Introduction Alargedeviationsprincipleforrandomevolutiondelayequationsinh¨olderspace JournaldeMath´ematiques,Math´ematiques&ApplicationsFondamentalesetInformatique

Math´ ematiques & Applications Fondamentales et Informatique

JMMAFI 2014, Vol.1,pp 17-32

A large deviations principle for random evolution delay equations in h¨ older space

S.H. RANDRIAMANIRISOA¹, T.J. RABEHERIMANANA² D´epartement de Math´ematiques et Informatique, B.P. 906, Ankatso 101,

Antananarivo, Madagascar

1hasinasran@gmail.com,²rabeherimanana.toussaint@gmx.f r

R´esum´e: Dans cet article, nous ´etudions un principe de grandes d´eviations con- cernant les ´equations d’ ´evolutions al´eatoires avec d´elai en norme h¨old´erienne.

Mots-cl´es: principe de grandes d´eviations, ´equations d’´evolution al´eatoire avec retard, espace h¨old´erien

Abstract: In this paper, we develop a large deviations principle for random evolution delay equations driven by small multiplicative white noise in h¨olderian space.

Keywords: large deviations principle,random evolution delay equation, h¨older space.

Introduction

For the last decades, Large deviations took the interests of many author. The re- sults found with large deviations principle have important applications to many areas including economics and finance field(see [7]), signal processing, statis- tical tests(see [3]), information theory, communication networks, risk-sensitive control and statistical mechanics(see [14], [18]). First rigourous results in large deviations have been found by H. Cramer [4], he has used it to simulate insurance operations.Then, S.R.S. Varadhan [18] developed fundamental work about large deviations principle in 1967. After, large deviations theory has been studied by more and more mathematicians, we can note the works of Schilder [17],Sanov [15] ,and Freidlin and Wentzell[7]which show large deviations principle basis tool.

Therefore, there is yet a little published papers about large deviations for system with memory. M Scheutzow [16] is the precursor of study of large devi- ations for stochastic system with memory, he worked in the context of additive white noise.

In this paper, we observe the case of large deviations for random evolution delay equations in strong topology. Our approach is similar to that in Mellouk [10], with taking account induction argument in to handle the delay.

This paper is organized as follow: basic setting and notation are given in the first section. In section 2, we have some definitions and general results on the Large deviations principle(LDP).Then, the main result concerning the LDP

for the solution of the stochastic differential delay equation (3.1) is stated in section 3. Section 4 is devoted to a general approximate contraction principle that meets our needs. In the last section(section 5), we prove that this version of the contraction principle can be applied to prove the main result.

Throughout this paper, we will use the following notational convention: for the proof, constants appearing are denoted by C, eventhough they may change from one line to the next one.

1. Basic setting and notation

LetW_t:= (W_t¹, W_t², ..., W_t^l) denote a standard l-dimensional Brownian motion on a complete filtered probability space (Ω,F,(Ft)_t≥0, P) withW₀ = 0 , Ω = C0([0, m],R^l) equipped with the usual topology of uniform convergence defined by the normkfk_∞ = sup_0≤t≤m|f(t)| . For 0< α < ¹₂ we define theα-H¨older spaceC^αψ([−1, m],R^d) as the space of continuous functions f such that

kfkα= sup

s,t∈[0,m]

|f(t)−f(s)|

|t−s|^α <∞.

Define the H¨olderian modulus of continuity of f by ωα(f, δ) = sup

0<|t−s|≤δ;s,t∈[0,m]

|f(t)−f(s)|

|t−s|^α . Letb= (b1, b2, ..., bd) : R⁺×R^d×R^d×R^d−→R^d and

σ= (σ_ij)i=1,...,d,j=1,...,l: R⁺×R^d×R^d×R^d−→R^d×R^lbe Borel measurable functions.

Letτ >0 be a fixed delay, andψ be a given continuous function on [−τ,0].

Consider the following differential delay equation (dde):

(I)

dX(t) =dXt=b(t, Xt, X_t−τ, Yt)dt t∈(0,∞)

X(t) =ψ(t) t∈[−τ,0]

and the associated perturbed stochastic differential delay equation (sdde) (II)

dX_t^ε=b(t, X_t^ε, X_t−τ^ε , Y(t))dt+ε¹²σ(t, X_t^ε, X_t−τ^ε , Zt)dWt t∈(0,∞)

X_t^ε=ψ(t) t∈[−τ,0]

Throughout this paper, we will assume, without loss of generality, that the delayτ is equal to 1.

For 0 < α < ¹₂, let C₀^α([−1, m],R^d) be the separable space of α-H¨older continuous functionsg: [0, m]−→R^l withg(0) = 0.

For 0< α < ¹₂, denote byC_ψ^α([−1, m],R^d) be the set of allα-H¨older contin- uous functionsf : [−1, m]−→R^d such thatf(t) =ψ(t) for allt∈[−1,0].

The goal of this paper is to derive an LDP in C_ψ^α([−1, m],R^d) for random evolution delay equations.

2. Definitions and general results

Let E be a topological space andF its Borel σ-field, and let{Pε, ε > 0} be a family of probability measures on (E,F). We begin by giving several definitions.

