Large deviations for neutral stochastic functional differential equations

Transcription

1 arxiv: v1 [math.pr] 15 Mar 219 Large deviations for neutral stochastic functional differential equations Yongqiang Suo and Chenggui Yuan Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, UK March 18, 219 Abstract In this paper, under a one-sided Lipschitz condition on the drift coefficient we adopt via contraction principle a exponential approximation argument to investigate large deviations for neutral stochastic functional differential equations. AMS Subject Classification: 6F5, 6F1, 6H1. Keywords: large deviations, neutral stochastic functional differential equations 1 introduction As is well known, Large deviation principle LDP for short is a branch of probability theory that deals with the asymptotic behaviour of rare events, and it has a wide range of applications, such as mathematic finance, statistic mechanics, biology and so on. So the large deviation principle for SDEs has been investigated extensively; see, e.g.;[2, 1, 16] and reference therein. From the literature, we know there are two main methods to investigate the LDPs, one method is based on contraction principle in LDPs, that is, it relies on approximation arguments and exponential-type probability estimates; see e.g.,[3, 9, 1, 11, 12, 13, 16, 17] and references therein. [9, 13, 17] concerned about the LDP for SDEs driven by Brownian motion or Poisson measure, [11] investigated how rapid-switching behaviour of solutionx ǫ t affects the small-noise asymptotics of X ǫ t-modulated diffusion processes on the certain interval. [1] investigated the LDP for invariant distributions of memory gradient diffusions. The other one is weak convergence method, which has also been applied in establishing LDPs for a various stochastic dynamic systems; see e.g.,[2, 1, 4, 5, 6, 7]. According to the compactness argument in this method of the solution space of corresponding skeleton equation, the weak convergence is done for Borel measurable functions whose existence is based on Yamada-Watanabe theorem. In [4, 5, 7], the authors study a large deviation principle for SDEs/SPDEs. 1

2 Compared with the weak convergence method, there are few literature about the LDP for SFDEs, [16] gave result about large deviations for SDEs with point delay, and large deviations for perturbed reflected diffusion processes was investigated in [3]. The aim of this paper is to study the LDP for NSFDEs, which extends the result in [16]. The structure of this paper is as follows. In section 2, we introduce some preliminary results and notation. In section 3, we state the main result about LDP for NSFDEs and give its proof. Before giving the preliminaries, a few words about the notation are in order. Throughout this paper, C > stipulates a generic constant, which might change from line to line and depend on the time parameters. 2 Preliminaries Let R d,,, be the d-dimensional Euclidean space with the inner product, which induces the norm. Let M d d denote the set of all d d matrices, which equipped with the Hilbert-Schimidorm HS. A standsforthetransposeofthematrixa. Forasub-interval U R, CU;R d be the family of all continuous functions f : U R d. Let τ > be a fixed number and C = C[,];R d, endowed with the uniform norm f := θ fθ. Fixed t, let f t C be defined by f t θ = ft+θ,θ [,]. In terminology, f t t is called the segment or window process corresponding to ft t. In this paper, we are interested in the following neutral stochastic functional differential equation NSFDE 2.1 d{x ǫ t GX ǫ t } = bxǫ t dt+ ǫσx ǫ t dwt, t [,T], Xǫ = ξ C, where G,b : C R d, σ : C R d R d and {Wt} t is a d-dimensional Brownian motion on some filtered probability space Ω, F,F t t,p. The proof of main result Theorem 3.1 will be based on an extension of the contraction principle in [8, Theorem ]. To make the content self-contained, we recall it as follows: Lemma 2.1. Let {µ ǫ } be a family of probability measures that satisfies the LDP with a good rate function I on a Hausdorff topological space X, and for m = 1,2,, let f m : X Y be continuous functions, with Y, d a metric space. Assume there exists a measurable map f : X Y such that for every α <, 2.2 lim m {x:ix α} df m x,fx =. Then any family of probability measures { µ ǫ } for which {µ ǫ fm 1 } are exponentially good approximations satisfies the LDP in Y with the good rate function I y = inf{ix : y = fx}. We now state the classical exponential inequality for stochastic integral, which is crucial in proving the exponential approximation. For more details, please refer to Stroock [18, lemma 4.7]. 2

