3 Useful results

Fondamentales et Informatique JMMAFI, ISSN 2411-7188, Vol.3, 2016, pp 40-61,

www.madarevues.gov.mg

Moderate Deviations Principle for stochastic system with fixed delay in strong topology

S.H. RANDRIAMANIRISOA¹, T.J. RABEHERIMANANA²

1 Facult´e des Sciences , D´epartement de Math´ematiques et Informatique, B.P 906, Ankatso, 101, Antananarivo, Madagascar

2 Facult´e des Sciences , D´epartement de Math´ematiques et Informatique, B.P 906, Ankatso, 101, Antananarivo, Madagascar

1hasinasran@gmail.com, ²rabeherimanana.toussaint@gmx.fr

R´esum´e : Dans cet article, nous ´etudions un principe de d´eviations mod´er´ees v´erifi´e par les syst`emes stochastiques avec m´emoire. Il s’agit d’un syst`eme d´ecrit par l’´equation diff´erentielle stochastique avec m´emoire suivante :

dX_t^ε =b(t, X_t^ε, X_t−τ^ε )dt+√

εσ(t, X_t^ε, X_t−τ^ε )dW_t t∈(0,∞)

X_t^ε =ψ(t) t∈[−τ,0]

o`uW est un mouvement brownien standardd−dimensionnel, b et σ v´erifient les conditions lipschitziennes et sont `a croissance lin´eaire ; ψ est une fonction continue donn´ee d´efinie sur [−τ,0].L’´etude se fait avec la topologie h¨olderienne.

On pr´esente ici l’´etude des processusY^ε := (Y_t^ε = X_t^ε−X_t⁰

√ε )t∈[−τ,m]etZ^ε := (Z_t^ε = X_t^ε−X_t⁰

√εh(ε) )t∈[−τ,m]

dans le cadre du Th´eoreme de la limite centrale et du Principe de d´eviation mod´er´ee respecti- vement. On utilisera la propri´et´e de l’´equivalence exponentielle et l’in´egalit´e de transportation de Talagrand `a cette fin.

Mots-cl´es : principe de d´eviations mod´er´ees , syst`emes stochastiques, retard, in´egalit´e de transportation de Talagrand, ´equivalence exponentielle, norme h¨olderienne.

Abstract : In this paper, we develop a moderate deviations principle for stochastic system with memory driven by small multiplicative white noise. The system is described by :

dX_t^ε =b(t, X_t^ε, X_t−τ^ε )dt+√

εσ(t, X_t^ε, X_t−τ^ε )dW_t t∈(0,∞)

X_t^ε =ψ(t) t∈[−τ,0]

where W is a d−dimensional brownian motion, b and σ are Lipschitzian and have a linear growth ;ψ is a given continuous function on [−τ,0]. One work in H¨older topology.

One focus on the study of Y^ε := (Y_t^ε = X_t^ε−X_t⁰

√ε )t∈[−τ,m] and Z^ε := (Z_t^ε = X_t^ε−X_t⁰

√εh(ε) )t∈[−τ,m]

in order to get a central limit theorem and a moderate deviation principle respectively. For the proof, we will use a theorem of exponential equivalence and Talagrand’s transportation inequality.

Keywords : moderate deviations principle, stochastic system, delay, Talagrand’s transpor- tation inequality, exponential equivalence, h¨olderian norm.

Introduction

LetW_t:= (W_t¹, W_t², ..., W_t^l) denote a standard l-dimensional Brownian motion on a complete filtered probability space (Ω,F,(F_t)_t≥0, P) with W₀ = 0 and W_t takes values in R^d , Ω = C0([0, m],R^l) equipped with the usual topology of uniform convergence defined by the norm

kfk∞= sup

0≤t≤m

|f(t)|.

Let b : R⁺ ×R^d×R^d −→ R^d and σ : R⁺×R^d ×R^d −→ R^d⊗R^l be Borel measurable functions with linear growth.

Let τ >0 be a fixed delay, and ψ be a given continuous function on [−τ,0].

Consider the stochastic delay differential equation (sdde) dXt =b(t, Xt, Xt−τ)dt t∈(0,∞)

X_t =ψ(t) t∈[−τ,0] (1)

and the perturbed stochastic delay differential equation associated dX_t^ε =b(t, X_t^ε, X_t−τ^ε )dt+√

εσ(t, X_t^ε, X_t−τ^ε )dW_t t∈(0,∞)

X_t^ε =ψ(t) t∈[−τ,0] (2)

We study deviation ofX^εfrom the deterministic solutionX, that is the asymptotic behavior of the trajectory of

Z^ε(t) := 1

√εh(ε)(X_t^ε−X_t)

where h(ε) is some deviation scale which strongly influences the asymptotic behavior ofZ^ε. If one puth(ε) = 1

√ε, one get large deviations (LD) estimates. T. Zhang and S. E.A Mohammed in [12] give the Large Deviations principle (LDP) for the family of{µ_ε;ε >0}law of{X^ε;ε >0}

in uniform topology. In our recent works in S. H. Randriamanirisoa and T.J. Rabeherimanana [8], we provide an extension of the LDP to H¨older space if we consider the case Z ≡ 0 and Y ≡0.

In the classical work, Friedlin-Wentzell [5] studied the Large deviations for small noise, limit of stochastic reaction-diffusion equations.