Definition 2.1: A function I :E −→[0,∞] is said to be a rate function if it is lower semicontinuous(lsc) i.e. ∀xn →x, I(x)≤limI(xn) Furthermore,

if, for each a <∞,Γa={x∈E;I(x)≤a} is compact. We will say that I is a good rate function.

Definition 2.2:For some function I, the probabilities{Pε}ε>0satisfy a large deviations principle if the following hold:

1. I is a good rate function .

2. (lower bound)for every open subset G of E lim inf

ε→0 εlnP_ε(G)≥ −I(G).

3. (upper bound) For every closed subset F of E lim sup

ε→0

εlnP_ε(F)≤ −I(F).

It is well known thatC_ψ^α([−1, m]; R^d) is not separable but its closed subspace defined by C_ψ^α,0([−1, m]; R^d) = {f ∈ C_ψ^α([−1, m]; R^d); lim_δ→0ωα(f, δ) = 0} is separable.

BothC_ψ^α([−1, m]; R^d) andC_ψ^α,0([−1, m]; R^d) are Banach spaces for the norm k.k_αandk.k_∞≤ k.k_α. It is well known thatP(kWk_α<∞) = 1 for 0< α < ¹₂. We shall need the following version of the Arzel`a-Ascoli theorem .

Theorem 2.3: A set A⊂C_ψ^α,0([−1, m]; R^d)has compact closure in C_ψ^α,0([−1, m]; R^d)if and only if the following two conditions hold :

sup

f∈A

kfkα<∞ and

limδ↓0sup

f∈A

ωα(f, δ) = 0.

LetHdenote the Cameron-Martin space i.e H={h(t) =

Z t 0

h˙sds: [0, m]→R^l: Z m

0

|h(s)|˙ ²ds <+∞}.

This is an Hilbert space endowed with the inner product defined by :

hf, gi_H= Z m

0

f˙_sg˙_sds. (2.1)

Ifg ∈C₀^α([0, m]; R^l) is absolutely continuous, that is,g: [0, m]−→R^lwith g(0) = 0, set

e(g) =Rm

0 |g(s)|˙ ²ds. Otherwise, definee(g) = +∞. (2.2) Theorem 2.4: A good rate function I(f) = inf{¹₂e(g);F(g) = f}, f ∈ C_ψ^α,0([−1, m]; R^d)and F(g) is a solution to the sdde (II).

Theorem 2.5: The probability measures induced by√

εW onC₀^α,0([0, m]; R^d) equipped with the norm k.k_∞ satisfy the LDP with the good rate function I(.)

defined by I(g) =¹₂kgk²_H. (2.3)

It extends the LDP proved by P. Baldi and al[1] which extends the classical Schilder theorem. We now state the H¨older version of classical exponential

inequality for stochastic integrals, which is crucial in proving the exponential approximations.

Lemma 2.6: Let f : [0,1]×Ω −→ R^l×R^d and g : [0,1]×Ω −→ R^l be bounded(Ft)-progressively measurable functions, and set

U(t) =Rt

0f(s)dWs+Rt

0g(s)ds; 0≤t≤1.

Define A = sup_t,ωtr(f(t, ω)f^T(t, ω)) and B = sup_t,ωg(t, ω). Then, for every s≥0,T ≥0,0< α < ¹₂ andr > lBT^1−α

P( sup

s≤t≤s+T

|U(t)−U(s)|

|t−s|^α )≤2lexp[−(r−lBT^1−α)²

2Al²T^1−2α ] (2.4)

3. The main result

In this section, we give conditions under which the solution of the sdde defined by

X_t^ε=x+

Z t 0

b(s, X_s^ε, X_s−1^ε , Ys)ds+ε¹² Z t

0

σ(s, X_s^ε, X_s−1^ε , Zs)dWs (3.1) (where x ∈ R^d, W is a standard Brownian motion) satisfies an LDP in any H¨older norm with the exponentα < ¹₂.

LetY ={Y(t),0≤t≤m}be a R^d-valued process which is{Ft}-progressively measurable. In order to make explicit the LDP rate function for the law of (I) in theα-H¨older topology , we suppose that Y is a random variable with values in L^1−α1 ([0, m],R^d) . Let Z ={Z(t); 0≤t ≤m} be an Ft-progressively mea- surable process taking values in R^d. We assume that suppZ is a compact subset inC^α,0([0, m],R^d), and that (Y, Z) and W are independent.

From now on, we suppose that the coefficientsσand b satisfy the following conditions:

(H0)b= (b1, b2, ..., bd) : R⁺×R^d×R^d×R^d−→R^d and

σ= (σij)i=1,...,d,j=1,...,l: R⁺×R^d×R^d×R^d−→R^d×R^lare Borel measurable functions.