3 Lemma 2.2. Let α : [, Ω R d R d and β : [, Ω R d be F t t -progressively measurable processes. Assume that α HS A and β B. Set ξt := αsdws+ t βsds for t. Let T > and R > satisfy 2BT < R. Then d1 2.3 P 3 LDP for NSFDE Let H denote the Cameron-Martin space, i.e. H = { ht = 1 R d ξt R 2dexp 2A 2 dt 2BT 2. } ḣsds : [,T] R d ; ḣs 2 ds < +, which is an Hilbert space endowed with the inner product as follows: We define f,g H = fsġsds. 3.1 L T h = { 1 2 ḣt 2 dt, if h H, + otherwise. The well-known Schilder theorem states that the laws µ ǫ of { ǫwt} t [,T] satisfies the LDP on C[,T];R d with the rate function L T. To investigate the LDP for the laws of {X ǫ t} t [,T], we give the following assumptions about coefficients. H1 There exists a constant L > such that 2 ξ η+gη Gξ,bξ bη L ξ η 2, and σξ ση 2 HS L ξ η 2, ξ,η C; H2 There exists a constant κ,1 such that 3.2 Gξ Gη κ ξ η, G =, ξ,η C. Remark 3.1. The one-sided Lipschitz condition on the drift coefficient in H1 is different from the global Lipschitz condition in [2]. Moreover, our method below is different from that of [2]. 3

4 Remark 3.2. From H1, H3, it is easy to see that ξ Gξ,bξ L 2 1+ ξ 2, Gξ 2 κ 2 ξ 2, ξ C. Remark 3.3. Let µdθ P[,] and let Gξ = α 1 ξθµdθ, σξ = α 2 ξθµdθ, bξ = α 3 ξ α 4 ξ α 1 for some constants α i,i = 1,,5 such that α 1 κ, 1/3 ξθµdθ +α5 ξθµdθ, α 3 α 1 1+α 5 1+α 1 α 2 2 L, then the assumptions H1 and H2 hold true. In fact, by the Hölder inequality, one has Gξ Gη 2 α 2 1 noting that ξθ ηθ 2 µdθ α 2 1 ξ η 2 µdθ = α 2 1 ξ η 2, α 4 ξ η Gξ Gη,ξ Gξ 1/3 η Gη 1/3, so ξ η Gξ Gη,bξ bη α 3 ξ η 2 +α 3 ξ η Gξ Gη +α 5 ξ η α 3 α 1 1+α 5 1+α 1 ξ η 2, σξ ση 2 HS α 2 2 ξθ ηθ µdθ α 5 Gξ Gη ξθ ηθ 2 µdθ α 2 2 ξ η 2. ξθ ηθ µdθ Therefore, the assumptions hold if the constants α i,i = 1,...,5 satisfy the conditions above. Let Fh be the unique solution of the following deterministic equation: Fht GF t h = Fh GF h+ t F b s h ds F σ s hḣsds, t [,T], F h = ξθ, θ [,]. Herein, F t hθ = Fht+θ, θ [,]. The main result of this section is stated as follows. Theorem 3.1. Under the assumptions H1-H2, it holds that {µ ǫ,ǫ > }, the law of X ǫ on C[,T];R d, satisfies the large deviation principle with the rate function below { } 3.5 If := inf L T h;fh = f,h H, f C[,T];R d, where L T h is defined as in 3.1. That is, 4

