

2 j′+jP[|N (0; 1)|¿u√(1 + j)(1 + j′) ];

where

k0 = inf{k¿1; (1 + k)2−k6(16v=u)2}:Hence, proceeding as for P1, we �nally get

P3622g−1((u=16v)2) exp

(− u2

log 2log( u16v

));

which, together with (18), give the desired estimate.

For ease of notation, we denote in the following lemma, �(z; x) for �(z; �; x) etc.

Lemma 12. Let X�and Y solve the following stochastic di�erential equations:

X�z = xz +

√�∫Rz

�(z; X��)W (d�) +

∫Rz

�(z; X��)h� d�+

∫Rz

b(z; X��) d�; (19)

where W is Brownian sheet and

Yz = xz +∫Rz

�(z; Y�)h� d�+∫Rz

b(z; Y�) d�:

Set

I �z =√�∫Rz

�(z; X��)W (d�):

Then there exists a constant C = C(h; L)¿ 0 such that

||X � − Y ||∗∗6C||I �||∗∗:

Proof. First note that I � is null on the axes. Thus, expression (8) yields

||I �||6||I �||∗∗:On the other hand, using Gronwall’s lemma for the two-parameter case, we get

||X � − Y ||6||I �|| exp(L∫ 1

0

∫ 1

0(1 + |h�| d�)

);

Hence,

||X � − Y ||6||I �||∗∗ exp(L(1 + ||h||H)) :=||I �||∗∗C: (20)

We have

||X � − Y ||∗∗6||I �||∗∗ +∣∣∣∣∣∣∣∣∫R·(�(·; X �

�; Y�) + (h�; ·; X ��; Y�)) d�

∣∣∣∣∣∣∣∣∗∗

; (21)

where

�(z; x; y) = b(z; x)− b(z; y)


and

(h·; z; x; y) = (�(z; x)− �(z; y))h· :

Let us �rst give an estimate of the second term of Eq. (21). From the de�nition of|| : ||∗∗ we have∣∣∣∣

∣∣∣∣∫R·(�(·; X �

�; Y�) + (h�; ·; X ��; Y�)) d�

∣∣∣∣∣∣∣∣∗∗6

6∑r=1

Tr;

where

T16 sup06s1¡s261

∫ s2s1

∫ 10 |�((s2; 1); X �

�; Y�)| d�!(s2 − s1)

+ sup06s1¡s261

∫ s10

∫ 10 |�((s2; 1); X �

�; Y�)− �((s1; 1); X��; Y�)| d�

!(s2 − s1)

and

T36 sup06s1¡s261

∫ s2s1

∫ 10 | (h�; (s2; 1); X

��; Y�)| d�

!(s2 − s1)

+ sup06s1¡s261

∫ s10

∫ 10 | (h�; (s2; 1); X

��; Y�)− (h�; (s1; 1); X

��; Y�)| d�

!(s2 − s1):

Furthermore, Assumption (H2) and (20) imply that

T16 sup06s1¡s261

L∫ s2s1

∫ 10 |X �

� − Y�| d�!(s2 − s1)

+ sup06s1¡s261

L∫ s10

∫ 10 ’(s2 − s1)|X �

� − Y�| d�!(s2 − s1)

6 LC||I �||∗∗ sup06s1¡s261

’(s2 − s1) + (s2 − s1)!(s2 − s1)

and

T36 sup06s1¡s261

L∫ s2s1

∫ 10 |X �

� − Y�||h�| d�!(s2 − s1)

+ sup06s1¡s261

L∫ s10

∫ 10 ’(s2 − s1)|X �

� − Y�||h�| d�!(s2 − s1)

6 2LC||I �||∗∗(1 + ||h||H) sup06s1¡s261

’(s2 − s1) +√s2 − s1

!(s2 − s1):

Since

sup06s1¡s261

’(s2 − s1) + (s2 − s1)!(s2 − s1)

:=D1

and

sup06s1¡s261

’(s2 − s1) +√s2 − s1

!(s2 − s1):=D2


are �nite we �nd that

T1 + T362LC||I �||∗∗(1 + ||h||H)(D1 + D2):

In a similar way we get

T2 + T462LC||I �||∗∗(1 + ||h||H)(D1 + D2):

Since T5 can be deduced from T6 by replacing h with 1, it su�ces to estimate T6.Using the notation z1 = (s1; t1), z2 = (s2; t2) with z1¡z2 and the fact that∣∣∣∣

