bounds, but Mao [{4] only dscussed te mean square (te case of p = ) almost sure exponental stablty. Due to te new tecnques developed n ts paper, te re

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1 Asymptotc Propertes of Neutral Stocastc Derental Delay Equatons Xuerong Mao Department of Statstcs Modellng Scence Unversty of Stratclyde Glasgow G XH, Scotl, U.K. Abstract: Ts paper dscusses asymptotc propertes, especally asymptotc stablty of neutral stocastc derental delay equatons. New tecnques are developed to cope wt te neutral delay case, te results of ts paper are more general tan te autor's earler work wtn te delay equatons. Key Words: Lyapunov functon, supermartngale convergence teorem, It^o's formula, asymptotc stablty.. Introducton Stocastc derental equatons ave been studed for more tan fty years tere are many books n te lterature e.g. Arnold [], Elworty [], Fredman [3], Has'mnsk [6], Mao [,] Moammed [5]. Motvated by te cemcal engneerng systems te teory of aeroelastcty, Kolmanovsk et al. [7, 8] ntroduced a class of neutral stocastc functonal derental equatons. But so far suc equatons ave been dscussed a lttle unlke te determnstc neutral equatons tat ave been well studed (cf. Hale & Lunel [4], Hale & Meyer [5]). In 995 Mao [] ntated te study of exponental stablty of a neutral stocastc functonal derental equaton d[x(t)? G(xt )] = f(x t ; t)dt + g(x t ; t)db(t); (:) wle Mao [3] employed te Razumkn tecnque to nvestgate te exponental stablty of ts equaton. Te results of Mao [, 3] were edted by te autor's recent book Mao [4] were some new results were also gven. In ts paper we consder a neutral stocastc derental delay equaton d[x(t)? G(x(t? ))] = f(x(t); x(t? ); t)dt + g(x(t); x(t? ); t)db(t): (:) Clearly equaton (.) s a specal case of equaton (.) dscussed n Mao [{4], but delay equatons are one of te most mportant classes of functonal derental equatons need specal attenton (cf. Hale & Lunel [4], Kolmanovsk et al. [7, 8]). Te mportant features of ts paper are: It s due to ts specal case tat we are able to develop new tecnques, wc are completely derent from tose used n Mao [{4], to nvestgate te asymptotc propertes of te solutons for te neutral delay equatons. Ts paper consders not only te pt (p > ) moment almost sure exponental stablty but also te oter asymptotc propertes e.g. asymptotc polynomal Partally supported by te Royal Socety te EPSRC/BBSRC.

2 bounds, but Mao [{4] only dscussed te mean square (te case of p = ) almost sure exponental stablty. Due to te new tecnques developed n ts paper, te results obtaned n ts paper are very general useful. Te teory developed ere gves a uned treatment for varous asymptotc estmates e.g. exponental polynomal bounds. It s also wort to pont out tat altoug te equatons dscussed n Mao [], namely equatons of type (.), are more general tan te neutral delay equaton (.), te condtons mposed tere make te teory not applcable to te delay equaton (please see Secton 5 below for te detaled explanaton). On te oter, te results obtaned n Mao [3, 4] can be appled to te delay equaton but are not so ecent as te new results n ts paper (please agan see secton 5 below for detals). In oter words, by developng new tecnques applcable to te delay equaton we mprove our earler results. From tese mportant features we see clearly sgncant contrbutons of ts paper.. Key Lemmas Trougout ts paper, unless oterwse speced, we let (; F; ff t g t ; P ) be a complete probablty space wt a ltraton ff t g t satsfyng te usual condtons (.e. t s rgt contnuous F contans all P -null sets). Let B(t) = (B (t); ; B m (t)) T be an m-dmensonal Brownan moton dened on te probablty space. Let j j denote te Eucldean norm n R n. If A s a vector or matrx, ts transpose s denoted by A T. If A s a matrx, ts trace norm s denoted by jaj = p trace(a T A). Let > C([?; ]; R n ) denote te famly of all contnuous R n -valued functons on [?; ]. Let CF b ([?; ]; R n ) be te famly of all F -measurable bounded C([?; ]; R n )-valued rom varables = f() :? g. Consder an n-dmensonal neutral stocastc derental delay equaton d[x(t)? G(x(t? ))] = f(x(t); x(t? ); t)dt + g(x(t); x(t? ); t)db(t) (:) on t wt ntal data fx() :? g = C b F ([?; ]; R n ). Here f : R n R n R +! R n, g : R n R n R +! R nm G : R n! R n. We sall mpose te followng stng ypoteses: (H) Bot f g satsfy te local Lpsctz condton te lnear growt condton. Tat s, for eac k = ; ;, tere s a c k > suc tat jf(x; y; t)? f(x; y; t)j _ jg(x; y; t)? g(x; y; t)j c k (jx? xj + jy? yj) for all t tose x; y; x; y R n wt jxj _ jyj _ jxj _ jyj k, tere s moreover a c > suc tat for all (x; y; t) R n R n R +. (H) Tere s a constant (; ) suc tat jf(x; y; t)j _ jg(x; y; t)j c( + jxj + jyj) jg(x)? G(y)j jx? yj for x; y R n