5 i for any closed subset C C[,T];R d, ii for any open subset G C[,T];R d, lim logµ ǫ C inf If, ǫ f C lim inf ǫ logµ ǫ G inf f G If. Before giving the proof of Theorem 3.1, we prepare some lemmas. We construct X ǫ,n by exploiting an approximate scheme, that is, for a real positive number s, let [s] = {k Z : k s} be its integer part. For any n N, we consider the following NSFDE 3.6 d{x ǫ,n t GX ǫ,n t } = bx ǫ,n t dt+ ǫ,n ǫσ X t dwt, t, X ǫ,n = ξ, where, for t, X ǫ,n t θ := X ǫ,n t+θ, := [nt]/n,n 1,θ [,]. According to [14, Theorem 2.2, p.24], 3.6 has a unique solution by solving piece-wisely with the time length 1/n. In the sequel, we consider two cases separately. Case 1:. We assume that b,σ are bounded, i.e. H3 There exists a constant M > such that bξ σξ HS M, ξ C. Next, we show that {X ǫ,n,ǫ > } defined by 3.6 approximates to {X ǫ,ǫ > }. Lemma 3.2. Assume H1, H2, and H3 hold, then for any δ >, one has 3.7 lim limǫlogp X ǫ t X ǫ,n t > δ =. n ǫ t T Proof. For notation brevity, we set Z ǫ,n t := X ǫ t X ǫ,n t,t and Y ǫ,n t := X ǫ t X ǫ,n t GXt GX ǫ ǫ,n t,t. Noting X ǫ,n = X n = ξ, we write Y ǫ,n t as follows: Y ǫ,n t = It is easy to see from 3.2 that bx ǫ s bx ǫ,n s ds+ ǫ σx ǫ s σ Z ǫ,n t Y ǫ,n t + GX ǫ t GXǫ,n t Y ǫ,n t +κ X ǫ t X ǫ,n t, 5 X ǫ,n s dws.

6 and noting X ǫ = X ǫ,n = ξ,ξ C, it yields that 3.8 Z ǫ,n t 1 1 κ Y ǫ,n t. For ρ >, we define τn ǫ ρ = inf{t : X ǫ,n t ξn ǫ ρ = inf{t : Z ǫ,nρ t δ}, and compute X ǫ,n t > ρ}, Z ǫ,nρ = Z ǫ,n t τ ǫ n ρ, 3.9 P Observe that X ǫ,n t θ Z ǫ,n t > δ = P Pτ ǫ n ρ T+P Z ǫ,n t > δ,τn ǫ ρ T Pτ ǫ n ρ T+Pξ ǫ n ρ T. +P Z ǫ,n t > δ,τ ǫ n ρ > T Z ǫ,n t > δ,τn ǫ ρ > T X ǫ,n t θ = X ǫ,n t+θ X ǫ,n t+θ = X ǫ,n t+θ X ǫ,n t+θi {t+θ<tn} +X ǫ,n t+θ X ǫ,n I {tn t+θ} This, together with 3.2, yields 3.1 X ǫ,n t = X ǫ,n t+θ X ǫ,n I {tn t+θ} = GX ǫ,n t+θ t+θ GXǫ,n + bxs ǫ,n ds+ X ǫ,n t 1 1 κ t θ +θ +θ ǫσ Xǫ,n s dws. +θ bxs ǫ,n ds+ ǫσ Xǫ,n s dws. Taking H3 into consideration and utilizing Lemma 2.2, one gets that P X ǫ,n t ǫ,n X t ρ 2dexp nρ1 κ dm 2 2nM 2 1 κ 2 dǫ provided that dm 1 κn < ρ. Which, together with the definition of stopping time τǫ n ρ, it follows that 3.11 lim limǫlogpτn ǫ n ρ T =. ǫ For λ >, let φ λ y = ρ 2 + y 2 λ, an application of Itô s formula yields 3.12 φ λ Y ǫ,nρ t = ρ 2λ +M ǫ,nρ t+ τ ǫ nρ γ ǫ λ sds,, 6

