LARGE DEVIATIONS FOR STOCHASTIC SYSTEMS WITH MEMORY

In this paper, we develop a large deviations principle for stochastic delay equations 
driven by small multiplicative white noise. Both upper and lower large deviations estimates are established.

1. Introduction.Large deviations were studied by many authors beginning with the fundamental work of Donsker and Varadhan [4], [5], [6].Subsequently several issues concerning large deviation principles and their applications to stochastic differential equations were studied by many authors, e.g.Freidlin and Wentzell [8], Stroock [18], Deuschel and Stroock [3], den Hollander [2], and others.However, there is little published work on large deviations for stochastic systems with memory.The problem of large deviations for such systems was first studied by M. Scheutzow [16] within the context of additive white noise.
In this paper we examine the question of small random perturbations of systems with memory and the associated problem of large deviations.Our analysis allows for multiplicative noise with possible dependence on the history in the diffusion coefficient.Our approach is similar to that in [1] and [18], but introduces a new induction argument in order to handle the delay.

We introduce the following conditions:
The research of the first author is supported in part by NSF grants DMS-9703596, DMS-9980209 and DMS-0203368.
(A1) The functions b, σ satisfy a Lipschitz condition.That is, there exist constants L 1 , L 2 such that for all x 1 , x 2 , y 1 , y 2 and t ∈ [0, ∞), Let τ > 0 be a fixed delay, and ψ be a given continuous function on [−τ, 0].Consider the following differential delay equation (dde): and the associated perturbed sdde: with solution X ε .
Throughout this paper, we will assume, without loss of generality, that the delay τ is equal to 1.

3.
Statement of the Main Theorem and Proofs.Let C 0 [0, m], R l denote the space of all continuous Otherwise, define e(g) = ∞.Let F (g) be the solution to the dde Denote by That is, The rest of the paper is devoted to the proof of this result.The proof is split into several lemmas.
For any ε > 0 and any n ≥ 1, denote by X ε n (•) the solution to the sdde: We need the following lemma from Stroock [18] (p.81).
Lemma 3.3.In addition to (A.1) and (A.2), assume that b, σ are bounded.Then for any m ≥ 1, δ > 0, the following is true: Proof.We prove (13) by induction on m.We first prove it for m Observe that Using Lemma 3.2, there exists a constant c ρ > 0 such that Hence, For λ > 0, define φ λ (y is a martingale with initial value zero, where, By uniform continuity, there exists an integer N so that Choose λ = 1 ε and take expectations in (19) to obtain Hence, we have Therefore, lim sup Given M > 0, first choose ρ sufficiently small so that log( ρ 2 ρ 2 +δ 2 ) + C ≤ −2M , and then use (18) to choose N so that lim sup ε→0 ε log P (τ ε n,ρ ≤ 1) ≤ −2M for n ≥ N .Combining these two facts gives Since M is arbitrary, we have proved (13) for m = 1.Assume now (13) holds for some integer m.We will prove it is also true for m + 1.Let Y ε n , τ ε n,ρ be defined as before.In addition, introduce two new stopping times: As in the proof of (18), By the induction hypothesis, Again by Itô's formula, is a martingale with M n,ρ 0 = 0, where , and sufficiently large n.
Using (25), (26) and following the proof of the case for m = 1, we see that ( 13) is also true for m + 1.
This completes the proof of the lemma.
For n ≥ 1, define the map Proof.Note that for g with e(g) ≤ α, By the linear growth condition on b and σ, we have Using Grownwall's inequality, this implies that In particular, Again by the linear growth condition and (30), we have uniformly over the set {g; e(g) ≤ α}.Thus, This proves the lemma.
Proof of Theorem 3.1 when b, σ are bounded.Notice that where X ε (•) is the solution to equation (6).
Proof.We use induction on m.We first prove (40) for m = 1.For λ > 0, set φ λ (y By Itô's formula, the process is a martingale with initial value zero, where Hence, This implies P ( sup Assume now that (40) holds for some m.We will prove that it is also true for m + 1.For R 1 > 0, set = P ( sup As before, by Itô's formula, is a martingale with initial value zero, where Using (51) and the proof of (46), we get lim sup Thus it follows from (49) that lim sup Using the induction hypothesis and letting R 1 → ∞ we obtain (40) for m + 1.This completes the proof of the proposition.Let X ε R (•) be the solution to the sdde Proof.Again we will use induction.We omit the proof for the case m = 1 since it is similar to that of Lemma 3.3.Let us assume that (55) holds for some m.We will prove that it also holds for m + 1. Set For ρ > 0 , let φ λ (y) : By the induction hypothesis, is a martingale with initial value zero, where, as in the proof of Lemma 3.3, for s As before, this implies that lim sup Hence, it follows from (57), ( 58) and (60) that lim R→∞ lim sup ε→0 ε log P ( sup For g with e(g) < ∞, let F R (g) be the solution to the dde  (66) Using the large deviation principle for {µ R ε , ε > 0}, we obtain lim sup  Remark.The results in this paper can be easily extended to the case, where different delays τ 1 , τ 2 are allowed in (6): . Then b R (t, x, y) = b(t, x, y), σ R (t, x, y) = σ(t, x, y), for t ∈ [0, m], |x| ≤ R, |y| ≤ R. Furthermore, b R and σ R satisfy the Lipschitz condition (A.1) with the same Lipschitz constant.