∫ s2

0

∫ t2

0 (h�; (s2; t2); X

��; Y�) d�−

∫ s1

0

∫ t2

0 (h�; (s1; t2); X

��; Y�) d�

−∫ s2

0

∫ t1

0 (h�; (s2; t1); X

��; Y�) d�+

∫ s1

0

∫ t1

0 (h�; (s1; t1); X

��; Y�) d�

∣∣∣∣is dominated by∣∣∣∣

∫ s1

0

∫ t1

0 (h�; Rz1 ; z2 ; X

��; Y�) d�

∣∣∣∣+∣∣∣∣∫ s2

s1

∫ t2

t1 (h�; (s2; t2); X

��; Y�) d�

∣∣∣∣+∣∣∣∣∫ s1

0

∫ t2

t1( (h�; (s2; t2); X

��; Y�)− (h�; (s1; t2); X

��; Y�)) d�

∣∣∣∣+∣∣∣∣∫ s2

s1

∫ t1

0( (h�; (s2; t2); X

��; Y�)− (h�; (s2; t1); X

��; Y�)) d�

∣∣∣∣ :Assumption (H2) implies that

T66 sup06t1¡t26106s1¡s261

’((s2 − s1)(t2 − t1))!(s2 − s1)!(t2 − t1)

L∫ s1

0

∫ t1

0|X �

� − Y�||h�| d�

+ sup06t1¡t26106s1¡s261

L∫ s2s1

∫ t2t1|X �

� − Y�||h�| d�!(s2 − s1)!(t2 − t1)

+ sup06t1¡t26106s1¡s261

’(s2 − s1)!(s2 − s1)!(t2 − t1)

L∫ s1

0

∫ t2

t1|X �

� − Y�||h�| d�

+ sup06t1¡t26106s1¡s261

’(t2 − t1)!(s2 − s1)!(t2 − t1)

L∫ s2

s1

∫ t1

0|X �

� − Y�||h�| d�:

Set

D3 := sup06t1¡t26106s1¡s261

’((s2 − s1)(t2 − t1))!(s2 − s1)!(t2 − t1)

;

D4 := sup06t1¡t26106s1¡s261

’(s2 − s1) +√t2 − t1

!(s2 − s1)!(t2 − t1)

and

D5 := sup06t1¡t26106s1¡s261

√(s2 − s1)(t2 − t1)

!(s2 − s1)!(t2 − t1):


Thus,

T662LC||I �||∗∗(1 + ||h||H)(D3 + 2D4 + D5);

whence,

||X � − Y ||∗∗6C||I �||∗∗:The proof of the lemma is now complete.

Lemma 13. For every R¿ 0 and �¿ 0 large enough; there exist �0¿ 0 and �¿ 0such that

P[∣∣∣∣∣∣∣∣√�

∫R·

�(·; �; X �� )W (d�)

∣∣∣∣∣∣∣∣∗∗¿�; ||√�W ||6�

]6exp−

(R�

);

for every �∈ (0; �0].

Proof. For any N ¿ 0, we consider the approximating sequence of the process X �·

de�ned by

X �;N� = X �

(j=2N ;k=2N ) if �∈�Njk =

[j2N

;j + 12N

[[k2N

;k + 12N

[

for all j = 0; : : : ; 2N − 1 and k = 0; : : : ; 2N − 1.Following similar arguments as in Corollary 3.8 in [RS] and using some exponential

inequalities (see also N’zi (1994) or Priouret (1982)), it is easy to check that for allR¿ 0 and all �¿ 0, there exist �1 and N1 such that for every 0¡�¡�1 and N¿N1,

P[||X � − X �;N ||¿�]6 13 exp(−R=�):

We have

P[∣∣∣∣∣∣∣∣√�

∫R·

�(·; �; X�)W (d�)∣∣∣∣∣∣∣∣∗∗¿�; ||√�W ||6�

]6P[||X �

· − X �;N· ||¿�]

+P

[2∣∣∣∣∣∣∣∣√�

∫R·(�(·; �; X �

� )− �(·; �N ; X �;N� ))W (d�)

∣∣∣∣∣∣∣∣∗∗

¿�; ||X �· − X �;N

· ||¡�

]

+P[2∣∣∣∣∣∣∣∣√�

∫R·

�(·; �N ; X �;N� )W (d�)

∣∣∣∣∣∣∣∣∗∗¿�; ||√�W ||6�

]:

By the Lipschitz condition on � i.e. (H2) and Propositions 6 and 7 the second termin the right-hand side is dominated by