7 X ǫ,n s dws is a mar- where M ǫ,nρ t := 2λ τnρ ǫ ρ 2 + Y ǫ,n s 2 λ 1 ǫ Y ǫ,n s,σxs ǫ,n σ tingale. Moreover, by H1, we see that 3.13 γλ ǫ s : = 2λρ2 + Y ǫ,n s 2 λ 1 Y ǫ,n s,bxs ǫ bxǫ,n s +2λλ 1ǫρ 2 + Y ǫ,n s 2 λ 2 σxs σ ǫ ǫ,n X s Y ǫ,n s 2 +λǫρ 2 + Y ǫ,n s 2 λ 1 σxs ǫ ǫ,n σ X s 2 HS 2Lλρ 2 + Y ǫ,n s 2 λ 1 Z ǫ,n s 2 +λ2λ 1ǫρ 2 + Y ǫ,n s 2 λ 1 σx ǫ s σ X ǫ,n s 2 HS C 1 ρ 2 + Y ǫ,n s 2 λ 1 Z ǫ,n s 2 +C 2ρ 2 + Y ǫ,n s 2 λ 1 X ǫ,n s X ǫ,n s 2, where C 1 = 2Lλ[2λ 1ǫ+1], C 2 = 2Lλǫ2λ 1. Using the Burkholder-Davis-Gundy BDG for short inequality, we obtain 3.14 E 8 2ǫλ E 1 2 E 1 2 E M ǫ,nρ t τ ǫ nρ τ ǫ nρ τnρ ǫ τ ǫ nρ +128λ 2 ǫe 1 2 E τnρ ǫ τ ǫ nρ +128Lλ 2 ǫe ρ 2 + Y ǫ,n s 2 2λ 2 Y ǫ,n s 2 σx ǫ s σ ρ 2 + Y ǫ,n s 2 λ +64λ 2 ǫe ρ 2 + Y ǫ,n s 2 λ +128Lλ 2 ǫe τ ǫ nρ τ ǫ nρ ρ 2 + Y ǫ,n s 2 λ 1 σxs ǫ,n σ ρ 2 + Y ǫ,n s 2 λ +128Lλ 2 ǫe ρ 2 + Y ǫ,n s 2 λ 1 X ǫ,n s τ ǫ nρ 1 ǫ,n X s 2 2 HSds ρ 2 + Y ǫ,n s 2 λ 1 σxs ǫ ǫ,n σ X s 2 HS ds ρ 2 + Y ǫ,n s 2 λ 1 Z ǫ,n s 2 ds X ǫ,n s 2 HS ds ρ 2 + Y ǫ,n s 2 λ 1 Z ǫ,n s 2 ds X ǫ,n s 2 ds. 7

8 Combining 3.13 and 3.14 and reformulating 3.12, one has E φ λ Y ǫ,nρ t ρ 2λ +4Lλ66λǫ ǫ+1 Eρ 2 + Y ǫ,nρ s 2 λ 1 Zs ǫ,nρ 2 ds +4Lλǫ68λ 1 Eρ 2 + Y ǫ,nρ s 2 λ 1 Xs ǫ,nρ 2ρ 2λ +4Lλ66λǫ ǫ+1 +4Lλǫ68λ 1 2ρ 2λ +C 3 +C 4 E E u s u s E u s ǫ,nρ X s 2 ds ρ 2 + Y ǫ,nρ u 2 λ 1 Z ǫ,nρ ρ 2 + Y ǫ,nρ u 2 λ 1 X ǫ,nρ ρ 2 + Y ǫ,nρ u 2 λ ds, u u 2 ds ǫ,nρ X u 2 ds L where C 3 = 4λ66λǫ ǫ+1, C 1 κ 2 4 = 4Lλǫ68λ 1. In the last step, we utilized the fact that Y ǫ,nρ t =,t [,] and 3.8. Choosing λ = 1 and setting t := ρ 2 + Y ǫ Φǫ,nρ ǫ,nρ t ξn ǫ ρ 2 1/ǫ, by the Gronwall inequality, we obtain where C 5 = L so then we have E κ 2 Φ ǫ,nρ t 2ρ 2λ e C 3+C 4 T 2ρ 2/ǫ e C5T/ǫ,. Noting that Φ ǫ,nρ t = ρ 2 + Y ǫ,nρ t 2 1/ǫ I {t ξ ǫ nρ } +ρ 2 + Y ǫ,nρ ξ ǫ n ρ 2 1/ǫ I {ξn ǫ ρ <t}, ρ 2 +1 κ 2 δ 2 1/ǫ Pξn ǫ ρ T E Φ ǫ,nρ t, Pξn ǫ 2 ǫ ρ 2 1/ǫ ρ T e C 5 T/ǫ. ρ 2 +1 κ 2 δ 2 Thus, lim ǫlogpξn ǫ ρ 2 ρ T log +C ǫ ρ 2 +1 κ 2 δ 2 5 T. ρ Finally, given L >, choose ρ sufficiently small such that log 2 + C ρ 2 +1 κ 2 δ 2 5 T 2L. Next, utilizing 3.11, choose N such that lim ǫ ǫlogpτn ǫ ρ T 2L for n N. Then, for n N there is an < ǫ n < 1 such that Pτn ǫ ρ T e L/ǫ and Pξn ǫ ρ T e L/ǫ for < ǫ ǫ n, so 3.9 leads to P Z ǫ,n t δ 2e L/ǫ, < ǫ ǫ n. 8

