Large Deviations for Stochastic Systems with Memory

Souther Illiois Uiversity Carbodale OpeSIUC Articles ad Preprits Departmet of Mathematics 7-6 Large Deviatios for Stochastic Systems with Memory Salah-Eldi A. Mohammed Souther Illiois Uiversity Carbodale, salah@sfde.math.siu.edu Tusheg Zhag Uiversity of Machester Follow this ad additioal works at: http://opesiuc.lib.siu.edu/math_articles Part of the Mathematics Commos This is a pre-copy-editig, author-produced PDF of a article accepted for publicatio i Discrete ad Cotiuous Dyamical Systems-Series B followig peer review. The defiitive publisher-autheticated versio "Large Deviatios for Stochastic Systems with Memory", Discrete ad Cotiuous Dyamical Systems-Series B, 64, 881-893 is available olie at: http://aimscieces.org/jourals/dcdsb/ dcdsb_olie.jsp. Recommeded Citatio Mohammed, Salah-Eldi A. ad Zhag, Tusheg. "Large Deviatios for Stochastic Systems with Memory." Jul 6. This Article is brought to you for free ad ope access by the Departmet of Mathematics at OpeSIUC. It has bee accepted for iclusio i Articles ad Preprits by a authorized admiistrator of OpeSIUC. For more iformatio, please cotact opesiuc@lib.siu.edu.

Mauscript submitted to Website: http://aimscieces.org AIMS Jourals Volume, Number, Xxxx XXXX pp. LARGE DEVIATIONS FOR STOCHASTIC SYSTEMS WITH MEMORY SALAH-ELDIN A. MOHAMMED AND TUSHENG ZHANG Abstract. I this paper, we develop a large deviatios priciple for stochastic delay equatios drive by small multiplicative white oise. Both upper ad lower large deviatios estimates are established. 1. Itroductio. Large deviatios were studied by may authors begiig with the fudametal work of Dosker ad Varadha [4],[5],[6]. Subsequetly several issues cocerig large deviatio priciples ad their applicatios to stochastic differetial equatios were studied by may authors, e.g. Freidli ad Wetzell [8], Stroock [18], Deuschel ad Stroock [3], de Hollader [], ad others. However, there is little published work o large deviatios for stochastic systems with memory. The problem of large deviatios for such systems was first studied by M. Scheutzow [16] withi the cotext of additive white oise. Stochastic systems with memory or stochastic differetial delay equatios sdde s serve as viable models i a variety of applicatios, ragig from ecoomics ad fiace to sigal processig Elsaosi, ksedal ad Sulem [7], Kolmaovskii ad Myshkis [1]. The origis of the qualitative theory of stochastic systems with memory goes back to work by Itô ad Nisio [9], Kusher [11], Mizel ad Trutzer [13], Mohammed [14], Scheutzow [17], Mao [1] ad others. I this paper we examie the questio of small radom perturbatios of systems with memory ad the associated problem of large deviatios. Our aalysis allows for multiplicative oise with possible depedece o the history i the diffusio coefficiet. Our approach is similar to that i [1] ad [18], but itroduces a ew iductio argumet i order to hadle the delay.. Basic Settig ad Notatio. Let W t := Wt 1, Wt,..., Wt l deote a stadard l-dimesioal Browia motio o a complete filtered probability space Ω, F, F t t, P, with W =. Let b = b 1, b,..., b d : R + R d R d R d, σ = σ ij i=1, d,j=1,,l : R + R d R d R d R l be Borel measurable fuctios. We itroduce the followig coditios: The research of the first author is ported i part by NSF grats DMS-973596, DMS-9989 ad DMS-3368. 1