K1exp(− �2

4��2K2

);

with K1 and K2 are some positive constants. Moreover, in view of the arguments usedin Appendix B, by replacing d(h� − k�) with W (d�), it follows immediately that∣∣∣∣

∣∣∣∣∫R·

�(·; �N ; X �;N� )W (d�)

∣∣∣∣∣∣∣∣∗∗622N M ||W ||∗∗:


Hence, in view of Lemma 11 we conclude that

P[∣∣∣∣∣∣∣∣√�

∫R·

�(·; �N ; X �;N� )W (d�)

∣∣∣∣∣∣∣∣∗∗¿�; ||√�W ||6�

]

662g−1((�=(22N M)4�)2) exp

(− �2

�(22N M)2log 2log(

�

(22N M)16�

)):

For a given R¿ 0 and �¿ 0 we choose �¿ 0 small enough so that �2=4�2K2¿R,then we choose

�2¡1

log(3K1)

(�2

4�2K2− R

)

and then we choose N large enough and � su�ciently small such that

62g−1((�=(22N M)4�)2) exp

(− �2

�(22N M)2log 2log(

�

(22N M)16�

))613exp(−R

�

);

for 0¡�6�0 = min(�1; �2), which completes the proof of the lemma.

Combining Lemmas 12 and 13 we get the following:

Lemma 14. For every R¿ 0; a¿ 0 and �¿ 0 large enough there exist �¿ 0 and�0 ∈ ]0; 1] such that for every h∈L(a) and �∈ ]0; �0] we have

P[||X � − Sx(h)||�;! ¿�; ||√�W ||6�]6exp−(R�

);

where X�satis�es Eq. (19) and W :=W − h=

√�.

Proof of Proposition 10. Let h∈L(a) and de�ne on (;F; (Fz)z∈ I2 ) the probabilityQ� by

dQ�

dP:= exp

(1√�

∫I2h�W (d�)− �(h)

�

):

Then, under Q�, W :=W − h=√� is a Wiener process. Now, set

A := {||X � − Sx(h)||∗∗¿�; ||√�W − h||6�}and

Z� = exp(− 1√

�

∫I2h�W (d�)

):

Thus,

P(A)6 EQ�

[dPdQ� ; A ∩

(Z� ¡ exp

(��

))]+ P

(Z�¿exp

(��

))

6 exp(a+ �

�

)Q�(||X � − Sx(h)||∗∗¿�; ||√�W ||6�) + exp

(a− �

�

):

Now, choose �0 = �0(R; a) such that

exp(a− �

�

)612exp(−R

�

)for �¿�0;


then apply Lemma 14 (with Q� instead of P) to choose � and �0 such that

exp(a+ �

�

)Q�(||X � − Sx(h)||∗∗¿�; ||√�W ||6�)6

12exp(−R

�

)

for all �∈ (0; �0], which ends the proof of the proposition.

5. For Further Reading

The following reference is also of interest to the reader: Roynette, 1993

Acknowledgements

We would like to thank the two referees of the paper for several insightful comments.

Appendix A

In this appendix we sketch the proofs of Propositions 5–7. In the sequel, we denote

Kj; j′ = {(n; n′); 2 j + 16n62 j+1; 2 j′ + 16n′62 j′+1}:

Proof of Proposition 5. Let M (s1; t1) = M ([0; s1] × [0; t1]) =∫ s10

∫ t10 H (s; t)W (d s; dt).

We have

P[||M ||∗¿u; ||H ||61]6A0 + A1 + A2 + A3;

where

A0 = P[|M (1; 1)|¿u; ||H ||61] ;A1 = P[||M ||∗1¿u; ||H ||61] ;A2 = P[||M ||∗2¿u; ||H ||61] ;A3 = P[||M ||∗3¿u; ||H ||61]:

To estimate Ai for i = 0; : : : ; 3, we note that, since M (·; ·) vanishes on the axes,Proposition 4 yields

A064 exp(−u2=18): (A.1)

Moreover,

A1 + A26∑j¿0

2 j+1∑n=2 j+1

P[|Cn(M (:; 1))|¿u√1 + j; ||H ||61]

+∑j¿0

2 j+1∑n=2 j+1

P[|Cn(M (1; :))|¿u√1 + j; ||H ||61]:


Now,

P[|Cn(M (:; 1))|¿u√1 + j; ||H ||61]6B1 + B2;

P[|Cn(M (1; :))|¿u√1 + j; ||H ||61]6B′

1 + B′2;

where, for r = 1; 2,

Br = P

[∣∣∣∣∣√2 j

∫ (2k−r+1)=2 j+1

(2k−r)=2 j+1

∫ 1

0H (s; t)W (d s; dt)