9 Thus, lim ǫlogp ǫ The proof of the lemma is complete. Z ǫ,n t > δ L, n N. For n 1, define the map F n : C [,T],R d C ξ [,T],R d by F n ωt GFω = Fn ω GF tn ω+ bfs nωds +σ F s n ωωt ω, t + 1, n F n ωt = ξt, t, where Fs nωθ = Fn ωs+θ and F s nωθ = F n ωs+θ s n. Notice that, X ǫ,n t = F n ǫwt, which is a continuous map. Herein, W is a standard Brownian motion. For h H, we define 3.16 F n ht GF n t h = Fn h GF n h+ b F n s h ds F n h = ξ C. + σ Fn s hḣsds,t [,T], The next lemma shows that the measurable map Fh can be approximated well by the continuous maps F n h. Lemma 3.3. Under the assumptions of Theorem 3.1, we have 3.17 lim F n ht Fht =, where α < is a constant. n {h:l T h α} t T Proof. Fornotationbrevity, wesetm n t := F n ht GF h, by fundamental inequality a+b 2 [1+η]a 2 + b2 and H2, we derive η F n ht 2 = F n ht GF h+gfn t h 2 GF n 1+η t h 2 + F n ht GF h p η κ 2 F 1+η h 2 + F n ht GF h. 2 η Letting η = κ, we then have 1 κ 3.18 F n ht 2 κ 1 1 κ ξ κ M n t 2. 2 On the other hand, it is easy to see that 3.19 M n t 2 1+κ 2 F n t h 2. 9

10 By H1, H2, we obtain from 3.16 that M n t 2 1+κ 2 ξ M n s,bfs n h+σ F hḣs ds 1+κ 2 ξ 2 +L 2 1+ Fs n h 2 ds+ 1+κ 2 ξ 2 +L 2 +L 2 1+ F n t h 2 ḣs 2 ds. M n s 2 ds+ 1+ Fs n h 2 ds+ M n s 2 ds σ F n t hḣs 2 ds Noting that F n t h = θ F n ht + θ θ F n ht + θ, which together with 3.18,3.19, yields that F n ht 2 t T ξ 2 + F n ht 2 ξ 2 + κ 1 κ ξ κ M n t κ+1+κ2 1 κ 2 ξ κ 2 [ L 2 +1+κ 2 +L 2 Fs n h 2 ḣs 2 ds+l 2 by the Gronwall inequality, we get F n ht 2 1 κ+1+κ 2 1 κ 2 n 1 t T where C 1 = In particular, 1 κ+1+κ 2 1 κ M α = h;l T h α ] ḣs 2 ds, F n s h 2 ds ξ 2 + 2L 2L T h { L2 +1+κ 2 T +2L 2 L T h } exp 1 κ 2 1 κ 2 C 1 1+L T hexp{c 2 1+L T h}, ξ 2 2L 2, C 1 κ 2 2 = L2 +1+κ 2 T 1 κ 2 n 1 t T 2L 2. 1 κ 2 F n ht 2 C 1 1+αexp{C 2 1+α} <. Hence, in the same way as the argument of 3.1, we arrive at 3.21 F n h F t h 1 1 κ 1 1 κ t θ +θ bf n s h ds+ C α M α 1 n 1/2, as n 1 bf n s hds+ +θ σ F s n hḣsds σ F s n hḣs ds