SALAH-ELDIN A. MOHAMMED AND TUSHENG ZHANG A1 The fuctios b, σ satisfy a Lipschitz coditio. That is, there exist costats L 1, L such that for all x 1, x, y 1, y ad t [,, σt, x 1, y 1 σt, x, y R d R L 1 x l 1 x + y 1 y, 1 bt, x 1, y 1 bt, x, y R d L x 1 x + y 1 y. A The fuctios b, x, y, σ, x, y are cotiuous o [,, uiformly i x, y R d, i.e., bs, x, y bt, x, y =, 3 s t x,y R d σs, x, y σt, x, y =. 4 s t x,y R d Let τ > be a fixed delay, ad ψ be a give cotiuous fuctio o [ τ, ]. Cosider the followig differetial delay equatio dde: dxt = bt, Xt, Xt τ dt, t, Xt = ψt, t [ τ, ], ad the associated perturbed sdde: dx t = b t, X t, X t τ dt + 1 σ t, X t, X t τ dw t, t, X t = ψt, t [ τ, ], with solutio X. 5 6 Throughout this paper, we will assume, without loss of geerality, that the delay τ is equal to 1. 3. Statemet of the Mai Theorem ad Proofs. Let C [, m], R l deote the space of all cotiuous fuctios g : [, m] R l with g =. If g C [, m], R l is absolutely cotiuous, set eg = m ġt dt. Otherwise, defie eg =. Let F g be the solutio to the dde F gt = F g + + F gt = ψt, 1 t. b s, F gs, F gs 1 ds σ s, F gs, F gs 1 ġsds, < t m Deote by C ψ [ 1, m], R d the set of all cotiuous fuctios f : [ 1, m] R d such that ft = ψt for all t [ 1, ]. 7 Theorem 3.1. Let µ be the law of X o C ψ [ 1, m], R d, equipped with the uiform topology. The family {µ, > } satisfies a large deviatio priciple with the followig good rate fuctio { } 1 If := if eg; F g = f, f C ψ [ 1, m], R d. 8 That is,

LARGE DEVIATIONS FOR STOCHASTIC SYSTEMS WITH MEMORY 3 i for ay closed subset C C ψ [ 1, m], R d, ii for ay ope subset G C ψ [ 1, m], R d, log µ C if If, 9 f C if log µ G if If. 1 f G The rest of the paper is devoted to the proof of this result. The proof is split ito several lemmas. For ay > ad ay 1, deote by X the solutio to the sdde: X t = X + + 1 X t = ψt, t [ 1, ]. b s, Xs, Xs 1 ds [s] [s] σ, X [s], X 1 dw s, t >, 11 We eed the followig lemma from Stroock [18] p. 81. Lemma 3.. Let α : [, Ω R d R l ad β : [, Ω R d be F t t -progressively measurable processes. Assume that α A ad β B, where the orm of α is the Hilbert-Schmidt orm ad the orm of β is the usual orm i R d. Set ξt := αsdw s + βsds for t. Let T > ad R > satisfy d 1 BT < R. The P ξt R d exp R d 1 BT /A dt. 1 t T Lemma 3.3. I additio to A.1 ad A., assume that b, σ are bouded. The for ay m 1, δ >, the followig is true: log P X t X t > δ =. 13 Proof. We prove 13 by iductio o m. We first prove it for m = 1. Set Y t := X t X t, t. The For ρ >, defie τ,ρ Y t = [ b s, X s, X s 1 b s, X s, X s 1 ] ds + 1 ξ,ρ := if{t, Y,ρt δ}. The [ σ s, X s, X s 1 ] [s] X 1 [s] σ, X dw s, t. [s] := if{t ; X t X [t] ρ}, ad set Y,ρt := Y t τ,ρ, t, P Y t > δ = P Y t > δ, τ,ρ 1 t 1 t 1 + P Y t > δ, τ,ρ > 1 t 1 P τ,ρ 1 + P ξ,ρ 1., 14 15

4 SALAH-ELDIN A. MOHAMMED AND TUSHENG ZHANG Observe that P τ,ρ 1 k=1 P k 1 t k Usig Lemma 3., there exists a costat c ρ > such that Hece, X t X k 1 ρ. 16 P τ,ρ 1 exp c ρ /, 1. 17 For λ >, defie φ λ y := ρ + y λ, y R d. By Itô s formula, M,ρ t is a martigale with iitial value zero, where, log P τ,ρ 1 =. 18 := φ λ Y,ρt τ,ρ γ λs := λρ + Y s λ 1 Y s, bs, X s, X s 1 γ λsds ρ λ 19 b s, X s, X s 1 + λλ 1ρ + Y s λ σ s, X s, X s 1 [s] [s] σ, X, X [s] 1 Y s + λ ρ + Y s λ 1 σ s, X s, X s 1 [s] [s] [s] σ, X, X 1 H.S. for s t τ,ρ. Noticig that X u = ψu ad X u = ψu for u, we see that γ λs cλφ λ Y s + { 4λλ 1 ρ + Y s λ Y s + λ ρ + Y s λ 1} { σs, X s, X s 1 [s] σ, X s, X s 1 + Y s + [s] X s X + [s] } ψs 1 ψ 1. By uiform cotiuity, there exists a iteger N so that ad for s 1 ad all N. Thus for N, σs, X s, X s 1 σ [s], X s, X s 1 < ρ ψs 1 ψ [s] 1 < ρ γ λs cλ + λ + λ φ λ Y s. 1 Choose λ = 1 ad take expectatios i 19 to obtai E[ρ + Y,ρt 1/ ] ρ / + C E[ρ + Y,ρs 1/ ]ds.