∣∣∣∣∣¿u2

√1 + j; ||H ||61

];

B′r = P

[∣∣∣∣∣√2 j

∫ 1

0

∫ (2k−r+1)=2 j+1

(2k−r)=2 j+1H (s; t)W (d s; dt)

∣∣∣∣∣¿u2

√1 + j; ||H ||61

]:

Hence, in view of Proposition 4, we get

B1 + B268 exp(−u2(1 + j)

18

)and

B′1 + B′

268 exp(−u2(1 + j)

18

):

Therefore,

A1 + A2616∑j¿0

2 j exp(−u2(1 + j)

18

):

Since u¿ 12√log 2, we get

A1 + A2632exp(− u2

18× 4)

: (A.2)

To estimate A3, we remark that, for every u¿ 12√log 2, we have

A364∑

j; j′¿0

∑(n;n′)∈Kj; j′

P[|Cn;n′(M)|¿u√(1 + j)(1 + j′); ||H ||61]:

Using Proposition 4 and the fact that |Cn;n′(M)| is dominated by four terms of theform

√2 j+j′

∫ k=2 j

(k−1)=2 j

∫ k′=2 j′

(k′−1)=2 j′H (s; t)W (d s; dt)

whose quadratic variation is bounded by 1, we get

A36 4:4∑

j; j′¿0

2 j′+j exp(−u2(1 + j)(1 + j′)

42 × 18)

6 16 exp(− u2

122

)∑j¿0

2 j exp(− ju2

2× 122)

2

6 64 exp(− u2

122

);

which together with Eqs. (A.1) and (A.2) yield the estimate of the lemma.


Proof of Proposition 6. We will only prove (i). The proof of (ii) follows in a similar way.Set G(z) =

∫Rz

H (z; �)W (d�). Clearly G(:) vanishes on the axes. We have

P[||G||∗¿�; ||H ||6�]6A′0 + A′

1 + A′2 + A′

3;

whereA′0 = P[|G(1; 1)|¿�; ||H ||6�];

A′1 = P[||G||∗1¿�; ||H ||6�];

A′2 = P[||G||∗2¿�; ||H ||6�];

A′3 = P[||G||∗3¿�; ||H ||6�]:

We want to estimate A′i for i = 0; : : : ; 3. The exponential inequality in Theorem 2.4 in

[RS] yields

A′064 exp− (�2=18C0�2); (A.3)

for � large enough, where C0 is a numerical constant which can be computed by theGarcia–Rodemich–Rumsey’s lemma. Furthermore,

A′1 + A′

26∑j¿0

2 j+1∑n=2 j+1

P[|Cn(G(:; 1))|¿�√1 + j; ||H ||6�]

+∑j¿0

2 j+1∑n=2 j+1

P[|Cn(G(1; :))|¿�√1 + j; ||H ||6�]:

Now,P[|Cn(G(:; 1))|¿�

√1 + j; ||H ||6�]6B1 + B2;

P[|Cn(G(1; :))|¿�√1 + j; ||H ||6�]6B′

1 + B′2;

where, for r = 1; 2,

Br = P

[√2 j

∣∣∣∣∣∫ (2k−r+1)=2 j+1

0

∫ 1

0H((

2k − r + 12 j+1 ; 1

); �)

W (d�)

−∫ (2k−r)=2 j+1

0

∫ 1

0H((

2k − r2 j+1 ; 1

); �)

W (d�)

∣∣∣∣∣¿�2

√1 + j; ||H ||6�

]

6 P

[√2 j

∣∣∣∣∣∫ (2k−r+1)=2 j+1

(2k−r)=2 j+1

∫ 1

0H((

2k − r + 12 j+1 ; 1

); �)

W (d�)

∣∣∣∣∣¿

�4

√1 + j; ||H ||6�

]

+P

[√2 j

∣∣∣∣∣∫ (2k−r)=2 j+1

0

∫ 1

0

(H((

2k − r + 12 j+1 ; 1

); �)

−H((

2k − r2 j+1 ; 1

); �))

W (d�)

∣∣∣∣∣¿

�4

√1 + j; ||H ||6�

];


and

B′r = P

[√2 j

∣∣∣∣∣∫ 1

0

∫ (2k−r+1)=2 j+1

0H((1;2k − r + 12 j+1

); �)

W (d�)