11 uniformly over the set {h;l T h α}. For notation brevity, we set D n ht := F n ht Fht GFh GF th, similarly, it is easy to see from H1,H2 that 3.22 F n ht Fht 2 and 1 1 κ D n ht 2, D n ht 2 1+κ 2 F n t h F th 2. Using 3.4 and 3.16, we deduce 3.24 D n ht 2 + L +L 2 D n hs,bf n s h bf s h ds 2 D n hs,[σ F s n h σfn s h+σfn s h σfsh]ḣs ds Fs n h F sh 2 ds+ D n hs 2 ds F n hs Fhs 2 ḣs 2 ds+l F n s h F n s h 2 ḣs 2 ds, which, together with 3.21, 3.22 and 3.23, yields that F n ht Fht 2 1 { L+1+κ 2 F n t T 1 κ 2 s h F s h 2 ds +L it follows from the Gronwall inequality that, F n ht Fhs 2 t T Hence, the desired assertion is followed by taking n. Proof of Theorem 3.1 in case 1 F n s h F sh 2 ḣs 2 ds+2lαc α M α 1 n 1/4 1 2LαC α M α { n L+1+κ 2 T +2Lα } exp. 1 κ 2 1 κ 2 1/4 }, Proof. Notice that X ǫ,n s = F n ǫ 1/2 Ws, where W is the Brownian motion. Then by the contraction principle in large deviations theory, we get that the law of X ǫ,n s satisfies an LDP. Then Lemma 3.2 states that X ǫ,n s approximates exponentially to X ǫ s. Furthermore, Lemma 3.3 shows that the extension of contraction principle to measurable maps Fh can be approximated well by continuous maps F n h, i.e. Lemma 3.2, so the proof of case 1 of Theorem 3.1 follows from Lemma

12 Next, we consider Case 2: b, σ are unbounded. Lemma 3.4. Under H1, H2, and for R >, we have 3.25 lim lim ǫlogp X ǫ t > R =. R ǫ t T Proof. For notation brevity, we set Y ǫ t := X ǫ t GX ǫ t, from H2 and fundamental inequality, it yields that 3.26 Y ǫ t 2 1+κ 2 X ǫ t 2, and 3.27 X ǫ t t T 1 κ ξ κ Y ǫ t 2. 2 For λ >, applying the Itô formula, H1,H2 and 3.3 yield Y ǫ t 2 λ 1+1+κ 2 ξ 2 λ +λ +2λλ 1ǫ +λǫ 1+ Y ǫ s 2 λ 1 2 Y ǫ s,bx ǫ s ds 1+ Y ǫ s 2 λ 2 σx ǫ sy ǫ s 2 ds 1+ Y ǫ s 2 λ 1 σx ǫ s 2 HS ds+mǫ,λ t 1+1+κ 2 ξ 2 λ +M ǫ,λ t +λl 2 1+2λǫ ǫ 1+ Y ǫ s 2 λ 1 1+ X ǫ s 2 ds 1+1+κ 2 ξ 2 λ +M ǫ,λ t +λl 2 C 1 1+2λǫ ǫ 1+ Y ǫ u 2 ds, λ u s where C 1 = 1+ ξ 2 1 κ 1 1 κ 2, M ǫ,λ t = 2λǫ 1+ Y ǫ s 2 λ 1 Y ǫ s,σx ǫ sdws, and in the last step, we used Noting that X ǫ s 2 ξ 2 + u s X ǫ u 2, by H1, 3.27 and the BDG inequality, we obtain E ǫλ E 1 2 E 1 2 E M ǫ,λ t 1+ Y ǫ s 2 2λ 1 σx ǫ s 2 HS ds 1/2 1+ Y ǫ s 2 λ +64L 2 λ 2 ǫe 1+ Y ǫ s 2 λ +64L 2 λ 2 C 1 ǫe Y ǫ s 2 λ 1 1+ X ǫ s 2 ds u s 1+ Y ǫ u 2 λ ds.