LARGE DEVIATIONS FOR STOCHASTIC SYSTEMS WITH MEMORY 5 Hece, E[ρ + Y,ρt 1/ ] ρ / e Ct. Sice we have Therefore, ρ + δ 1 P ξ,ρ 1 E[ρ + Y,ρ1 1/ ], 1 ρ P ξ,ρ 1 ρ + δ e C. ρ log P ξ,ρ 1 log ρ + δ + C. 3 Give M >, first choose ρ sufficietly small so that log ρ +δ + C M, ad the use 18 to choose N so that log P τ,ρ 1 M for N. Combiig these two facts gives log P 1 t 1 ρ X t X t > δ M. Sice M is arbitrary, we have proved 13 for m = 1. Assume ow 13 holds for some iteger m. We will prove it is also true for m + 1. Let Y, τ,ρ be defied as before. I additio, itroduce two ew stoppig times: τ 1,,ρ := if{t ; X t 1 Xt 1 ρ}, { τ,ρ, := if t ; [t] Xt 1 X 1 ρ}, ad defie Z,ρt := Y t τ 1,,ρ τ,,ρ τ,ρ ad ξ,ρ := if{t ; Z,ρt δ}. We the have As i the proof of 18, P Y t > δ P τ,ρ 1, τ,ρ, τ,ρ m + 1 t m+1 + P Y t > δ, τ,ρ 1, τ,ρ, τ,ρ > m + 1 t m+1 P τ 1,,ρ m + 1 + P τ,ρ τ,,ρ m + 1 + P ξ,ρ m + 1. 4 By the iductio hypothesis, Agai by Itô s formula, log P τ,ρ τ,ρ, m + 1 =. 5 log P τ,ρ 1, m + 1 log P M,ρ t τ := φ λ Z,ρt,ρ X t X t > ρ =. 1,, τ,ρ τ,ρ 6 γ λsds ρ λ 7

6 SALAH-ELDIN A. MOHAMMED AND TUSHENG ZHANG is a martigale with M,ρ =, where γ λs λρ + Y s λ 1 Y s Y s + X s 1 X s 1 + { 4λλ 1ρ + Y s λ Y s + λρ + Y s λ 1} { σs, X s, X s 1 [s] σ, X s, X s 1 [s] + Y s + X s X + } X [s] s 1 X 1 + X s 1 Xs 1 λρ + Y s λ 1 Y s Y s + ρ 8 + { 4λλ 1ρ + Y s λ Y s + λρ + Y s λ 1} Y s + 4ρ cλ + λ + λ φy s for s τ,ρ τ 1,,ρ τ,,ρ m + 1, ad sufficietly large. Usig 5, 6 ad followig 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 1, defie the map F : C [, m], R l C ψ [ 1, m], R d by F ωt := ψt, 1 t k t F ωt := F ω + bs, F ωs, F ωs 1ds k k k k k + σ, F ω, F ω 1 ωt ω for k t k+1. It is easy to see that F : C [, m], R l C ψ [ 1, m], R d is cotiuous. 9 Lemma 3.4. {g;eg α} F gt F gt =. Proof. Note that for g with eg α, Thus, F gt =F g + b s, F gs, F gs 1 ds [s] [s] [s] + σ, F g, F g 1 ġsds. F gt F gt = + [ bs, F gs, F gs 1 b s, F gs, F gs 1 ] ds [ [s] σ, F g [s] [s], F g 1 3 31 σ s, F gs, F gs 1 ] ġsds.