If one consider h(ε)≡1, we are in the case of the Central Limit Theorem(CLT).

Set Y^ε(t) := 1

√ε(X^ε−X). One shall show that Y^ε converge as ε↓ 0 to a random field. Then, we will study Moderate Deviations (MD), that is when deviation scale satisfies h(ε)−→+∞, and √

εh(ε)−→0 as ε↓0.

In this paper, we establish its moderate deviation principle (MDP) in the context of H¨ooder norm. For this end, we will use Talagrand’s transportation inequality on path space established in Djellout and al. [4] and measure concentration see in Bobkov and G¨otze [1], Djellout and al.

[4], and Villani [9].

The paper is organized as follow : in the first section, we give basic settings and some useful general results. We find the main results in section 2. In section 3 we will provide some useful results. The last section is devoted to prove the results.

1 Basic Settings

Let us consider (2). W_t is defined on some well filtered probability space (Ω,F,(F_t)_t≥0, P).

We always assume :

(H₁) : σ, b are locally Lipschitzian.

That means, there exists some constants L_b and L_σ such that for all x₁, x₂, y₁, y₂ ∈R^d and t∈[0,∞)

kσ(., x₁, y₁)−σ(., x₂, y₂)k_R^d_⊗R^l ≤L_σ(|x₁−x₂|+|y₁−y₂|) (3) kb(., x₁, y₁)−b(., x₂, y₂)k_R^d ≤L_b(|x₁−x₂|+|y₁−y₂|) (4) (H2) : σ(., x, y), b(., x, y) are uniformly continuous in x, y ∈ R^d on[0,∞),

that means :

lims→t sup

x,y∈R^d

|σ(s, x, y)−σ(t, x, y)|= 0.

lims→t sup

x,y∈R^d

|b(s, x, y)−b(t, x, y)|= 0.

(H₃) : b is C¹ .

(H₄) : There existsC > 0 such that

max{tr(σσ∗(t, x, y)),hx, b(t, x, y)i,hy, b(t, x, y)i} ≤C(1 +|x|²+|y|²)

where hx, yi=x.y is the inner product with hx, xi=|x|² and (H₅) : There exist a constant K such that

sup{|b(t, x, y) +σ(t, x, y)|} ≤K(1 +|x|+|y|) (5) (2) has a unique non-explosive solution denoted by X^ε = (X_t^ε)t∈[−τ,m] under the conditions (H₁), (H₂). LetH denote the Cameron-Martin space associated with the Brownian motionW.

H={h(t) = Z t

0

h(s)ds˙ : ˙h∈L²([0, T]), h is absolutely continuous}

When ε goes to 0, we get : limε→0P( sup

t∈[−τ,∞)

|X_t^ε−X_t⁰|> R) = 0, ∀ R >0

where X⁰ is the solution of the non-perturbed dynamical system (1). Letµ^ε the law of X^ε, µ^ε converges tightly to the degenerated measureδ_X⁰.

Let C_ψ([−τ, m],R^d) be the Banach space of continuous functions f : [−τ, m]→ R^d, with f(t) =ψ(t) for t∈[−τ,0] ; equipped with the sup-norm

kfk:= sup

t∈[0,m]

|f(t)|.

Let C_ψ([0, m],R^d) be the Banach space of continuous functions g : [0, m]→R^d, with g(0) = 0.

For a precise asymptotic estimate of deviations ofX^ε from the dynamical systemX⁰, T. Zhang and S.E.A Mohammed in [12] claim that the law of {X_t^ε, ε > 0} satisfies a LDP in uniform topology onC_ψ([−τ, m],R^d) with the good rate function given by

I_LDP(f) =





 inf{1

2 Rm

0 |g(s)|˙ ²ds; g ∈C_ψ([0, m],R^l),

g absolutely continuous, F(g) =f} +∞ otherwise

(6) with F is solution to

F(h)(t) =Rt

0b(s, F(h)(s), F(h)(s−τ))ds+Rt

0σ(s, F(h)(s), F(h)(s−τ))dh(s) t ∈(0,+∞)

F(h)(t) =ψ(t) t∈[−τ,0]

(7) Otherwise, ILDP = +∞.

The subject of this paper is the problem of MDP for {X_t^ε}_{t∈[−τ,m]} in H¨older norm.

Let α > 0 and denote by C₀^α([−τ, m],R^d) the set of α− H¨older norm continuous functions i.e of continuous functions f : [−τ, m]−→R^d such that f(t) = 0, fort ∈[−τ,0] , and

kfkα = sup

t

|f(t)|+ sup

|s−t|≤1,s6=t

|f(t)−f(s)|

|t−s|^α <∞

The normk.k_α is called theα−H¨older norm. It is well known that the trajectories of X areα−

H¨older continuous for α∈[0,1 2[.

LetC_ψ^α([−τ, m],R^d) denote the set ofα−H¨older norm continuous functions i.e of continuous functions f : [−τ, m]−→R^d such that f(t) =ψ(t), if t∈[−τ,0] and

kfkα = sup

t

|f(t)|+ sup

|s−t|≤1,s6=t

|f(t)−f(s)|

|t−s|^α <∞.