−∫ 1

0

∫ (2k−r)=2 j+1

0H((1;2k − r2 j+1

); �)

W (d�))

∣∣∣∣∣¿�2

√1 + j; ||H ||6�

]

6 P

[√2 j

∣∣∣∣∣∫ 1

0

∫ (2k−r+1)=2 j+1

(2k−r)=2 j+1H((1;2k − r + 12 j+1

); �)

W (d�)

∣∣∣∣∣¿

�4

√1 + j; ||H ||6�

]

+P

[√2 j

∣∣∣∣∣∫ 1

0

∫ (2k−r)=2 j+1

0

(H((1;2k − r + 12 j+1

); �)

−H((1;2k − r2 j+1

); �))

W (d�)

∣∣∣∣∣¿

�4

√1 + j; ||H ||6�

]:

Hence, in view of the exponential inequality of Theorem 2.4 in [RS], we get

B1 + B2616 exp−(

�2(1 + j)72(�2 + L2)C0

)

and

B′1 + B′

2616 exp−(

�2(1 + j)72(�2 + L2)C0

):

Therefore,

A′1 + A′

2632∑j¿0

2 j exp−(

�2(1 + j)72(�2 + L2)C0

):

Since �¿ 6√�2 + L2

√2C0 log 2, we get

A′1 + A′

2664 exp−(

�2

72(�2 + L2)C0

): (A.4)

To estimate A′3, we remark that for every � su�ciently large,

A′364

∑j; j′¿0

∑(n;n′)∈Kj; j′

P[|Cn;n′(G)|¿�√(1 + j)(1 + j′); ||H ||6�]

and that |Cn;n′(G)| is dominated by four terms of the form√2 j+j′(Z1 + Z2 + Z3 + Z4);


where

Z1 =

∣∣∣∣∣∫ (k−1)=2 j

0

∫ (k′−1)=2 j′

0H (R((k−1)=2 j ; k=2 j);((k′−1)=2 j′ ; k′=2 j′ ); �)W (d�)

∣∣∣∣∣ ;

Z2 =

∣∣∣∣∣∫ (k−1)=2 j

0

∫ k′=2 j′

(k′−1)=2 j′

(H((

k2 j ;

k ′

2 j′

); �)− H

(((k − 1)2 j ;

k ′

2 j′

); �))

W (d�)

∣∣∣∣∣ ;

Z3 =

∣∣∣∣∣∫ k=2 j

(k−1)=2 j

∫ k′=2 j′

(k′−1)=2 j′H((

k2 j ;

k ′

2 j′

); �)

W (d�)

∣∣∣∣∣ ;

Z4 =

∣∣∣∣∣∫ k=2 j

(k−1)=2 j

∫ (k′−1)=2 j′

0

(H((

k2 j ;

k ′

2 j′

); �)− H

((k2 j ;

k ′ − 12 j′

); �))

W (d�)

∣∣∣∣∣ :Since the quadratic variation of each of Zi’s for i = 1; 2; 4 is bounded by L22−(j+j′)

and the quadratic variation of Z3 is bounded by �22−(j+j′), the quadratic variation ofCn;n′(G) is dominated by 4(4�2 + 12L2), the exponential inequality for the uniformnorm (Theorem 2.4 in [RS]) yields

A′36 4:16

∑j; j′¿0

2 j′+j exp−(

�2(1 + j)(1 + j′)18(4�2 + 12L2)C0

)

6 64 exp−(

�2

18(4�2 + 12L2)C0

)∑j¿0

2 j exp(− j�2

18(4�2 + 12L2)C0

)2

:

Therefore, for �¿ 6√�2 + 3L2

√6C0 log 2

A′364:64 exp−

(�2

72(�2 + 3L2)C0

);

which together with Eqs. (A.3) and (A.4) yield the estimate of (i).

Proof of Proposition 7. We will only prove (i). The proof of (ii) follows using thesame arguments. Set M (s; t) =

∫ s0

∫Rs; t

H (�; r)W (d�) dr. We have

P[||M (·; ·)||∗∗ ¿�; ||H ||6�]

6P[

sup06s1¡s261

|M (s2; 1)−M (s1; 1)|!(s2 − s1)

¿�3; ||H ||6�

]

+P[

sup06t1¡t261

|M (1; t2)−M (1; t1)|!(t2 − t1)

¿�3; ||H ||6�

]

+P

sup06t1¡t26106s1¡s261

|M (Rz1 ; z2 )|!(s2 − s1)!(t2 − t1)