13 Substituting 3.29 into 3.28, and reformulating 3.28, we arrive at E 1+ Y ǫ t 2 λ κ 2 ξ 2 λ +2L 2 C 1 λ[66λǫ+1 ǫ] E u s 1+ Y ǫ u 2 λ ds. For R >, we define ξr ǫ = inf{t : Xǫ t > R}, utilising BDG s inequality yields that E 1+ Y ǫ t ξr ǫ 2 λ 21+1+κ 2 ξ 2 λ exp{2l 2 C 1 λ[66λǫ+1 ǫ]t}, which implies that { } E 1+ Y ǫ t ξr ǫ 2 I λ {ξ ǫ R T} 21+1+κ 2 ξ 2 λ exp{2l 2 C 1 λ[66λǫ+1 ǫ]t}, P t T choosing λ = 1 ǫ ǫlogp X ǫ t > R t T yields that PξR ǫ T 21+1+κ2 ξ 2 λ exp{2l 2 C 1 λ[66λǫ+1 ǫ]t} λ, 1+[R κ 1 κ ξ 2 ]1 κ2 X ǫ t > R log lim lim R 21+1+κ 2 ξ 2 +ǫ2l 2 C 1 λ[66λǫ+1 ǫ]t 1+[R κ 1 κ ξ 2 ]1 κ κ 2 ξ 2 log +2L 2 C 1 67 ǫt, 1+[R κ 1 κ ξ 2 ]1 κ2 ǫ The proof is therefore complete. ǫlogp t T X ǫ t > R =. For R >, define m R = { Gx, bx, σx HS ; x R}, and G R i = m R 1 G i m R +1, b R i = m R 1 b i m R + 1, σ R i,j = m R 1 σ i,j m R + 1, 1 i,j d. Let G R = G R 1,GR 2,,GR d, b R = b R 1,bR 2,,bR d and σ R = σ R i,j 1 i,j d. Then for x R, G R x = Gx, b R x = bx, σ R x = σx. Also, G R, b R and σ R satisfy the assumptions H1 and H2. Let X ǫ,r be the solution to the NSFDE d{x ǫ,r t GX ǫ,r t } = b R X ǫ,r t dt+ ǫσ R X ǫ,r t dwt,t >, with the initial datum X ǫ,r = ξθ, θ [,]. We recall a Lemma in [8], which is a key point in the proofs of following Lemmas. 13

14 Lemma 3.5. Let N be a fixed integer. Then, for any a i ǫ, 3.31 lim ǫlog ǫ N i=1 a i ǫ = max N lim i=1 ǫ ǫloga i ǫ. The lemma below states that X ǫ,r is the uniformly exponential approximation of X ǫ on the interval [,T]. Lemma 3.6. Assume H1, H2 hold, then for any T >, δ >, one has that: 3.32 lim lim ǫlogp X ǫ t X ǫ,r t > δ =. R ǫ t T Proof. For notation simplicity, we set Z ǫ,r t := X ǫ t X ǫ,r t and Y ǫ,r t := X ǫ t X ǫ,r t GXt ǫ GXǫ,R t. From H2, it is easy to see that 1 Z ǫ,r t 1 κ Y ǫ,r t. Define ξ ǫ R 1 := inf{t : X ǫ t R 1 }. For any R R 1, we have 3.33 Y ǫ,r t ξ ǫ R 1 = ξ ǫ R1 b R X ǫ s b R X ǫ,r s ds+ ǫ ξ ǫ R1 σ R X ǫ s σ R X ǫ,r s dws. Setting Z ǫ R 1 t := Z ǫ,r t ξ ǫ R 1, Y ǫ R 1 t := Y ǫ,r t ξ ǫ R 1 and ξ ǫ R,δ := inf{t : Zǫ R 1 t δ}. Then, we have 3.34 P = P t T t T Z ǫ,r t > δ Z ǫ,r t ξ ǫ R 1 > δ,i {ξ ǫ R1 T} PξR ǫ 1 T+PξR,δ ǫ T P X ǫ t > R 1 +PξR,δ ǫ T. t T +P t T By mimicking the argument in Lemma 3.2 for t T ξ ǫ R 1, one gets E ρ 2 + Y ǫ R 1 t 2 1/ǫ 2ρ 2/ǫ e CT/ǫ. Z ǫ,r t ξ ǫ R 1 > δ,i {ξ ǫ R1 T} This implies that PξR,δ ǫ 2 ǫ ρ 2 1/ǫ T e CT/ǫ. ρ 2 +1 κ 2 δ 2 14