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

0<|t−s|≤δ;s,t,∈[0,m]

|f(t)−f(s)|

|t−s|^α

C_ψ^α,0([−τ, m],R^d) denote a closed subspace of C_ψ^α([−τ, m],R^d) defined by C_ψ^α,0([−τ, m],R^d) = {f inC_ψ^α([−τ, m],R^d); limδ→0ω_α(f, δ) = 0}is separable.

Set

Y^ε:= (Y_t^ε)t∈[−τ,∞) := (X_t^ε−X_t⁰

√ε ) (8)

and

Z^ε:= (Z_t^ε)t∈[−τ,∞) with Z_t^ε:= Y_t^ε

h(ε) (9)

Here, we shall study the asymptotic behaviour of : Z^ε where h(ε)−→+∞ and √

εh(ε)−→0, as ε−→0 (10)

Through this paper, we always assume that h(ε) satisfies (10).

2 Main result

In addition to (H1)−(H5), one assume that b is differentiable with respect to the second and last variable. And ∂b

∂x,∂b

∂y are also uniformly Lipschitz. More precisely, There exists some constants L_∂xb and L_∂yb such that :

|∂b

∂x(t, u, y)− ∂b

∂x(t, v, y)| ≤L_∂xb|u−v| (11)

|∂b

∂y(t, x, r)− ∂b

∂y(t, x, s)| ≤L∂yb|r−s| (12)

∀t∈R⁺, u, v, x, y, r, s∈R^d.

Combined with the uniform Lipschitz continuity of b, one obtain that :

|∂b

∂x(t, u, y)| ≤L_b

|∂b

∂y(t, x, r)| ≤L_b

∀t∈R⁺, u, x, y, r∈R^d.

The purpose of this paper is to prove results concerning CLT and MDP which are satisfied byY_t^ε and Z_t^ε respectively in H¨older space.

First, let us recall a LDP for X^ε in H¨older norm. This result is an adapted version of the result found in our previous work [8] .

Theorem 1 (LDP) : Let µ_ε be the law of X^ε solution of (1) on C_ψ([−τ, m];R^d) , equipped with the norm k.k_α. The family {µ_ε, ε >0} satisfies a LDP with the following rate function

I˜^∗(f) = lim

a→0 inf

ρ∈Bα(f,a)

I(ρ)˜

where I(ρ) = inf{˜ 1 2

Z m 0

|h(s)|˙ ²ds;h∈ H:F(h, r, u)≡ρ}

with F is solution to the differential delayed equation :







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

0 b(s, F(h)(s)), F(h)(s−τ))ds +Rt

0 σ(s, F(h)(s), F(h)(s−τ)) ˙h(s)ds t ∈[0,∞)

F(h)(t) = ψ(t) t∈[−τ,0]

That is ,

i) For any closed subset C ⊂C_ψ^α,0([−τ, m];R^d) lim sup

ε→0

εlogµε(C)≤ −inf

f∈C

I˜^∗(f).

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

ε→0

εlogµ_ε(G)≤ −inf

f∈G

I˜^∗(f).

where I˜^∗ is a good rate function with respect to the topology ofC_ψ^α,0([−τ, m];R^d),0≤α < 1 2. Proof :The proof is a particular case of the main result in our paper [8] withY ≡0 andZ ≡0.

Next result is about the CLT.

Theorem 2(Central Limit Theorem) : Assume (H₁)−(H₅). Let suppose ∂b

∂x , ∂b

∂y and σ verify all assumptions before. Then, ∀α∈ [0,¹₄), r ≥1, the random field Y^ε converge in L^r to a random field Y ≡Y⁰ as ε→0 on C_ψ^α([0, m],R^d) determined by :







dY_t = [∂b

∂x(t, X_t, Xt−τ)Y_t+ ∂b

∂y(t, X_t, Xt−τ)Yt−τ]dt+σ(t, X_t, Xt−τ)dW_t t∈(0,∞)

Y_t = 0 t∈[−τ,0]

(13)

where ∂b

∂x = ( ∂b

∂x_i)_1≤i≤d and ∂b

∂y = ( ∂b

∂y_j)_1≤j≤d. By theorem 4 in the next section and in view of √

εh(ε) → 0, h(ε) → ∞, as ε tends to 0.

One see thatZ^ε obeys LDP on C_ψ^α([0, m],R^d) with speed h²(ε), and the rate function I_{M DP}(f) = inf{1

2 Z m

0

|h(t)|˙ ²dt, h ∈ H, G(h) =f} (14) where Gis solution to the following deterministic equation :











dG(h)(t) = ∂b

∂x(t, F(h)(t), F(h)(t−τ))G(h)(t)dt+ ∂b

∂y(t, F(h)(t), F(h)(t−τ))G(h)(t−τ)dt +σ(t, F(h)(t), F(h)(t−τ)) ˙h(t)dt t∈(0,∞)

G(h)(t) = 0 t∈[−τ,0]

where F is solution to (7). Hence, formally one get the following result.

Theorem 3(MDP) :Assume (H₁)−(H₅) . Let suppose σ is continuous, b, ∂_xb, ∂_yb and σ satisfy assumptions above.

Then, for all α∈[0,¹₄) , Z^ε obeys a LDP on C_ψ^α([0, m],R^d) with speed h²(ε) with rate function I_{M DP} defined by (14).

3 Useful results

Let us recall some useful results.

Theorem 4( found in [3]) : If an LDP with a good rate function I(.) holds for the probability measure{µ_ε, ε >0}which are exponentially equivalent to {˜µ_ε, ε >0}, then the same LDP holds for µ˜_ε.