(H1)The functionb(., x,x, y) is jointly measurable in (x,˜ x, y) and there exists˜ a constantC >0 such that

kb(., x,x, y)k ≤˜ C(1 +|x|) f or any(x,x, y)˜ ∈R^d×R^d×R^d

(H2)The functionσ(., x,x, z) is jointly measurable in (x,˜ x, z) and there ex-˜ ists a constant C >0 such that

kσ(., x,x, z)k ≤˜ C f or any(x,x, z)˜ ∈R^d×R^d×R^d

(H₃) The functions b, σ satisfy a Lipschitz condition. That is, there exist constantsL₁, L₂ such that for allx₁,x˜₁, x₂, y₁,y˜₁, y₂,and t∈[0,∞)

kb(t, x1,x˜1, y1)−b(t, x2,x˜2, y2)k_Rd≤L1(|x1−x2|+|x˜1−x˜2|+|y1−y2|)

kσ(t, x1,x˜1, y1)−σ(t, x2,x˜2y2)k_Rd×R^l≤L2(|x1−x2|+|x˜1−x˜2|+|y1−y2|)

The existence of a unique solution of (3.1) which is (Ft)-adapted and hasα- H¨older continuous sample paths, is ensured by (H0),(H1),(H2) , and standard results on existence and uniqueness of solutions of sdde with random coefficients by (H₃). For h ∈ H, r ∈ L^1−α1 ([0, m],R^d) and u ∈ suppZ , let F(h, r, u)(.) denote the unique solution to the dde

F(h)(t) =F(h)(0) + Z t

0

b(s, F(h)(s), F(h)(s−1), r(s))ds +

Z t 0

σ(s, F(h)(s), F(h)(s−1), u(s)) ˙h(s)ds t∈[0,∞) (3.2) The existence of a unique solution of (3.2) which is (F_t)-adapted is a conse- quence of the Lipschitz continuity ofσand b and is standard.

Define ˜I:C_ψ^α,0([−1, m]; R^d)−→[0,∞] by I(˜˜h) = inf{1

2 Z m

0

|h(s)|˙ ²ds;h∈ H:

∃(r, u)∈suppY ×suppZ such that F(h, r, u)≡h}˜ Otherwise, ˜I(˜h) = +∞

Since ˜I is not necessarily lsc, we introduce its lsc regularization (see for example Bezuidenhout [2], p 651) ˜I^∗ defined by

I˜^∗(˜h) = lim

a→0 inf

ρ∈Bα(˜h,a)

I(ρ)˜ (3.3)

whereBα( ˜ρ, a) is the ball of radius acentred at ˜ρwith respect to the norm k.kα. The existence of the limit on the right hand side of (3.3) is ensured by the fact that inf_ρ∈B

α(˜h,a)I(ρ) is a decreasing function of˜ a.

The main result of the paper is the following.

Theorem 3.1: Letµ_εbe the law ofX^εsolution of (II) onC_ψ^α,0([−1, m]; R^d), equipped with the normk.k_α. The family{µ_ε, ε >0} satisfies a large deviations principle with the following rate function

I˜^∗(f) = lim

a→0 inf

ρ∈Bα(f,a)

I(ρ)˜ whereI(ρ) = inf{˜ ¹₂Rm

0 |h(s)|˙ ²ds;h∈ H:

∃(r, u)∈suppY ×suppZ such that F(h, r, u)≡ρ}

That is ,

(i) For any closed subsetC⊂C_ψ^α,0([−1, m],R^d) lim sup

ε→0

εlnµε(C)≤ − inf

f∈C

I˜^∗(f).

(ii) For any open subset G⊂C_ψ^α,0([−1, m],R^d) lim sup

ε→0

εlnµ_ε(C)≥ − inf

f∈G

I˜^∗(f).

whereI˜^∗ is defined in (3.3)andI˜^∗ is a good rate function with respect to the topology ofC_ψ^α,0([−1, m],R^d),0< α < ¹₂.

The proof is split into several lemmas. The rest of the paper is devoted to the proof of this result.

4. An extension of the contraction principle

Let (EX, dX),(EY, dY),(EZ, dZ),(E⁰, d⁰) denote Polish spaces and (Ω,F, P) be a probability space. Suppose that {X^ε, ε >0}is a family of random variables with value in E_X. Y is a random variable with values in E_Y. Z is a random variable with values in E_Z .Given a rate function I on E_X and a > 0 , set Γ_a ={x∈E_X:I(x)≤a} and Γ_∞=∪aΓ_a.

Theorem 4.1 : Let I be a good rate function on E_X, F_N, F : Γ_∞×E_Y × E_Z −→E⁰, X_N^ε, X^ε: Ω−→E⁰, be applications such that the following hold : (a)

(i)For alla >0 andN ≥1 ,FN|Γ_a×suppY×suppZ is continuous.

FN|Γ_a×suppY×suppZ converges to F|Γ_a×suppY×suppZ uniformly asN −→ ∞.

(b) For eacha >0 andN≥1,FN({I≤a} ×suppY ×suppZ)and F({I≤a} ×suppY ×suppZ)are relatively compact in(E⁰, d⁰).

(c) For all N ≥1,{X_N^ε :ε >0} satisfies an LDP (as ε→0 ) on E’ with good rate function

I_N^∗(ζ) = lim

ρ→0 inf

ξ∈B⁰(ζ,ρ)IN(ξ)

where B⁰(ζ, ρ)denotes the ball of radius ρcentred atζ in(E⁰, d⁰) , and IN(ξ) = inf{I(x) :∃(y, z)∈suppY ×suppZ such that FN(x, y, z) =ξ}

Proof: See Mellouk [10].

5. Proof of theorem 3.1

This section is devoted to proving main theorem by means of theorem 4.1. In order to apply theorem 4.1, we use the following notation :

For 0< α < ¹₂, set

(EX, dX) = (C₀^α,0([0;m]; R^l),k.kα), (E_Y, d_Y) = (L^(1−α)1 ([0, m],R^d),k.k 1

L(1−α)), (EZ, dZ) = (C^α,0([0;m]; R^d),k.kα) , (E⁰, d⁰) = (C_ψ^α,0([−1;m]; R^d),k.kα).