15 Taking Logarithmic function into consideration, we have lim ǫ ǫlogpξ ǫ R,δ T log ρ 2 ρ 2 +1 κ 2 δ 2 +CT. This, together with 3.25,3.31 and 3.34, implies lim lim ǫlogp Z ǫ,r t > δ R ǫ t T lim lim ǫlog P X ǫ t > R 1 + lim lim PξR,δ ǫ T R ǫ t T R ǫ { limǫlogp X ǫ ρ t > R 1 2 } log +CT. ǫ ρ 2 +1 κ 2 δ 2 t T The conclusion follows from letting first ρ and then R 1 by Lemma 3.4. For h with L T h <, let F R h be the solution of the equation below F R ht GFt R h = FR h GF R h+ b R Fs R hds+ σ R Fs R hḣsds with the initial datum F R h = ξθ, θ [,]. Define for each f C[,T];R d. If { 1 T I R f = inf 2 t T Fht If = I R f, for all f with Proof of Theorem 3.1 in case 2 } ḣt 2 dt; F R h = f, R, then Fh = F R h. t T ft R. Proof. For R >, and a closed subset C C[,T];R d, set C R := C {f; f R}. CR δ denotes the δ-neighborhood of C R. Denote by µ ǫ,r the law of XR ǫ. Then we have µ ǫ C = µ ǫ C R1 +µ ǫ C, X ǫ t > R 1 µ ǫ C R1 +P P t T t T X ǫ t > R 1 t T X ǫ t X ǫ,r t > δ +µ R ǫ C δr1 +P t T X ǫ t > R 1. 15

16 Taking the large deviation principle for {µ R ǫ,ǫ > } yields from 3.5 that lim ǫlogµ ǫ C ǫ { limǫlog P ǫ t T } +P X ǫ t > R 1 t T inf I R f f CR δ 1 X ǫ t X ǫ,r t > δ + inf I R f f CR δ 1 lim ǫlogp X ǫ t > R 1 ǫ t T lim ǫlogp X ǫ t X ǫ,r t > δ. ǫ t T Then we obtain the upper bound i in Theorem 3.1, that is lim ǫlogµ ǫ C inf If, ǫ f C bytakingfirst R, andδ, thenr 1. LetGbeanopensubset ofc[,t];r d. Then for any φ G, and taking δ >, we define Bφ,δ = {f; f φ δ} G. Then using the large deviation principle for {µ R ǫ ;ǫ > }, one gets I R φ liminfǫlogµ R ǫ Bφ, δ ǫ 2 { liminfǫlog P X ǫ,r t φ δ ǫ 2, X ǫ t X ǫ,r t δ t T 2 +P lim infǫlogµ ǫ G ǫ t T X ǫ,r t φ δ 2, X ǫ t X ǫ,r t δ t T 2 lim infǫlogp X ǫ t X ǫ,r t δ ǫ t T 2. t T Noting that I R φ = Iφ provided that φ R. Then we have Iφ liminf ǫ Owing to the arbitrary of φ, it follows that ǫlogµ ǫ G, as R. inf If liminfǫlogµ ǫ G, f G ǫ which is the lower bound i in Theorem 3.1, thus, the proof of Theorem 3.1 is complete. References [1] Bao, J., Yin, G., Yuan, C., Asymptotic analysis for functional stochastic differential equations, Springer, Cham, }

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