Lemma 5 : Assume (5).

Then, for any p≥2, there exist C such that :

E|X^ε(t)|^p ≤C(1 + sup|ψ(t)|^p)

Proof : One use a similar argument adapted as in Walsh [10], Theorem 3.2.

Proposition 6 : Assume (5) and ψ bounded.

Then, for all p≥2, there exist C(p, K, ∂_xb, ∂_yb, T, ψ) such that :

E(|X^ε−X|^T_∞)^p ≤ε^p2C(p, K, ∂_xb, ∂_yb, T, ψ)−→0 as ε →0

Proof : One use a similar argument adapted as in Wang and Zhang [11], Proposition 3.2.

We will use Talagrand’s transportation inequality.

Let

W_p(µ, ν) := inf(

Z Z

d(x, y)^pdπ(x, y))^1p 1≤p≤+∞

L^p−Wasserstein distance between µ, ν two probability measures on E.

µ satisfies Talagrand’sT₂−transportation inequality T₂(C_T) on (E, d) if

∀ν :W₂(ν, µ)² ≤2C_TH(ν|µ)

where H(ν|µ) is Kullback-Leibler information or relative entropy ofν with respect toµ.

H(ν|µ) =







R log dν

dµdν ν << µ

+∞ otherwise

Consider E = L²([0, T]; R^d) = {φ : [0, T] −→ R^d measurable ; kφk²₂ = RT

0 |φ(t)|²dt < +∞}

equipped with the distance d₂(φ₁, φ₂) := kφ₁−φ₂k₂. Let µ^ε the law ofX^ε = (X_t^ε)_t∈[0,T] a pro- bability measure on E .

Lemma 7 : Assume that σ, b are locally lipschitz functions, σ bounded such that : 1

2(|(σ(s, x, y)−σ(s, u, v))(σ(s, x, y)−σ(s, u, v))^∗|+hx−u, b(s, x, y)−b(s, u, v)i +hy−v, b(s, x, y)−b(s, u, v)i)≤L(|x−u|²+|y−v|²)

Then, for all ε ∈ [0,1] , µ^ε the law of X^ε := (X_t^ε)_t∈[0,T] satisfies on (L²([0, T];R^d);d₂) T₂−Talagrand’s inequality T₂(εC_T), where

C_T := kσk²_∞(exp((δ+ 2L)T)−1) δ(δ+ 2L)

δ >0 is taken arbitrary, and

kσk_∞ = sup

(t,x,y)∈R⁺×R^d×R^d,v∈R^d,kvk=1

kσ(t, x, y)vk

≤r

sup

u∈R⁺×R^d×R^d

tr(σσ^∗(u))

≤√ M

Proof : One use a similar argument as in Ma, Wang and Wu [7] lemma 4.2.

Lemma 8 : Assume that σ, b are locally Lipschitzian, and σ bounded such that : 1

2(|(σ(s, x, y)−σ(s, u, v))(σ(s, x, y)−σ(s, u, v))^∗|+hx−u, b(s, x, y)−b(s, u, v)i +hy−v, b(s, x, y)−b(s, u, v)i)≤L(|x−u|²+|y−v|²)

Then ∀ε∈(0,1], r >0 P((

Z T 0

|Y_t^ε|²dt)¹² −E( Z T

0

|Y_t^ε|²dt)¹² ≥r)≤exp{− r² 2C_T} where C_T = kσk²_∞(e^(δ+2L)T −1)

δ(δ+ 2L)

Proof : One use a similar adapted argument as in Ma, Wang and Wu [7] Lemma 4.3.

Set Φ(ϕ) = kϕ−X⁰

√ε k∞. Φ is a lipschitz function on (L^p([0, m];R^d);d₂) with the lipschitz constant 1

√ε. Since kY^εk∞ = Φ(X^ε), this inequality follow from lemma 7 by Bobkov-G¨otze’s criterion [1], Theorem 3.1.

4 Proofs

First of all, one suppose that b andσ are bounded . We will prove both theorems (CLT and MDP).

4.1 Proof of Theorem 2

One has to prove that

limε→0E(kY^ε−Y⁰k_α) = 0.

Y^ε(t) = 1

√ε(X^ε(t)−X⁰(t))

= 1

√ε Z t

0

{b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ )}ds+ Z t

0

σ(s, X_s^ε, X_s−τ^ε )dWs

and

Y⁰(t) = Z t

0

[∂b

∂x(s, X_s⁰, X_s−τ⁰ )Y⁰(s) + ∂b

∂y(s, X_s⁰, X_s−τ⁰ )Y⁰(s−τ)ds+ Z t

0

σ(s, X_s⁰, X_s−τ⁰ )dW_s Y^ε(t)−Y⁰(t) =

Z t 0

[σ(s, X_s^ε, X_s−τ^ε )−σ(s, X_s⁰, X_s−τ⁰ )]dW_s +

Z t 0

[b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ )

√ε

− {∂b

∂x(s, X_s⁰, X_s−τ⁰ )Y_s⁰(s) + ∂b

∂y(s, X_s⁰, X_s−τ⁰ )Y_s−τ⁰ (s)}]ds

=A^ε₁(t) +A^ε₂(t) +A^ε₃(t) (15)

where A^ε₁(t) =

Z t 0

[σ(s, X_s^ε, X_s−τ^ε )−σ(s, X_s⁰, X_s−τ⁰ )]dW_s A^ε₂(t) =

Z t 0

[b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ )

√ε − {∂b

∂x(s, X_s⁰, X_s−τ⁰ )Y_s^ε(s) + ∂b

∂y(s, X_s⁰, X_s−τ⁰ )Y_s−τ⁰ (s))}]ds A^ε₃(t) =

Z t 0

[∂b

∂x(s, X_s⁰, X_s−τ⁰ )(Y^ε(s)−Y⁰(s)) + ∂b

∂y(s, X_s⁰, X_s−τ⁰ )(Y^ε(s−τ)−Y⁰(s−τ))ds First, we shall estimate A^ε₁(t)A^ε₂(t), A^ε₃(t).

E|A^ε₁(t)|^p ≤ Z t

0

L_σE(|X^ε(t)−X⁰(t)|^p).

Thanks to proposition 6, we have for p > 2

E(|X^ε(t)−X⁰(t)|^p) ≤ε^p2C(p, K, L_σ, L_b)

E|A^ε₁(t)|^p ≤ε^p2C(p, K, L_σ, L_b). (16) Using Taylor formula, there exist a random field θ^ε(t) (0,1)^d−valued such that :

b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ )

√ε =∂b

∂x(s, X_s⁰ +θ^ε(s)(X_s^ε−X_s⁰), X_s−τ⁰ )Y_s^ε(s) + ∂b

∂y(s, X_s⁰, X_s−τ⁰ +θ^ε(s)(X_s−τ^ε −X_s−τ⁰ ))Y_s−τ^ε (s) Since ∂b

∂x and ∂b

∂y are uniformly continuous, we have :

|∂b

∂x(s, X_s⁰+θ^ε(s)(X_s^ε−X_s⁰, X_s−τ⁰ )− ∂b

∂x(s, X_s⁰, X_s−τ⁰ )|

≤L_∂xb(|X_s⁰ +θ^ε(s)(X_s^ε−X_s⁰)−X_s⁰|+|X_s−τ⁰ +θ^ε(s)(X_s−τ^ε −X_s−τ⁰ )−X_s−τ⁰ |)

≤L_∂xb(|θ^ε(s)(X_s^ε−X_s⁰)|+|θ^ε(s)(X_s−τ^ε −X_s−τ⁰ )|)

≤L∂xb(|X_s^ε−X_s⁰|+|X_s−τ^ε −X_s−τ⁰ |) Then,

|A^ε₂(t)| ≤ Z t

0

|(X_s^ε−X_s⁰)Y_s^ε|ds ≤√ εC

Z t 0

|Y_s^ε|²ds Thus, we get for p > 2

E(|A^ε₂(t)|)² ≤ε^p2C (17)

|A^ε₃(t)| ≤ Z t

0

[L3|Y_s^ε−Y_s⁰|+L4|Y_s−τ^ε −Y_s−τ⁰ |]ds (18)

Using Gronwall’s inequality, we get

E(|Y_s^ε−Y_s⁰|^p +|Y_s−τ^ε −Y_s−τ⁰ |^p)≤ε^p2C(p, K, L₁) For p >2, t∈[0, T] by Burkholder-Gandy-Devis’s inequality, we get

E|A^ε₁(t)−A^ε₁(s)|^p ≤C_pE( Z t

s

[σ(u, X_u^ε, X_u−τ^ε )−σ(u, X_u⁰, X_u−τ⁰ )]²du)^p2

≤C(p, L_b)E( Z t

s

|X_s^ε−X_s⁰|²du)^p2

Thanks to proposition 6, one has

E|A^ε₁(t)−A^ε₁(s)|^p ≤C(p, K, L_σ, L_b)ε^p2|t−s|^p−22 (19)

|A^ε₂(t)−A^ε₂(s)|^p ≤(

Z t s

|b(u, X_u^ε, X_u−τ^ε )−b(u, X_u⁰, X_u−τ⁰ )

√ε

− {∂b

∂x(u, X_u⁰, X_u−τ⁰ )Y_u^ε(u) + ∂b

∂y(u, X_u⁰, X_u−τ⁰ )Y_u−τ⁰ (u))}|du)^p E|A^ε₂(t)−A^ε₂(s)|^p ≤E(

Z t s

L₁ 1

√ε(|X_u^ε−X_u⁰|+|X_u−τ^ε −X_u−τ⁰ |)du+ Z t

s

[L₃|Y_u^ε|+L₄|Y_u−τ^ε |du)^p

≤C(p, K, L₁, L₂, L₃, L₄)|t−s|^p−22 (20) Similarly

E|A^ε₃(t)−A^ε₃(s)|^p ≤C(p, L₃, L₄)(

Z t s

[|Y_u^ε−Y_u⁰|+|Y_u−τ^ε −Y_u−τ⁰ |]du)^p

≤C(p, K, L₁, L₃, L₄)ε^p2|t−s|^p−22 (21) Thus, by (19), (20), (21) there exist a constant ˇC independent of ε such that :

E|Y_t^ε−Y_t⁰−(Y_s^ε−Y_s⁰)|^p ≤Cεˇ ^p2|t−s|^p−22 Then, for all α∈(0,¹₂), one can choose p∈(2,3) such that :

limε→0EkY^ε−Y⁰k_α = 0.

