Large deviations for neutral stochastic functional differential equations

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.

Before giving the preliminaries, a few words about the notation are in order. Throughout this paper, C > 0 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 is equipped with the Hilbert-Schimidt norm · HS . A * stands for the transpose of the matrix A. For a sub-interval U ⊆ R, C(U; R d ) means the family of all continuous functions f : U → R d . Let τ > 0 be a fixed number and C = C([−τ, 0]; R d ), endowed with the uniform norm f ∞ := sup −τ ≤θ≤0 |f (θ)|. For fixed t ≥ 0, let f t ∈ C be defined by f t (θ) = f (t + θ), θ ∈ [−τ, 0]. In terminology, (f t ) t≥0 is called the segment (or window) process corresponding to (f (t)) t≥−τ .
In this paper, we are interested in the following NSFDE where G, b : C → R d , σ : C → R d × R d and {W (t)} t≥0 is a d-dimensional Brownian motion on some filtered probability space (Ω, F , (F t ) t≥0 , P). The proofs of main results will be based on an extension of the contraction principle in [8,Theorem 4.2.23]. To make the content self-contained, we recall it as follows: Lemma 2.1. Let {µ } 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 < ∞, 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].
which is a Hilbert space endowed with the inner product as follows: (s)ġ(s)ds.

LDP FOR NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 2371
We define The well-known Schilder theorem (see [8]) states that the laws To investigate the LDP for the laws of {X (t)} t∈[−τ,T ] , we give the following assumptions about coefficients. (H1) There exists a constant L > 0 such that (H3) There exists a constant M > 0 such that Remark 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].
Let F (h) be the unique solution of the following deterministic equation: Herein, The first main result of this section is stated as follows.
where L T (h) is defined as in (4). That is, We can extend the result of Theorem 3.1 to the case that b, σ are not necessary to satisfy the bounded condition (H3). We only assume b is locally Lipschitz with polynomial growth, that is, there exist constants L > 0, q ∈ N, such that ∀ξ, η ∈ C , we have Let 0 denote the function such that 0(θ) = 0, θ ∈ [−τ, 0]. We can see that b is polynomial growth we state the second result as follows.
where L T (h) is defined as in (4).
In the sequel, we first finish the proof of Theorem 3.1. Before giving the proof of Theorem 3.1, we prepare some lemmas.
The next lemma shows that the measurable map F (h)(·) can be approximated well by the continuous maps F n (h)(·).
where α < ∞ is a constant.
If G(0) = 0, by (H2) and the fundamental inequality, for any ξ ∈ C and > 0, we have Taking sufficiently small such that κ(1 + ) < 1, the proof can be complete by repeating the one above.
We now complete the Proof of Theorem 3.1. Notice that X ,n (s) = F n ( √ W )(s), where W is the ddimensional Brownian motion. Then by the contraction principle in large deviations theory, we get that the law of X ,n (s) satisfies an LDP. Then Lemma 3.3 states that X ,n (s) approximates exponentially X (s). Furthermore, Lemma 3.4 shows that the extension of contraction principle to measurable maps F (h)(·) can be approximated well by continuous maps F n (h)(·), i.e. Lemma 3.3, so the proof of Theorem 3.1 follows from Lemma 2.1.
In the sequel, we will finish the proof of Theorem 3.2.
In order to prove our theorem, we shall use the truncated method. For R > 0, define In the following, we prove that b R and σ R satisfy the Lipschitz condition under the condition (9). We only give the proof for b R .
Similarly, we can show that b R satisfies the Lipschitz condition if η ∞ ≤ R, ξ ∞ > R.

LDP FOR NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 2381
Since b R and σ R satisfy the Lipschitz condition, it is easy to verify that b R and σ R satisfy the assumptions (H1) and (H3).
Let X ,R (·) be the solution to the NSFDE We recall a Lemma in [8], which is a key point in the proofs of following Lemmas. Lemma 3.6. Let N be a fixed integer. Then, for any a i ≥ 0, The lemma below states that X ,R (·) is the uniformly exponential approximation of X (·) on the interval [−τ, T ].

YONGQIANG SUO AND CHENGGUI YUAN
By mimicking the argument in Lemma 3.3 for t ≤ T ∧ ξ R1 , one gets This implies that Taking Logarithmic function into consideration, we have This, together with (30),(36) and (39), implies The conclusion follows from letting first ρ → 0 and then R 1 → ∞ by Lemma 3.5.
Proof of Theorem 3.2. For h with L T (h) < ∞, let F R (h) be the solution of the equation below Proof. For R > 0, and a closed subset C ⊂ C([−τ, T ]; R d ), set C R := C∩{f ; f ∞ ≤ R}. C δ R denotes the δ-neighborhood of C R . Denote by µ ,R the law of X R . Then we have µ (C) = µ (C R1 ) + µ C, sup
Owing to the arbitrary of φ 0 , it follows that which is the lower bound (i) in Theorem 3.1, thus, the proof of Theorem 3.2 is complete.