Throughout this section, Y ={Y(t),0 ≤t≤ m} and Z ={Z(t),0 ≤t ≤m}

are the processes defined in section 3. The rate function I is defined by I(h) ={1

2 Z m

0

|h(s)|˙ ²ds, if h∈ H}

Otherwise,I(h) = +∞

In the following, for a <∞, set Γa ={I ≤a}and Γ∞=∪aΓa. For ε >0 and N ≥1, sett_N = ^{[N t]}_N ([y] is the integer part of y).

LetX^ε={X^ε(t),−1< t <∞}be the solution of (II) andX_N^ε ={X_N^ε(t),−1<

t <∞} be the solution the sdde

dX_N^ε(t) =b(t, X_N^ε(t), X_N^ε(t−1), Y(t))dt

+ε¹²σ(t_N, X_N^ε(t_N), X_N^ε(t_N−1), Z(t_N))dW_t (5.1) We have to consider two cases.

case 1: b, σare bounded

For (r, u)∈L^(1−α)1 ([0, m],R^d)×C^α,0([0, m]; R^d) , define the map

FN(.) :C₀^α,0([0, m]; R^l)×L^(1−α)1 ([0, m]; R^d)×C^α,0([0, m]; R^d)−→C_ψ^α,0([−1, m]; R^d)

by















FN(ω, r, u)(t) =ψ(t) −1≤t≤0

F_N(ω, r, u)(t) =F_N(ω, r, u)(_N^k) +Rt

k N

b(s, FN(ω, r, u)(s), FN(ω, r, u)(s−1), r(s))ds

+σ(_N^k, FN(ω, r, u)(_N^k), FN(ω, r, u)(_N^k −1), u(_N^k))(ω(t)−ω(_N^k)) f or _N^k ≤t≤ ^k+1_N Letg∈ Hande(g) defined as in (2.2). For g withe(g)≤a















FN(g, r, u)(t) =FN(g, r, u)(_N^k) +Rt

0b(s, FN(g, r, u)(s), FN(g, r, u)(s−1), r(s))ds +Rt

0σ(^{[N s]}_N ), F_N(g, r, u)(^{[N s]}_N , F_N(g, r, u)(^{[N s]}_N −1), u(^{[N s]}_N )) ˙g(s)ds t∈[0,∞)

F_N(g, r, u)(t) =ψ(t) t∈[−1,0]

Notice that X_N^ε(s) =FN(ε¹²W, Y, Z)(s) where W is standard brownian.

DefineI_N(f) = inf{¹₂e(g);F_N(g) =f}for eachf ∈C_ψ^α,0([−1, m]; R^d).

Case 2: for any b and σ

We remove the boundedness assumption on b andσ.

ForR >0, definem_R:= sup{|b(t, x,˜x, y)|,|σ(t, x,x, y)|;˜ t∈[0, m],|x| ≤R,|˜x| ≤ R,|y| ≤R} andbⁱ_R:= (−mR−1)∨b_i∧(m_R+ 1),

σ_i,j^R := (−m_R−1)∨σ_i,j∧(m_R+ 1),1≤i, j≤d.

Put bR:= (b^R₁, b^R₂, ..., b^R_d) andσR:= (σ_i,j^R)1≤i,j≤d.

Then bR(t, x,x, y) =˜ b(t, x,x, y),˜ σR(t, x,x, y) =˜ σ(t, x,x, y), for˜ t ∈ [0, m],|x| ≤R,|˜x| ≤R,|y| ≤R. Furthermore, bR andσR satisfy the Lipschitz condition (H₃) with the same Lipschitz constant.

Ifg∈ His absolutely continuous, sete(g) as defined in (2.2).

For g with e(g)<∞, and for (r, u)∈L^(1−α)1 ([0, m],R^d)×C^α,0([0, m]; R^d) ,letF_R(g, r, u) be the solution to the dde











F_R(g, r, u)(t) =F_R(g, r, u)(0) +Rt

0b(s, FR(g, r, u)(s), FR(g, r, u)(s−1), r(s))ds +Rt

0σ(s, FR(g, r, u)(s), FR(g, r, u)(s−1), u(s)) ˙g(s)ds t∈(0,∞)

X_t^ε=ψ(t) t∈[−1,0]

Define IN(f) = inf{¹₂e(g);FR(g) =f}for eachf ∈C_ψ^α,0([−1, m]; R^d).

The existence and uniqueness of the solution of (5.1) follows from hypothe- sis (H0)- (H3) on the coefficients. Furthermore, the trajectories of X^ε andX_N^ε belong almost surely toC^α,0([−1, m]; R^d).

To prove the central result : theorem 3.1 we will follow step by step the assumptions of theorem 4.1. To complete the argument,we will remind some results found in Mellouk [10](see also Y.J. Hu [9]). Precisely, these are lemma 5.1,5.2, 5.3, 5.4 and 5.5.