It concludes the proof of Theorem 2(CLT).

4.2 Proof of theorem 3

Y_t^ε = X_t^ε−X_t⁰

√ε , t≥0. So Y_t^ε satisfies : dY_t^ε = 1

√ε(b(t, X_t^ε, X_t−τ^ε )−b(t, X_t⁰, X_t−τ⁰ ))dt+σ(t, X_t^ε, X_t−τ^ε )dW_t t∈(0,+∞)

Y_t^ε = 0 t ∈[−τ,0]

Since X_t^ε close to X_t⁰ and X_t−τ^ε close to X_t−τ⁰ , we have : dY_t^ε '[∂b

∂x(t, X_t⁰, X_t−τ⁰ )Y_t^ε+ ∂b

∂y(t, X_t⁰, X_t−τ^ε )Y_t−τ^ε ]dt+σ(t, X_t^ε, X_t−τ^ε )dW_t where ∂b

∂x = ( ∂b

∂x_i)1≤i≤d and ∂b

∂y = ( ∂b

∂y_j)1≤j≤d.

Y^ε should be close to Y⁰ determined by (13). Then, by Schilder’s theorem and the contraction principle , Z^ε obeys the LDP with the speed h²(ε) and the rate function I_{M DP}(h) given by (14).

Hence by Theorem 3, it is enough to show that Z^ε ish²(ε)−exponentially equivalent to Z⁰. In other words, we have to prove

lim sup

ε→0

h⁻²(ε) logP(kZ^ε−Z⁰k_α> δ) =−∞ ∀ δ >0 (22) As we first prove the case for b and σ bounded, we need the following Proposition.

Proposition 10 : b, σ are Lipschitz functions, there exist some constant C(p, K, Cb) which depend on p, K, C_b such that

E(kX^ε−X⁰k∞)^p ≤ε^p2C(p, K, C_b)−→0 as ε→0 Proof :

X_t^ε−X_t⁰ = Z t

0

(b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ ))ds+√ ε

Z t 0

σ(s, X_s^ε, X_s−τ^ε )dW_s kX^ε−X⁰k^p_∞≤( sup

t∈[0,T]

| Z t

0

[b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ )]|)^p +ε^p2( sup

t∈[0,T]

| Z t

0

σ(s, X_s^ε, X_s−τ^ε )dWs|)^p

LetJ₁^ε(t) = Z t

0

[b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ )]ds and J₂^ε(t) =

Z t 0

σ(s, X_s^ε, X_s−τ^ε )dWs.

Since b is lipschitz continuous function, by H¨older’s inequality : E(kJ₁^εk∞)^p ≤C_b^p(

Z t 0

1ds)^pqE Z t

0

(kX^ε−X⁰k∞)^p where 1

p+ 1 q = 1

E(kJ₂^εk∞)^p ≤ε^p2E| Z t

0

σ(s, X_s^ε, X_s−τ^ε )dW_s|^p

≤ε^p2E| Z t

0

σ²(s, X_s^ε, X_s−τ^ε )ds|^p2

≤ε^p2C(K, p)

Therefore, there exists some constant C(p, K, C_b) such that E(kX^ε−X⁰k∞)^p ≤C(p, K, C_b)(E

Z T 0

(kX^ε−X⁰k∞)^pdt+ε^p2) By Gronwall’s lemma, one obtain

E(kX^ε−X⁰k_∞)^p ≤ε^p2C(p, K, C_b)e^C(p,K,Cb) Recall

Y_t^ε=X_t^ε−X_t⁰

√ε = 1

√ε Z t

0

[b(s, X_s^ε, X_s−τ^ε )]ds+ Z t

0

[σ(s, X_s^ε, X_s−τ^ε )]dW_s Y_t⁰ =

Z t 0

[∂b

∂x(s, X_s⁰, X_s−τ⁰ )Z_s⁰+ ∂b

∂y(s, X_s⁰, X_s−τ⁰ )Z_s−τ⁰ ]ds +

Z t 0

[σ(s, X_s^ε, X_s−τ^ε )]dWs

d(Y_t^ε−Y_t⁰) =[∂b

∂x(t, X_t⁰, X_t−τ⁰ )(Z_t^ε−Z_t⁰) + ∂b

∂y(t, X_t⁰, X_t−τ⁰ )(Z_t−τ^ε −Z_t−τ⁰ )]dt + [σ(t, X_t^ε, X_t−τ^ε )−σ(t, X_t⁰, X_t−τ⁰ )]dW_t

+ 1

√ε[(b(t, X_t^ε, X_t−τ^ε )−b(t, X_t⁰, X_t−τ⁰ ))

−(∂b

∂x(t, X_t⁰, X_t−τ⁰ )(X_t^ε−X_t⁰) + ∂b

∂y(t, X_t⁰, X_t−τ⁰ )(X_t−τ^ε −X_t⁰))]dt

=[∂b

∂x(t, X_t⁰, X_t−τ⁰ )(X_t^ε−X_t⁰) + ∂b

∂y(t, X_t⁰, X_t−τ⁰ )(X_t−τ^ε −X_t−τ⁰ )]dt+dU_t^ε

U_t^ε = Z t

0

σ(s, X_s^ε, X_s−τ^ε )−σ(s, X_s⁰, X_s−τ⁰ )]dW_s + 1

√ε Z t

0

[(b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ ))