3 , moreover, G() =. It s known (cf. Mao []) tat under ypoteses (H) (H), equaton (.) as a unque contnuous soluton on t?, wc s denoted by x(t; ) n ts paper. Moreover, for every p >, E sup?st jx(s; )j p < on t : Let C ; (R n R + ; R + ) denote te famly of all nonnegatve functons V (x; t) on R n R + wc are twce contnuously derentable n x once n t. For eac V C ; (R n R + ; R + ), dene an operator LV from R n R n R + to R by LV (x; y; t) = V t (x? G(y); t) + V x (x? G(y); t)f(x; y; t) + trace g T (x; y; t)v xx (x? G(y); t)g(x; y; t) ; were V t (x; t) (x; t) ; V x (x; t) = V (x; t) V xx (x; j : (x; t) n Let us stress tat LV s dened on R n R n R + wle V on R n R +. We also denote by L (R + ; R + ) te famly of all functons : R +! R + suc tat R (t)dt <. Furtermore, let C(R n ; R + ) C(R n R + ; R + ) denote te famles of all contnuous functons from R n to R + from R n R + to R +, respectvely. To prove te key lemmas of ts paper we wll need te followng semmartngale convergence teorem establsed by Lptser & Sryayev [9] (Teorem 7 on page 39). Lemma. Let A(t) U(t) be two contnuous adapted ncreasng processes on t wt A() = U() = a.s. Let M(t) be a real-valued contnuous local martngale wt M() = a.s. Let be a nonnegatve F -measurable rom varable. Dene X(t) = + A(t)? U(t) + M(t) for t : If X(t) s nonnegatve, ten n o n o n o lm A(t) < lm X(t) < \ lm U(t) < a:s: were B D a.s. means P (B \ D c ) =. In partcular, f lm A(t) < a.s., ten for almost all! lm X(t;!) < lm U(t;!) < ; tat s bot X(t) U(t) converge to nte rom varables. Te followng two lemmas wll play an mportant role n ts paper. 3

4 Lemma. Let (H) (H) old. Assume tat tere are functons V C ; (R n R + ; R + ), L (R + ; R + ) w ; w C(R n R + ; R + ) suc tat LV (x; y; t) (t)? w (x; t) + w (y; t); (x; y; t) R n R n R + (:) w (x; t) w (x; t + ); (x; t) R n R + : (:3) Ten, for every C b F ([?; ]; R n ), we ave (a) lm sup EV (x(t; )? G(x(t? ; )); t) < ; (b) (c) (d) Z E w (x(t; ); t)? w (x(t; ); t + ) dt < ; lm V (x(t; )? G(x(t? ; )); t) + Z t? w (x(t; ); t)? w (x(t; ); t + ) dt < w (x(s; ); s + )ds < a:s:; Proof. Fx ntal data CF b ([?; ]; R n ) arbtrarly wrte x(t; ) = x(t) for smplcty. By It^o's formula, Note tat Hence V (x(t)? G(x(t? )); t) = V (x()? G(x(?)); ) + + a:s: LV (x(s); x(s? ); s)ds V x (x(s)? G(x(s? )); s)g(x(s); x(s? ); s)db(s): t? + w (x(s); s + )ds = V (x(t)? G(x(t? )); t) + = V (()? G((?)); ) +? + Z? w (x(s); s + )ds w (x(s); s + )? w (x(s? ); s) ds: t? Z? w (x(s); s + )ds w ((); + )d + (s)ds (s)? LV (x(s); x(s? ); s)? w (x(s); s + ) + w (x(s? ); s) ds V x (x(s)? G(x(s? )); s)g(x(s); x(s? ); s)db(s) (:4) V (()? G((?)); ) +? + Z? w (x(s); s)? w (x(s); s + ) ds w ((); + )d + (s)ds V x (x(s)? G(x(s? )); s)g(x(s); x(s? ); s)db(s); (:5) 4