LARGE DEVIATIONS FOR STOCHASTIC SYSTEMS WITH MEMORY 7 By the liear growth coditio o b ad σ, we have F gt ψ + C + C Usig Growwall s iequality, this implies that 1 + F gu ds 1 u s 1 + F gu ġ sds. 1 u s I particular, M α = 1 1 u m g;eg α 1 1 u m F g u C exp m + eg. 3 Agai by the liear growth coditio ad 3, we have F gu C expm + α <. 33 [s] F gt F g + σ [t] t b s, F gs, F gs 1 ds [t] [s] [s] [s], F g, F g 1 ġ sds 1 1 C α M α 34 uiformly over the set {g; eg α}. Thus, F gt F gt C + C + F gs F gs ds + C F gs 1 F gs 1 ds [s] σ, x, y σs, x, y ġ sds x,y [ F gs 1 F gs 1 + F gs F gs ] ġ sds [ F [s] + gs F g + ] [s] F gs 1 F g 1 ġ sds 35 [ 1 1 [s] C α + σ s x,y, x, y σs, x, y ] + + 1 u s 1 u s F gu F gu ds F gu F gu ġ sds. 36

8 SALAH-ELDIN A. MOHAMMED AND TUSHENG ZHANG This gives, Hece, F gu F gu 1 u m CC α [ 1 1 + s {g;eg α} [s] σ x,y, x, y σs, x, y ]. F gt F gt =. 37 This proves the lemma. Proof of Theorem 3.1 whe b, σ are bouded. Notice that X s = F 1 W s, where W is the Browia motio. The theorem follows from Lemma 3.3, Lemma 3.4 ad the geeralized cotractio priciple Theorem 4..3 [1] i large deviatios theory.. Next, we remove the boudedess assumptios o b ad σ. We begi with Propositio 3.5. Assume that σt, x, y C1 + x + y, 38 bt, x, y C1 + x + y, 39 for all x, y R d. The for each iteger m 1, R where X is the solutio to equatio 6. log P X t > R = 4 Proof. We use iductio o m. We first prove 4 for m = 1. For λ >, set φ λ y := 1 + y λ, y R d. By Itô s formula, the process M λ t := φ λ X t γ λsds 1 + x λ, t, 41 is a martigale with iitial value zero, where γλs = λ 1 + X s λ 1 X s, b s, X s, X s 1 + λλ 1 1 + X s λ σs, X s, X s 1 X s + λ 1 + X s λ 1 σ s, X s, X s 1 H.S,

LARGE DEVIATIONS FOR STOCHASTIC SYSTEMS WITH MEMORY 9 for s 1. Sice X u = ψu for u, it follows that γ λs λ 1 + X s λ 1 X s [ 1 + X s + 1 t + λλ 1 1 + X s λ X s [ 1 + X s + ψt ] + λ 1 + X s λ 1[ 1 + X s 1 t + ψt ] Cψ λ + λλ + 1 φλ X s. 1 t ψt ] 4 Let ξr := if{t, X t > R}. Choosig λ = 1, it follows from 4 that E [ 1 + X t ξr 1 ] Hece, 1 + x 1 + C E [ 1 + X s ξr ] 43 1 ds. E [ 1 + X t ξr ] 1 1 + x 1 C e. 44 This implies Hece, R P X t > R P ξr 1 1 t 1 45 1 + R 1 1 + x 1 C e. log P 1 t 1 X t > R =. 46 Assume ow that 4 holds for some m. We will prove that it is also true for m + 1. For R 1 >, set ξ 1 := if{t, X t 1 R 1 } ad X 1t := X t ξ 1. Defie ξ R := if{t, X 1t R}. The, P As before, by Itô s formula, +1 X t > R 47 P ξ 1 m + 1 + P ξ R m + 1 48 = P X t > R 1 + P ξr m + 1. 49 M λ t ξ := φ λ X1t 1 γλsds 1 + x λ, t, 5 is a martigale with iitial value zero, where γ λs λ 1 + X s λ 1 X s [ 1 + X s + R 1 ] + 4λλ 11 + X s λ X s [1 + X s + R 1 ] + λ 1 + X s λ 1[ 1 + X s + R 1 ] 51 C R1 λ + λλ + 1 φλ X s. for s 1 ξ 1.

1 SALAH-ELDIN A. MOHAMMED AND TUSHENG ZHANG Usig 51 ad the proof of 46, we get log P ξr m + 1 log1 + R + log1 + x + C R1 m + 1. 5 Thus it follows from 49 that log P X t R t m+1 log P X t R 1 t m log1 + R + log1 + x + C R1 m + 1. 53 Hece, R log P X t R t m+1 log P X t R 1. t m Usig the iductio hypothesis ad lettig R 1 we obtai 4 for m + 1. This completes the proof of the propositio. For R >, defie m R := { bt, x, y, σt, x, y ; t [, m], x R, y R} ad 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. Put b R := b R 1, b R,..., b R d ad σ R := σ R i,j 1 i,j d. The b R t, x, y = bt, x, y, σ R t, x, y = σt, x, y, for t [, m], x R, y R. Furthermore, b R ad σ R satisfy the Lipschitz coditio A.1 with the same Lipschitz costat. Let XR be the solutio to the sdde X Rt = X R + + 1 X Rt = ψt, t [ 1, ]. b R s, X Rs, X Rs 1ds σ R s, X Rs, X Rs 1dW s, t >, 54 Propositio 3.6. Fix m 1. The R log P X t X Rt > δ =. 55 Proof. Agai we will use iductio. We omit the proof for the case m = 1 sice 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