5.1 Continuity of F

We prove that

FN(.) :C₀^α,0([0, m]; R^l)×L^(1−α)1 ([0, m]; R^d)×C^α,0([0, m]; R^d)−→C_ψ^α,0([−1, m]; R^d) (resp. FR) is continuous . FixN ≥1 and let (h1, r1, u1),(h2, r2, u2)∈C₀^α,0([0, m]; R^l)×

L^(1−α)1 ([0, m]; R^d)×C^α,0([0, m]; R^d) ; then setF_N⁽ⁱ⁾(.) = FN(hi, ri, ui), i= 1,2 and Ψ_N(.) =F_N⁽¹⁾(.)−F_N⁽²⁾(.).

Lemma 5.1: GivenC >0, there exists a constantC_N >0(depending on N and C) such that , for kh1kα∨kh2kα≤C

kΨN(.)kα≤CNkh1−h2kα (5.2)

5.2 Uniform convergence of F

to F on Γ

× suppY × suppZ

To verify assertion (a)(ii) of theorem 4.1, we use the following:

Lemma 5.2: For any a >0, sup

N

sup

khkH≤a

(kFN(h)(.)k_∞∨ kF(h)(.)k_∞)<∞ (5.3)

lim

N→∞ sup

khkH≤a

(kFN(h)(.)−F(h)(.)kα) = 0 (5.4)

5.3 Relative compactness

We prove the condition (b) of theorem 4.1 from the following:

Lemma 5.3:Let I be the good rate function defined by (2.3),0< α < ¹₂ and K be a relatively compact subset ofC₀^α,0([0, m]; R^l). Then for each

N ≥1, a >0, the setsFN(K×suppY×suppZ), FN({I≤a} ×suppY ×suppZ) andF({I≤a} ×suppY ×suppZ)are relatively compact inC_ψ^α,0([0, m]; R^d)

5.4 Large Deviation principle for X

as ε → 0

For N ≥ 1, we prove that the family X_N^ε ≡ FN(ε¹²W, Y, Z) defined by (5.1) satisfies on C₀^α,0([0, m]; R^d) an LDP, and show that the rate function is of the form (4.1). Since FN is continuous on C₀^α,0([0, m]; R^l)×L^(1−α)1 ([0, m]; R^d)× C^α,0([0, m]; R^d) we use a version of the contraction principle. Schilder’s theo- rem implies thatε¹²W satisfies an LDP onC₀^α,0([0, m]; R^l) with rate function I defined by (2.3).

ForN ≥1, defineI_N(f) = inf{I(h);h∈ H:

∃(r, u)∈suppY ×suppZ such that F_N(h, r, u)≡f}, and letI_N^∗ be its lsc regularization, ie

I_N^∗(f) = lim

a→0 inf

g∈Bα(f,a)IN(g)

An argument similar to that in the proof of theorem 4.1 shows thatI_N^∗ is a good rate function and we check that {X_N^ε, ε > 0} satisfies an LDP with rate functionI_N^∗.

To prove lower and upper bound we use these following lemmas.

Lemma 5.4: Let G be an open subset of C_ψ^α,0([−1, m]; R^d); then lim inf

ε→0 εlnP(X_N^ε ∈G)≥ −inf{I_N^∗(f);f ∈G}

Lemma 5.5: Let A be a closed subset ofC_ψ^α,0([−1, m]; R^d); then lim sup

ε→0

εlnP(X_N^ε ∈A)≤ −inf{I_N^∗(f);f ∈A}

5.5 Exponential approximations

Finally, we show that {X_N^ε, ε > 0} defined by (5.1) are exponentially good approximations of{X^ε, ε >0}defined by

X^ε(t) =X^ε(0) + Z t

0

b(s, X^ε(s), X^ε(s−1), Y(s))ds

+ε¹² Z t

0

σ(s, X^ε(s), X^ε(s−1), Z(s))dWs

Let us at first establish the following approximation.

Lemma 5.6:For any δ >0 lim sup

N→∞

lim sup

ε→0

εlnP( sup

t∈[0,m]

|X_N^ε(t)−X^ε(t)|> δ

N^α) =−∞ (5.5)

Proof:We prove (5.5) by induction on m. We first prove it form= 1 . Since the drift coefficient b is not necessarily bounded, to prove (5.5) let us introduce some auxiliary results.

Let 0< α < β < ¹₂ and 0< γ < β−α; then by theorem 2.5 lim sup

ε→0

εlnP( sup

1≤k≤N

|ε¹²(Wk

N −Wk−1

N )| ≥N^γ−β)

≤lim sup

ε→0

εlnP(kε¹²Wk_β≥N^γ)

≤ −inf{1

2khk_H;khkβ ≥N^γ} ≤ −1

2N^2γ (5.6)

Indeed, ifh∈ Hsatisfieskhkβ≥N^γ, the Cauchy-Schwarz inequality implies khk_H ≥N^γ. Define the set

Bβ,γ,ε={ sup

1≥k≥N

|ε¹²(Wk

N −W^k−1

N )| ≤N^γ−β}\

{kε¹²Wkβ≤N^γ} (5.7) by

lim sup

ε→0

εlnP( sup

1≥k≥N

|ε¹²(Wk

N −Wk−1

N )| ≤N^γ−β)≤ −1 2N^2γ

N→∞lim lim sup

ε→0

εlnP(B_β,γ,ε^c ) =−∞ (5.8)

Furthermore, on the setB_β,γ,ε , by Gronwall’s lemma and the assumptions of the coefficients b, σ for t ∈ [−1, m], we deduce the existence of a constant C >0 such that