¿�3; ||H ||6�

=P1 + P2 + P3

where

M (Rz1 ; z2 ) =M (s2; t2)−M (s2; t1)−M (s1; t2) +M (s1; t1):


Let us �rst estimate P1. We have

|M (s2; 1)−M (s1; 1)|

6

∣∣∣∣∣∫ s2

s1

∫Rs2 ;1

H (�; r)W (d�)dr

∣∣∣∣∣+∣∣∣∣∣∫ s2

s1

∫ 1

0

(∫ s1

0H (�; r)dr

)W (d�)

∣∣∣∣∣ :Thus,

P16 P

[sup

06s1¡s261

| ∫ s2s1

∫ 10 (∫ s10 H (�; r)dr)W (d�)|!(s2 − s1)

¿�6; ||H ||6�

]

+P

[sup

06s1¡s261

| ∫ s2s1(∫Rs2 ;1

H (�; r)dr)W (d�))dr|!(s2 − s1)

¿�6; ||H ||6�

]

= P11 + P21 :

Using Proposition 5, we get

P116100 exp−(

�2

36 · 144�2)

:

Now,

P21 6 P

[sup

06r; s261

∣∣∣∣∣∫Rs2 ;1

H (�; r)W (d�)

∣∣∣∣∣¿�; ||H ||6�

]

+P

[sup

06s1¡s261

| ∫ s2s1(∫Rs2 ;1

H (�; r)W (d�))dr|!(s2 − s1)

¿�6;

sup06r; s261

∣∣∣∣∣∫Rs2 ;1

H (�; r)W (d�)

∣∣∣∣∣6�

]:

Choosing 3�¡� and applying Proposition 5, we get

P21 6 P

[sup

06r; s261

∣∣∣∣∣∫Rs2 ;1

H (�; r)W (d�)

∣∣∣∣∣¿�; ||H ||6�

]

6 exp(− �2C24�2 + 12L2

):

P2 is estimated in a similar way as P1. It remains to estimate P3. We note that

|M (Rz1 ; z2 )| =∣∣∣∣∣∫ s2

0

(∫Rs2 ; t2

H (�; r)W (d�)dr −∫Rs2 ; t1

H (�; r)W (d�)

)dr

+∫ s1

0

(∫Rs1 ; t1

H (�; r)W (d�)dr −∫Rs1 ; t2

H (�; r)W (d�)

)dr

∣∣∣∣∣6

∣∣∣∣∣∫ s2

s1

(∫Rs2 ; t2

H (�; r)W (d�)−∫Rs2 ; t1

H (�; r)W (d�)

)dr

∣∣∣∣∣+

∣∣∣∣∣∫ s1

0

∫Rz1 ; z2

H (�; r)W (d�)dr

∣∣∣∣∣ :


Thus,

P36 P

sup06t1¡t26106s1¡s261

| ∫ s2s1(∫ s20

∫ t2t1

H (�; r)W (d�)) dr|!(s2 − s1)!(t2 − t1)

¿�6; ||H ||6�

+P

sup06t1¡t26106s1¡s261

| ∫Rz1 ; z2(∫ s10 H (�; r)dr)W (d�))|

!(s2 − s1)!(t2 − t1)¿

�6; ||H ||6�

= P13 + P23 :

But, for 6�¡�,

P13 6 P

sup06t1¡t26106r; s261

| ∫ s20

∫ t2t1

H (�; r)W (d�)|!(t2 − t1)

¿�; ||H ||6�

6 exp(− �2C24�2 + 12L2

);

by Proposition 6. Furthermore on the set {||H ||6�}; || ∫ ·0 H (·; r)dr||6�. Hence, by

Proposition 6, we get

P236exp(− �2C24�2 + 12L2

):

for � su�ciently large.

Appendix B

In this section we shall prove inequality (17) above. The proof of the inequality∣∣∣∣∣∣∣∣∫R·

�(·; �N ; X �;N� )W (d�)

∣∣∣∣∣∣∣∣∗∗622N M ||W ||∗∗

that has been used in the proof of Lemma 13 is omitted, since it can proved usingsimilar arguments for the appropriate norm, by replacing d(h� − k�) with W (d�).Set

�(z) =∫Rz

�(z; �N ; Sx(h)N� )d(h� − k�);

where

Sx(h)N� = Sx(h)(j=2N ; k=2N ) if �∈�Njk =

[j2N

;j + 12N

[×[

k2N

;k + 12N

[and

�N =(

j2N

;k2N

)if �∈�N

jk :