LARGE DEVIATIONS FOR STOCHASTIC SYSTEMS WITH MEMORY 11 YR t := X t XR t. For R 1 >, defie ξr 1 := if{t ; X t R 1 }. For ay R R 1 we have Y Rt ξ R 1 = ξ R 1 [ br s, X s, X s 1 b R s, X Rs, X Rs 1 ] ds + 1 ξ R 1 [ σr s, X s, X s 1 σ R s, X R s, X Rs 1 ] dw s, t. 56 For ρ >, let φ λ y := ρ + y λ ad τr,ρ := if{t ; X t 1 XR t 1 ρ}. Set Y R,ρ t := YR t ξ R 1 τr,ρ ad ξ R,ρ := if{t ; Y R,ρ t δ}. The P YRt > δ +1 P ξ R 1 m + 1 + P τr,ρ m + 1 + P ξr,ρ m + 1 P X t > R 1 +1 + P X t XRt > ρ + P ξr,ρ m + 1. 57 By the iductio hypothesis, By Itô s formula, log P X t XRt > ρ R =. 58 ξ φ λ YR,ρt R τ 1 R,ρ γλsds ρ λ = M R,ρ t 59 is a martigale with iitial value zero, where, as i the proof of Lemma 3.3, for s t ξ R 1 τ R,ρ, γ λs Cλ + λλ + 1φ λ Y Rs. 6 As before, this implies that Hece, it follows from 57, 58 ad 6 that ρ log P ξr,ρ m + 1 log ρ + δ + C. 61 R log P log P { ρ log ρ + δ +1 +1 } + C. Y Rt > δ X t > R 1 6 By Propositio 3.5, lettig first ρ ad the, R 1, we obtai 55 for m+1. The proof of Propositio 3.6 is complete.

1 SALAH-ELDIN A. MOHAMMED AND TUSHENG ZHANG For g with eg <, let F R g be the solutio to the dde F R gt = F R g + + F R gt = ψt, 1 t. b R s, FR gs, F R gs 1 ds σ R s, FR gs, F R gs 1 ġsds 63 Defie for each f C ψ [ 1, m] R d. If { } 1 I R f := if eg; F Rg = f F gt R, the F g = F R g. Therefore, 64 If = I R f, for all f with ft R. 65 Lemma 3.7. I is a good rate fuctio o C ψ [ 1, m], R d ; that is, for ay α, the level set {f; If α} is compact. Proof. As i Lemma 3.4, we ca show that R eg α F R g F g =. I particular, this implies that F is cotiuous o each level set {g; eg α}. Sice e is a good rate fuctio, this is sufficiet to coclude that I is also a good rate fuctio. Proof of Theorem 3.1 i the ubouded case. For R > ad a closed subset C C ψ [ 1, m], R l, set C R = C {f; f R}. C δ R deotes the δ-eighborhood of C R. Deote by µ R µ C µ C R1 + P X t > R 1 µ R C δ R 1 + P Usig the large deviatio priciple for {µ R, > }, we obtai Sedig R to ifiity gives log µ C X t X Rt > δ + P X t > R 1. if I R f log P f CR δ 1 log µ C log P X t X Rt > δ. if If log P f CR δ 1 Lettig first δ, ad the R 1, we obtai which is the upper boud 9 i Theorem 3.1. log µ C if If f C the law of XR. The we have X t > R 1 X t > R 1. 66 67 68