X_N^ε ≤C{

N

X

k=1

ε¹²|W k

N∧t −W^k−1

N∧t|+ Z t

0

{1 +|X_N^ε(s)|}ds}

≤C{N^γ+1−β+ Z t

0

|X_N^ε(s)|ds}

AsN is large enough, there exists a constantK >0 such thatRt

0|X_N^ε(s)|ds≤K andN^γ+1−β>> K. So,there exists a constant ˇC >0 such that

X_N^ε ≤CNˇ ^γ+1−β (5.9)

Set Ψ^ε_N(.) =X_N^ε(.)−X^ε(.) andt_N = ^{[N t]}_N then fort∈[0,1] , Ψ^ε_N(t) satisfies Ψ^ε_N(t) =

Z t 0

{b(s, X_N^ε(s), X_N^ε(s−1), Y(s))−b(s, X^ε(s), X^ε(s−1), Z(s))}ds

+ε¹² Z t

0

{σ(s_N, X_N^ε(s_N), X_N^ε(s−1), Y(s))−σ(s, X^ε(s), X^ε(s−1), Z(s))}dWs(5.10) Forρ >0 , we defineτ_N,ε^ρ (ω) = inf{t≥0;|X_N^ε(t, ω)−X_N^ε(t_N, ω)| ≥ _N^ρα} ∧1

θ^ρ_N,ε(ω) = inf{t≥0;|Ψ^ε_N(t, ω)| ≥ δ

N^α} ∧τ_N,ε^ρ (ω) and υ^ρ_N,ε(t) =R

Ω{_N^ρ2α² +|Ψ^ε_N(t∧θ_N,ε^ρ,δ(ω), ω)|²}^1εdP Then clearly,

P( sup

t∈[0,1]

|X_N^ε(t)−X^ε(t)|> δ

N^α)≤P(τ_N,ε^ρ <1) +P(θ_N,ε^ρ,δ <1) (5.11) Observe thatP(τ_N,ε^ρ <1)≤P(τ_N,ε^ρ <1, B_β,γ,ε) +P(τ_N,ε^ρ <1, B^c_β,γ,ε) First, we apply Stroock’s inequality (2.4) and expression (5.9), together with hypothesis (H₀)- (H₂) to obtain the existence of a constantC >0 such that P(τ_N,ε^ρ <1)≤

N

X

k=1

P( sup

k−1 N ≤t≤_N^k

|X_N^ε(t)−X_N^ε(k−1 N )| ≥ ρ

N^α, B_β,γ,ε) +P(B_β,γ,ε^c )

≤CdNexp{−(ρ−CdN^γ+1−β(_N^11−α))²

Cεd²(_N¹)^1−2α }+P(B_β,γ,ε^c )

≤CdNexp(−CN^1−2α

ε ) +P(B_β,γ,ε^c ) SinceN^γ+α−β −→0, asγ < β−α, thus, using (5.11), we deduce

lim

N lim sup

ε

εlnP(τ_N,ε^ρ <1) =−∞ (5.12)

Since suppZ is a compact subset of C₀^α,0([0, m]; R^d), for every ρ >0 there existsN0≥1 such that, forN≥N0,

sup

0≤t≤1

|Z(t)−Z(t_N)| ≤ρN^−α (5.13) For 0< ε <¹₂, set ρN =_N^ρα andfε,ρ(y) = (ρ²_N+|y|²)^1ε

Then, an application of Ito’s formula tof_ε,ρ(Ψ^ε_N(t)) yields that fε,ρ(Ψ^ε_N(t∧θ^ρ,δ_N,ε))−R^t∧θN,ε^ρ,δ

0 g_ε,N^ρ (s)ds−ρ

2 ε

N , is a martingale, where , if< ., . >

is the inner product in R^d ; g_ε,N^ρ (t) = 2

ε(ρ²_N +|Ψ^ε_N(t)|²)^ε−11

hΨ^ε_N(t), b(t, X_N^ε(t), X_N^ε(t−1), Y(t))−b(t, X^ε(t), X^ε(t−1), Y(t))i

+2 ε(1

ε −1)ε

k(σ(t, X_N^ε(t_N), X_N^ε(t_N−1), Z(t_N))−σ(t, X^ε(t), X^ε(t−1), Z(t)))^∗Ψ^ε_N(t)k² (ρ²_N +|Ψ^ε_N(t)|²)^ε−21

+kσ(t_N, X_N^ε(t_N), X_N^ε(t_N −1), Z(t_N))−σ(t, X^ε(t), X^ε(t−1), Z(t))k² (ρ²_N +|Ψ^ε_N(t)|²)^ε−11

For 0 ≤ t ≤τ_N,ε^ρ , using (5.13), we have, for N ≥N0 and 0 < ε < ¹₂ that there existsC >0 such that

|g_ε,N^ρ (t)| ≤C2

ε(ρ²_N +|Ψ^ε_N(t)|²)^1ε |Ψ^ε_N(t)|

ρ²_N+|Ψ^ε_N(t)|²|X_N^ε(t)−X^ε(t)|

+C|1

ε −1|{ρ²_N +|Z(t_N)−Z(t)|²} |Ψ^ε_N(t)|²

ρ²_N +|Ψ^ε_N(t)|²(ρ²_N +|Ψ^ε_N(t)|²)^ε−11

+C{ρ²_N+|Z(t_N)−Z(t)|²}(ρ²_N+|Ψ^ε_N(t)|²)^ε−11

≤C1

εfε,ρ(Ψ^ε_N(t)) |Ψ^ε_N(t)|ρN

ρ²_N+C|Ψ^ε_N(t)|²

+|1

ε−1| ρ²_N ρ²_N +|Ψ^ε_N(t)|²

|Ψ^ε_N(t)|²

ρ²_N +|Ψ^ε_N(t)|²fε,ρ(Ψ^ε_N(t))