In the rest of the proof we will sometimes use the following notation:

f(A) =∫Af � d� for A∈ I 2:


We shall prove that there exists a positive constant M such that

||�||�;!622N M ||h− k||�;!:For r ¿ 0 and z = (s; t), we have

|�(z + re1)− �(z)|6∣∣∣∣∫ s+r

s

∫ t

0�((s+ r; t); �N ; Sx(h)N� )d(h� − k�)

∣∣∣∣+∣∣∣∣∫ s

0

∫ t

0(�((s+ r; t); �N ; Sx(h)N� )− �((s; t); �N ; Sx(h)N� ))d(h� − k�)

∣∣∣∣ :Approximating the integrals by Riemann sums, we get

|�(z + re1)− �(z)|6∣∣∣∣∣∣2N−1∑j;l=0

�((s+ r; t);

(j2N

;l2N

); Sx(h)(j=2N ; k=2N )

)

×(h− k)([s; s+ r]× [0; t] ∩ �Njl)

∣∣∣∣∣∣+

∣∣∣∣∣∣2N−1∑j;l=0

(�((s+ r; t);

(j2N

;l2N

); Sx(h)(j=2N ; k=2N )

)

−(�((s; t);

(j2N

;l2N

); Sx(h)(j=2N ;k=2N )

))

×(h− k)([s; s+ r]× [0; t] ∩ �Njl)

∣∣∣∣∣∣ :Now, relation (H2) yields

|�(z + re1)− �(z)|62N−1∑j;l=0

M |(h− k)([s; s+ r]× [0; t] ∩ �Njl)|

+2N−1∑j;l=0

L’(r)|(h− k)([0; s]× [0; t] ∩ �Njl)|

6 22NM |(h− k)(z + re1)− (h− k)(z)|+ 22NL’(r)||h− k||:Similarly,

|�(z + re2)− �(z)|622NM |(h− k)(z + re2)− (h− k)(z)|+ 22NL’(r)||h− k||:Hence, for i = 1; 2,

||�r; i�||p622N+1−1=p(M ||�r; i(h− k)||p + L’(r)||h− k||):Therefore,

!p;i(�; t) = sup|r|6t

||�r; i�||p

6 22N+1−1=p(M sup

|r|6t||�r; i(h− k)||p + L’(t)||h− k||

):


and then

!�; i(�; t) = supp¿1

!p;i(�; t)√p

622N (M!�;i(h− k; t) + L’(t)||h− k||);

which implies that

sup0¡t61

!�; i(�; t)!(t)

622N(M sup

0¡t61

!�; i(h− k; t)!(t)

+ L sup0¡t61

’(t)!(t)

||h− k||)

: (B.1)

Now, for r = (r1; r2)∈R2+, we have|�(r1 ; r2)(�(z))|

6∣∣∣∣∫Rz

�(Rz; z+r ; �N ; Sx(h)bN� )d(h� − k�)

∣∣∣∣+

∣∣∣∣∣∫Rz; z+r

�(z + r; �N ; Sx(h)N� )d(h� − k�)

∣∣∣∣∣+∣∣∣∣∫ s

0

∫ t+r2

t(�(z + r; �N ; Sx(h)N� )− �((s; t + r2); �N ; Sx(h)N� ))d(h� − k�)

∣∣∣∣+∣∣∣∣∫ s+r1

s

∫ t

0(�(z + r; �N ; Sx(h)N� )− �((s+ r1; t); �N ; Sx(h)N� ))d(h� − k�)

∣∣∣∣ :Approximating the terms appearing in the right-hand side by the Riemann sum asabove, we get that

|�(r1 ; r2)(�(z))|6 22N (4L’(r1r2)||h− k||+M |�(r1 ; r2)(h− k)(z)|)+22N (L’(r1)|�(r2 ;2)(h− k)(z)|+ L’(r2)|�(r1 ;1)(h− k)(z)|):

Therefore,

2−2(N+1−1=p)!p(�; (t1; t2))

=2−2(N+1−1=p) sup|r1|6t1 ;|r2|6t2

||�(r1 ; r2)�||p

64L’(t1t2)||h− k||+M sup|r1|6t1 ;|r2|6t2

||�(r1 ; r2)(h− k)||p

+L’(t1) sup|r2|6t2

||�(r2 ;2)(h− k)||p + L’(t2) sup|r1|6t1

||�(r1 ;1)(h− k)||p;

where we have used the assumption that ’(·) is increasing. Hence,

!�(�; (t1; t2)) = supp¿1

!p(�; (t1; t2))√p

= supp¿1

sup|r1|6t1 ;|r2|6t2

||�(r1 ; r2)(�)||p√p

satis�es

2−2N sup0¡t1 ;t261

!�(�; (t1; t2))!(t1)!(t2)