−(∂b

∂x(s, X_s⁰, X_s−τ⁰ )(X_s^ε−X_s⁰) + ∂b

∂y(s, X_s⁰, X_s−τ⁰ )(X_s−τ^ε −X_s−τ⁰ ))]ds Y_t^ε−Y_t⁰ =U_t^ε+

Z t 0

[∂b

∂x(t, X_t⁰, X_t−τ⁰ ) + ∂b

∂y(t, X_t⁰, X_t−τ⁰ )]J(s, t)U_t^ε

where J(s, s) = Id and db

dt = (∂b

∂x(t, X_t⁰, X_t−τ⁰ ) + ∂b

∂y(t, X_t⁰, X_t−τ⁰ ))J(s, t)U_t^ε(s)ds for 0≤s≤t Since

|h∂b

∂x(t, X_t⁰, X_t−τ⁰ ) + ∂b

∂y(t, X_t⁰, X_t−τ⁰ ), x+yi| ≤L(|x|²+|y|²) One has

|J(s, t)u| ≤C(p, K, C_b)|u| ∀y∈R^d As

k∂b

∂x(t, X_t⁰, X_t−τ⁰ ) + ∂b

∂y(t, X_t⁰, X_t−τ⁰ )k∞= sup

x,y,v∈R^d,kuk≤1

k[∂b

∂x(t, X_t⁰, X_t−τ⁰ ) + ∂b

∂y(t, X_t⁰, X_t−τ⁰ )](u)k One has, for all t ∈[0, T],

|Y_t^ε−Y_t⁰| ≤ |U_t^ε|+k∂b

∂x(t, X_t⁰, X_t−τ⁰ ) + ∂b

∂y(t, X_t⁰, X_t−τ⁰ )k∞C(p, K, C_b) Z t

0

|U_s^ε|ds

≤(1 +k∂b

∂x(t, X_t⁰, X_t−τ⁰ ) + ∂b

∂y(t, X_t⁰, X_t−τ⁰ )k∞

C(p, K, C_b)

L )kU^εk∞ (23) One has to prove

Proposition 11 : For all r >0 lim sup

ε→0

h⁻²(ε) logP(kU^εk∞

h(ε) > r) = −∞

Proof

|U_t^ε| ≤|

Z t 0

(σ(s, X_s^ε, X_s−τ^ε )−σ(s, X_s⁰, X_s−τ⁰ ))dW_s| +

Z t 0

√1

ε|(b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ ))

−(∂b

∂x(s, X_s⁰, X_s−τ⁰ )(X_s^ε−X_s⁰) + ∂b

∂y(s, X_s⁰, X_s−τ⁰ )(X_s−τ^ε −X_s−τ⁰ ))|ds :=|M_t^ε|+

Z t 0

√1

ε|(b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ ))(X_s^ε−X_s⁰)|ds

where M_t^ε is a continuous martingale with hM^εi_t=

Z t 0

(σ(s, X_s^ε, X_s−τ^ε )−σ(s, X_s⁰, X_s−τ⁰ ))(σ(s, X_s^ε, X_s−τ^ε )−σ(s, X_s⁰, X_s−τ⁰ ))^∗ds For all η >0, one has :

P( sup

0≤t≤T

|M_t^ε| ≥ rh(ε)

2 )≤P( sup

0≤t≤T

|M_t^ε| ≥ rh(ε)

2 ;hM^εi_t≤η) +P(hM^εi_T ≥η)

≤2 exp (−

(rh(ε))² 2²

2η ) +P(hM^εi_T ≥η)

≤2 exp (−(rh(ε))²

2η ) +P(hM^εi_T ≥η) Since,

hM^εi_T = Z T

0

(σ(s, X_s^ε, X_s−τ^ε )−σ(s, X_s⁰, X_s−τ⁰ ))(σ(s, X_s^ε, X_s−τ^ε )−σ(s, X_s⁰, X_s−τ⁰ ))^∗ds

≤L Z T

0

|X_s^ε−X_s⁰|ds =εL Z T

0

|Y_s^ε|²ds

for some L. forε small enough, by proposition 1, one has : E(

Z T 0

|Y_s^ε|²ds)¹² ≤ 1 2( η

εL)¹²

Using lemma 8, one has : P(hM^εi_T ≥η)≤P(

Z T 0

|Y_s^ε|²ds≥ η εL)

≤P((

Z T 0

|Y_s^ε|²ds)¹² −E( Z T

0

|Y_s^ε|²ds)¹² ≥( η

εL)¹² − 1 2( η

εL)¹²)

≤P((

Z T 0

|Y_s^ε|²ds)¹² −E( Z T

0

|Y_s^ε|²ds)¹² ≥ 1 2( η

εL)¹²)

≤exp{− η 8εCTL}

Notice that εh²(ε)−→0 andη >0 is taken arbitrary, one get : lim sup

ε→0

h⁻²(ε) logP( sup

0≤t≤1

|M_t^ε| ≥ rh(ε)

2 ) =−∞ (24)