+C|1

ε−1||Z(t_N)−Z(t)|² ρ²_N +|Ψ^ε_N(t)|²

|Ψ^ε_N(t)|²

ρ²_N+|Ψ^ε_N(t)|²fε,ρ(Ψ^ε_N(t))

+Cρ²_N +|Z(t_N)−Z(t)|²

ρ²_N +|Ψ^ε_N(t)|² fε,ρ(Ψ^ε_N(t))

≤C{(1

ε+ 1) + (1 +|Z(t_N)−Z(t)|²

ρ²_N +|Ψ^ε_N(t)|² )}fε,ρ(Ψ^ε_N(t))

≤C1

εfε,ρ(Ψ^ε_N(t))

This together with Doob’s stopping theorem, shows that there exists a con- stant ˇC <∞independent ofN, ε, ρandN0 such that, forN ≥N0

υ_N,ε^ρ (t)≤ρ

2 ε

N+ Cˇ

ε Z t

0

υ^ρ_N,ε(s)ds t∈[0,1]

(see, for example, Deuschel and Stroock [6], p. 30)

Therefore, forN ≥N0, υ^ρ_N,ε(1)≤exp{¹₂( ˇC+ 2 lnρ−2αlnN)}

Since for allN≥1,P(θ_N,ε^ρ,δ <1)≤(^ρ_N^2+δ2α²)^−1ευ_N,ε^ρ (1) We conclude

ρ→0limsup

N

lim sup

ε→0

εlnP(θ_N,ε^ρ,δ <1) =−∞

This together with (5.11) and (5.12) implies (5.5) form= 1.

Assume now (5.5) holds for some integer m. We will prove it is also true for m+ 1.

Let Ψ^ε_N, τ_N,ρ^ε be defined as before.

Case 1(b and σ are bounded)

In addition, we introduce two new stopping times.

τ_N,ρ^1,ε := inf{t≥0;|X^ε(t−1)−X_N^ε(t−1)| ≥ ρ N^α}

τ_N,ρ^2,ε := inf{t≥0;|X_N^ε(t−1)−X_N^ε(t_N

N −1)| ≥ ρ N^α} and define Φ^ε_N,ρ(t) = Ψ^ε_N(t∧τ_N,ρ^1,ε ∧τ_N,ρ^2,ε ∧τ_N,ρ^ε ) and

θ¯^ε_N,ρ:= inf{t≥0,|Φ^ε_N,ρ(t)| ≥ _N^δα} Then,

P( sup

t≤m+1

|Ψ^ε_N(t)|> δ N^α)

≤P(τ_N,ρ^1,ε ∧τ_N,ρ^2,ε ∧τ_N,ρ^ε ≥m+ 1) +P( sup

t≤m+1

|X^ε(t)−X_N^ε(t)|> δ

N^α, τ_N,ρ^1,ε ∧τ_N,ρ^2,ε ∧τ_N,ρ^ε > m+ 1)

≤P(τ_N,ρ^1,ε ≤m+1)+P(τ_N,ρ^2,ε∧τ_N,ρ^ε ≤m+1)+P(¯θ^ε_N,ρ≤m+1) (5.14) Then,

lim

N→∞lim sup

ε→0

εlnP(τ_N,ρ^ε ∧τ_N,ρ^2,ε ≤m+1) =−∞ (5.15) By the induction hypothesis,

lim

N→∞lim sup

ε→0

εlnP(τ_N,ρ^1,ε ≤m+ 1)

≤ lim

N→∞lim sup

ε→0

εlnP( sup

−1≤t≤m

|X^ε(t)−X_N^ε(t)|> ρ

N^α) =−∞ (5.16)

Using (5.14) and (5.15) and following the argument form= 1, we see that (5.5) is also true form+ 1.

Case 2(b, σ are not bounded)

Set Ψ^ε_R(t) =X^ε(t)−X_R^ε(t) ForR1>0, defineθ^ε_R1 := inf{t≥0;|X^ε(t)| ≥ _N^R1α} For anyR≥R1, we have

Ψ^ε_R(t∧θ_R^ε₁) = Z t∧θ^ε_R

1

0

[bR(s, X^ε(s), X^ε(s−1), Y(s))−bR(s, X_R^ε(s), X_R^ε(s−1), Y(s))]ds

+ε¹² Z t∧θ^ε_R

1

0

[σ_R(s, X^ε(s), X^ε(s−1), Z(s))−σ_R(s, X_R^ε(s), X_R^ε(s−1), Z(s))]dW_s (5.17) Forρ >0, letυλ(y) = (ρ²+|y|²)^λ and