6M sup0¡t1 ;t261

!�(h− k; (t1; t2))!(t1)!(t2)

+ 4L||h− k|| sup0¡t1 ;t261

’(t1t2)!(t1)!(t2)

+L sup0¡t1 ;t261

’(t1)!(t1)

!�;2(h− k; t2)!(t2)

+ L’(t2)!(t2)

!�;1(h− k; t1)!(t1)

: (B.2)


Now, combining Eqs. (B.1) and (B.2), together with the de�nition of the norm || · ||�;!,we obtain

||�||�;!622N M ||h− k||�;!:as is claimed.

References

Azencott, R., 1980. Grandes d�eviations et applications. Ecole d’Et�e de Probabilit�es de Saint-Flour VIII, 1978.Lecture Notes in Mathematics, vol. 774. Springer, New York, pp. 1–176.

Ben Arous, G., Ledoux, M., 1994. Grandes d�eviations de Freidlin-Wentzell en norme h�olderienne. S�eminairede Probabilit�es 1583, 293–299.

Ciesielski, Z., 1993. Orlicz spaces, spline systems and Brownian motion. Const. Approx. 9, 191–208.Ciesielski, Z., Domsta, J., 1972. Construction of an orthonormal basis in Cm(Id) and Wm

p (Id). Const.

Approx. 41, 211–224.Ciesielski, Z., Kamont, A., 1995. Levy’s fractional Brownian random �eld and function spaces. Acta Sci.Math. (Szeged) 60, 99–118.

Ciesielski, Z., Kerkyacharian, G., Roynette, B., 1993. Quelques espaces fonctionnels associ�es �a des processusgaussiens. Stud. Math. 107, 171–204.

Doss, H., Dozzi, M., 1987. Estimations de grandes d�eviations pour les processus de di�usion �a param�etremultidimensionnel. Lecture Notes in Mathematics, vol. 1204. Springer, Berlin, pp. 68–90.

Dozzi, M., 1989. Stochastic Processes with a Multidimensional Parameter. Longman Scienti�c & Technical,Essex.

Eddahbi, M., 1997. Large deviations for solutions of hyperbolic SPDEs in H�older norm. Potential Anal. 7,517–537.

Fernique, X., 1971. R�egularit�e de processus gaussiens. Invent. Math. 12, 304–320.Jonsson, A., Wallin, H., 1984. Function spaces on subsets of Rn. Mathematical Reports, vol. 2, Part 1,pp. 1–221. (Issue Ed.), Peetre, J.

Kamont, A., 1994. Isomorphism of some anisotropic Besov and sequence spaces. Stud. Math. 110 (2),169–189.

Kamont, A., 1996. On the fractional anisotopic Wiener �eld. Probab. Math. Statist. 16 (1), 85–98.Mishura, S., 1987. Exponential estimates for two-parameter martingales. Ukrainian Math. J. 39, 275–279.Norris, J., 1995. Twisted sheets. J. Funct. Anal. 123, 273–334.N’zi, M., 1994. On the large deviation estimates for two-parameter di�usion processes and applications.Stochastics Stochastics Rep. 50, 65–83.

Peetre, J., 1976. New Thoughts on Besov spaces. Duke Univ. Math. Series, Durham, NC.Priouret, P., 1982. Remarques sur les petites perturbations de syst�emes dynamiques. S�eminaire de Probabilit�esXVI. Lecture Notes in Mathematics, vol. 920. Springer, New York, pp. 184–200.

Rao, M.M., Ren, Z.D., 1991. Theory of Orlicz Spaces. Marcel Dekker, New York.Ropela, S., 1976. Spline Basis in Besov spaces. Bul Acad. Polon. Scio. S�er. Sci. Math., Astronom. Phys.24, 319–325.

Roynette, B., 1993. Mouvement brownien et espaces de Besov. Stochastics Stochastics Rep. 43, 221–260.Rovira, C., Sanz-Sol�e, M., 1996. The law of the solutions to a nonlinear hyperbolic SPDE. J. Theory Probab.9 (4), 863–901.

Rovira, C., Sanz-Sol�e, M., 1997. Large deviations for stochastic Volterra equations in the plane, preprint.

large deviations for a stochastic volterra-type equation in the besov–orlicz space

Documents