Since b ∈ C¹ and ∂b

∂x, ∂b

∂x are uniformly continuous, for all η > 0 there exist some δ > 0 such that :

|b(s, x, y)−b(s, u, y)− ∂b

∂x(s, x, y)(x−u)| ≤ |x−u|

as |X_t^ε−X_t⁰| ≤δ

|b(s, x, y)−b(s, x, v)− ∂b

∂y(s, x, y)(y−v)| ≤ |y−v| as |X_t−τ^ε −X_t−τ⁰ | ≤δ

Hence, P(

Z T 0

√1

ε|b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ )− ∂b

∂x(s, X_s⁰, X_s−τ⁰ )(X_s^ε−X_s⁰)

−(∂b

∂y(s, X_s⁰, X_s−τ⁰ )(X_s−τ^ε −X_s−τ⁰ )|ds≥ rh(ε) 2 )

≤P(kX^ε−X⁰k_∞≥δ) +P( Z T

0

|Z_t^ε|dt ≥ rh(ε) 4η ) +P(

Z T 0

|Z_t−τ^ε |dt≥ rh(ε) 4η ) By Freidlin-Wentzell Theorem :

lim sup

ε→0

εlogP(kX^ε−X⁰k∞≥δ) = 0

and

lim sup

ε→0

εh⁻²(ε) logP(kX^ε−X⁰k_∞ ≥δ) = 0

Since ε is sufficiently small, E( Z T

0

|Z_t^ε|²dt)¹² ≤ rh(ε) 4η√

T by lemma 7.

Using Cauchy-Schwartz inequality and lemma 8, one get P(

Z T 0

|Z_s^ε|dt ≥ rh(ε)

2η )≤P((

Z T 0

|Z_s^ε|dt)¹²)≥ rh(ε) 2η )

≤P((

Z T 0

|Z_s^ε|dt)¹² −E( Z T

0

|Z_s^ε|dt)¹²)≥ rh(ε) 4η√

T)

≤exp{−(rh(ε))² 32η²C_T }

Analogously,

P( Z T

0

|Z_s−τ^ε |dt≥ rh(ε)

2η )≤exp{−(rh(ε))² 32η²C_T } Since η is taken arbitrary, one has :

lim sup

ε→0

h⁻²(ε) logP( Z T

0

√1

ε|b(s, X_s^ε, X_s−τ^ε )−b(s, X_s⁰, X_s−τ⁰ )− ∂b

∂x(s, X_s⁰, X_s−τ⁰ )(X_s^ε−X_s⁰))

−(∂b

∂y(s, X_s⁰, X_s−τ⁰ )(X_s−τ^ε −X_s−τ⁰ ))|ds≥ rh(ε)

2 ) = −∞ (25)

One complete the proof of proposition 11 by (23), (24) and (25).

Next, one has to prove in H¨older norm.

Lemma 12 : For allδ >0, lim sup

ε→0

h⁻²(ε) logP(kZ^ε−Z⁰kα > δ) = −∞

Proof :

I₃^ε(t) = Z t

0

[∂b

∂x(s, X_s⁰, X_s−τ⁰ )(Y_s^ε−Y_s⁰) + ∂b

∂y(s, X_s⁰, X_s−τ⁰ )(Y_s−τ^ε −Y_s−τ⁰ )]ds

Using H¨older’s inequality, one has

|I₃^ε(t)−I₃^ε(s)| ≤(

Z t s

|∂b

∂x(s, X_s⁰, X_s−τ⁰ )|^qdu)^1q( Z t

s

|Y_s^ε−Y_s⁰|^pdu)^1p +≤(

Z t s

|∂b

∂y(s, X_s⁰, X_s−τ⁰ )|^qdu)^1q( Z t

s

|Y_s−τ^ε −Y_s−τ⁰ |^pdu)^1p

≤C_∂^q

x,∂y|t−s|^1q[(

Z t s

|Y_u^ε|^pdu)^1p + ( Z t

s

|Y_u−τ^ε |^pdu)^1p]

Choose q ∈(2; 3), α= 1

q , one get kI₃^εk_α ≤C_∂x,∂y(

Z t s

(1 +u)^α(kY^ε−Y⁰k_α)^pdu)^1p Hence, for t∈[0,1], one has

kY_u^ε−Y_u⁰k_α ≤C(p, ∂_xb, ∂_yb)[kI₁^εk^p_α+kI₂^εk^p_α+ Z t

s

(kY_u^ε−Y_u⁰k_α)^pdu]^1p Applying Gronwall’s lemma to g(t) = (kY^ε−Y⁰k_α)^p, one has :

(kY^ε−Y⁰k_α)^p ≤C(p, C_∂x,∂y)(kI₁^εk_α+kI₂^εk_α)^pe^{C(p,T ,C∂xb,∂y b)} It suffices to show that for δ >0,

lim sup

ε→0

h⁻²(ε) logP(kI_i^εk_α

h(ε) ≤δ) = −∞ i= 1,2 Forε >0, η >0,

P(kI₁^εk_α ≥h(ε)δ)≤P(kI₁^εk_α ≥h(ε)δ,kX^ε−X⁰k∞ < η) +P(kX^ε−X⁰k∞ ≥η) Since

|(σ(s, x, y)−σ(s, u, v))(σ(s, x, y)−σ(s, u, v))^∗| ≤L(|x−u|²+|y−v|²) then, for each s ≥0,0< α < 1

2 and h(ε)δ≥ηdL,

P(kI₁^εk_α ≥h(ε)δ,kX^ε−X⁰k∞< η)≤2dexp(−(h(ε)δ)² 2η²ld² )