τ_R,ρ^ε := inf{t≥0,|X^ε(t−1)−X_R^ε(t−1)| ≥ ρ N^α}

θ^ε_R,ρ:= inf{t≥0,|Ψ^ε_R,ρ(t)| ≥ δ N^α} where Ψ^ε_R,ρ(t) := Ψ^ε_R(t∧θ^ε_R

1∧τ_R,ρ^ε ) Then,

P( sup

−1≤t≤m+1

|Ψ^ε_R(t)|> R1

N^α)≤P(θ_R^ε₁ ≤m+1)+P(θ^ε_R,ρ≤m+1)+P(τ_R,ρ^ε ≤m+1)

≤P( sup

−1≤t≤m+1

|X^ε(t)|> R1

N^α) +P( sup

−1≤t≤m+1

|X^ε(t)−X_R^ε(t)|> ρ

N^α) +P(θ^ε_R,ρ≤m+ 1) (5.18) By the induction hypothesis

R→∞lim lim sup

ε→0

εlnP( sup

−1≤t≤m

|X^ε(t)−X_R^ε(t)|> ρ

N^α) =−∞ (5.19)

By Ito’s formula Φ_λ(Ψ^ε_R,ρ(t))−

Z t∧θ_R^ε

1∧τ_R,ρ^ε 0

γ_λ^ε(s)ds−ρ^2λ (5.20)

is a martingale with initial value zero where γ_λ^ε(s) = 2λ(ρ²+|Ψ^ε_R(s)|²)^λ−1

hΨ^ε_R(s), b(s, X^ε(s), X^ε(s−1), Y(s))−b(s, X_R^ε(s), X_R^ε(s−1), Y(s))i + 2λ(λ−1)ε(ρ²+|Ψ^ε_R(s)|²)^λ−2

|(σ(s, X^ε(s), X^ε(s−1), Z(s))−σ(s_N, X_R^ε(s_N), X_R^ε(s_N−1), Z(s_N)))^∗Ψ^ε_R(s)|² +λε(ρ²+|Ψ^ε_R(s)|²)^λ−2

kσ(s, X^ε(s), X^ε(s−1), Z(s))−σ(s_N, X_R^ε(s_N), X_R^ε(s_N−1), Z(s_N))k²_H.S

Fors≤t∧θ_R^ε

1∧τ_R,ρ^ε

γ_λ^ε(s)≤C(λ+λ(λ+ 1)ε)(ρ²+|Ψ^ε_R(s)|²)^λ Chooseλ= ¹₂, and take expectations in (5.20) to obtain

E[(ρ²+|Ψ^ε_R,ρ(t)|²)^1ε]≤ρ^2ε +C ε

Z t 0

E[(ρ²+|Ψ^ε_R,ρ(s)|²)^1ε]ds Hence,

E[(ρ²+|Ψ^ε_R,ρ(t)|²)^1ε]≤ρ^2ε exp^Ctε Since,

(ρ²+δ²)^1εP(θ^ε_R,ρ≤m+ 1)≤E[(ρ²+|Ψ^ε_R,ρ(m+ 1)|²)^1ε] we have

P(θ_R,ρ^ε ≤m+ 1)≤( ρ²

ρ²+δ²)^1εexp^Cε(m+1) Therefore

lim sup

ε→0

εlnP(θ^ε_R,ρ≤1)≤ln( ρ²

ρ²+δ²) +C

Given M >0, chooseρsmall enough so that ln(_ρ2^ρ+δ^{2 2}) +C≤ −2M ChooseR0so that

lim sup

ε→0

εlnP(τ_R,ρ^ε ≤1)≤ −2M f orR≥R₀ So, we have

R→∞lim lim sup

ε→0

εlnP( sup

−1≤t≤m+1

|X^ε(t)−X_R^ε(t)|> δ

R^α)≤ −M Since M is arbitrary, it follows from (5.17), (5.18), (5.19) that

R→∞lim lim sup

ε→0

εlnP( sup

−1≤t≤m+1

|Ψ^ε_R(t)|> δ R^α)

≤lim sup

ε→0

εlnP( sup

−1≤t≤m+1

|X^ε(t)|> R1)∨ {ln( ρ²

ρ²+δ²) +C}

By using Proposition (3.5) in [12], letting firstρ→0 and thenR1, we obtain (5.5) form+ 1.

Lemma 5.8: For any δ >0 lim sup

N→∞

lim sup

ε→0

εlnP(kX_N^ε(t)−X^ε(t)k_α≥δ) =−∞

Proof: See Mellouk [10].

Remark:

1- If b and σ are independent to the random variables Y and Z, we have a generalization of LDP for stochastic systems with memory in H¨older topology (refer [10]).

2- If b andσare dependent to Y and Z, and the equation has no delay, we have

a LDP for random evolution equations ([8], [13]).

3- LDP found here can be extended to the case where there are several delays.

As example, two delays τ1, τ2 are allowed in the equation (II), and we have:

(III)







dX_t^ε=b(t, X_t^ε, X_t−τ^ε

1, Y_t, Y_t−τ1)dt+ε¹²σ(t, X_t^ε, X_t−τ^ε

2, Z_t, Z_t−τ2)dW_t t∈(0,∞)

X_t^ε=ψ(t) t∈[−τ,0]

Here, we considerτ=τ1∨τ2then we use the same argument.

It would be interesting to extend this result to another strong topology such as Besov space.

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