LARGE DEVIATIONS FOR STOCHASTIC SYSTEMS WITH MEMORY 13 Let G be a ope subset of C ψ [ 1, m] R d. Fix ay φ G ad choose δ > such that Bφ, δ = {f; f φ δ} G. The I R φ log µ R Bφ, δ log µ G log P X t XRt > δ Note that I R φ = Iφ for all R φ. So, lettig R i the above iequality, we get. 69 Sice φ is arbitrary, it follows that Iφ log µ G. 7 if If log µ G f G which is the lower boud 1 i Theorem 3.1. The proof of Theorem 3.1 is ow complete. Remark. The results i this paper ca be easily exteded to the case, where differet delays τ 1, τ are allowed i 6: dx t = b t, X t, X t τ 1 dt + 1 σ t, X t, X t τ dw t, X t = ψt, t [ τ 1 τ, ]. t,, 71 REFERENCES [1] Dembo, A. ad Zeitoui, A, Large Deviatios Techiques ad Applicatios, Spriger-Verlag, Berli Heidelberg, 1998. [] de Hollader, F., Large deviatios, Fields Istitute Moographs, 14. America Mathematical Society, Providece, RI,, pp.143 [3] Deuschel, J.-D., ad Stroock, D. W., Large Deviatios, Academic Press, Bosto, Sa Diego, New York, 1989. [4] Dosker, M., ad Varadha, S. R. S., Asymptotic evaluatio of certai Markov process expectatios for large time, I, Comm. Pure Appl. Math., 8, 1975, 1-47. [5] Dosker, M., ad Varadha, S. R. S., Asymptotic evaluatio of certai Markov process expectatios for large time, II, Comm. Pure Appl. Math., 8, 1975, 79-31. [6] Dosker, M., ad Varadha, S. R. S., Asymptotic evaluatio of certai Markov process expectatios for large time, III, Comm. Pure Appl. Math., 9, 1976, 389-461. [7] Elsaosi, I., ksedal, B., ad Sulem, A., Some solvable stochastic cotrol problems with delay, Stochastics Stochastics Rep. 71, o. 1-, 69-89. [8] Freidli, M. I., ad Wetzell, A. D., Radom Perturbatios of Dyamical Systems, Spriger-Verlag, New York, Berli, Heidelberg, Tokyo, 1984. [9] Itô, K., ad Nisio, M., O statioary solutios of a stochastic differetial equatio, J. Math. Kyoto Uiversity, 4-1 1964, 1 75. [1] Kolmaovskii, V., ad Myshkis, A., Itroductio to the theory ad applicatios of fuctioal-differetial equatios, Mathematics ad its Applicatios, 463, Kluwer Academic Publishers, Dordrecht 1999. [11] Kusher, H. J., O the stability of processes defied by stochastic differetial-differece equatios, J. Differetial Equatios, 4 1968, 44-443. [1] Mao, X., Stochastic differetial equatios ad their applicatios, Horwood Series i Mathematics & Applicatios, Horwood Publishig Limited, Chichester, 1997, pp. 366. [13] Mizel, V. J. ad Trutzer, V, Stochastic hereditary equatios: existece ad asymptotic stability, Joural of Itegral Equatios 7 1984, 1 7. [14] Mohammed, S.-E. A., Stochastic Fuctioal Differetial Equatios, Research Notes i Mathematics, 99, Pitma Advaced Publishig Program, Bosto, Lodo, Melboure 1984.

14 SALAH-ELDIN A. MOHAMMED AND TUSHENG ZHANG [15] Mohammed, S.-E. A., Stochastic differetial systems with memory: Theory, examples ad applicatios, Stochastic Aalysis ad Related Topics VI, The Geilo Workshop, 1996, eds. L. Decreusefod, J. Gjerde, B. ksedal ad A.S. Ustuel, Progress i Probability, vol. 4, Birkhauser 1998. [16] Scheutzow, M., Qualitative behaviour of stochastic delay equatios with a bouded memory, Stochastics 1 1984, o. 1, 41 8. [17] Scheutzow, M., Statioary ad Periodic Stochastic Differetial Systems: A study of qualitative chages with respect to the oise level ad asymptotics, Habilitatiosschrift, Fachbereich Mathematik, Uiversity of Kaiserslauter, Germay, 1988. [18] Stroock, D.W., A Itroductio to the Theory of Large Deviatios, Spriger-Verlag, Berli, 1984. [19] Zhag, T.S., O the small time asymptotics of diffusios o Hilbert spaces, Aals of Probability 8: 537-557. [] Zhag, T.S., A large deviatio priciple of diffusios o cofiguratio spaces, Stochastic Processes ad Applicatios 91 1 39-54. Departmet of Mathematics, Souther Illiois Uiversity, Carbodale, Illiois, U.S.A. Departmet of Mathematics, Uiversity of Machester, Oxford Road, Machester M13 9PL, Eglad, U.K.