5 were we ave used te followng fact from (.) (.3) tat (s)? LV (x(s); x(s? ); s)? w (x(s); s + ) + w (x(s? ); s) w (x(s); s)? w (x(s); s + ) : Snce s bounded all V, G w are contnuous, bot V (()? G((?)); ) w ((); + ) (? ) must be bounded, we denote by C ter bound. It ten follows from (.5) tat EV (x(t)? G(x(t? )); t) C( + ) + Z wc mples concluson (a). It also follows from (.5) tat (s)ds < Z E w (x(s); s)? w (x(s); s + ) ds C( + ) + (s)ds < wc mples concluson (b) by lettng t!. Moreover, applyng Lemma. to (.4) one sees tat lm V (x(t)? G(x(t? )); t) + w (x(s); s + )ds < a:s: t? wc s te requred concluson (c). Fnally, applyng Lemma. to (.5) we obtan concluson (d). Te proof s complete. Lemma.3 Let (H) (H) old. Assume tat tere are functons V C ; (R n R + ; R + ), L (R + ; R + ) w C(R n R + ; R + ) suc tat LV (x; y; t) (t)? w(x? G(y); t); (x; y; t) R n R n R + : (:6) Ten, for every C b F ([?; ]; R n ), we ave (a) lm sup EV (x(t; )? G(x(t? ; )); t) < ; (b) (c) (d) Z Ew(x(t; )? G(x(t? ; )); t)dt < ; lm V (x(t; )? G(x(t? ; )); t) < a:s:; Z w(x(t; )? G(x(t? ; )); t)dt < a:s: Ts teorem can be proved n te same way as Lemma. so te detals are left to te reader. 3. Asymptotc Sample Propertes Te results obtaned n te prevous secton can be appled to sow te asymptotc sample propertes, especally to establs very useful sucent crtera for te almost 5

6 sure asymptotc stablty of te neutral stocastc derental delay equaton (.). Let us prepare anoter lemma. Lemma 3. Let (H) old. Let : R +! (; ) z : [?; )! R n be contnuous functons. Assume tat Ten Proof. tat := lm sup (t) (t? ) < (3:) := lm sup (t)jz(t)? G(z(t? ))j < : (3:) lm sup (t)jz(t)j? : (3:3) For any " (;? ), by assumptons (3.) (3.), tere s a T suc (t) (t? ) + " (t)jz(t)? G(z(t? ))j + " wenever t T. From ts (H) we can ten derve tat (t)jz(t)j (t)jz(t)? G(z(t? ))j + (t)jg(z(t? ))j for all t T. Hence, for any T > T, (t)jz(t)j + " + ( + ") Ts mples sup T t T sup T tt (t)jz(t)j + " + ( + ")(t? )jz(t? )j (3:4) + " + ( + ") sup T t T sup T?tT + ( + ") sup T tt (t)jz(t)j :? ( + ") + " + ( + ") (t? )jz(t? )j (t)jz(t)j sup T?tT (t)jz(t)j : Lettng T! we see tat sup T t< (t)jz(t)j < ence lm sup (t)jz(t)j < : We can terefore derve from (3.4) tat lm sup (t)jz(t)j + " + ( + ") lm sup (t? )jz(t? )j = + " + ( + ") lm sup (t)jz(t)j : 6

7 Ts yelds lm sup (t)jz(t)j + "? ( + ") : Lettng "! we obtan te requred asserton (3.3). Te proof s complete. We wll need a few more notatons. Let K denote te class of contnuous (strctly) ncreasng functons from R + to R + wt () =. Let K denote te class of functons n K wt (r)! as r!. Functons n K K are called class K K functons, respectvely. If K, ts nverse functon s denoted by?, t clearly belongs to K. Moreover, let H denote te class of contnuous functons from R + to R + suc tat (r)! as r!. Teorem 3. Under te assumptons of eter Lemma. or Lemma.3, for every gven ntal data C b F ([?; ]; R n ), te soluton of equaton (.) as te followng propertes: () If tere are two functons H K suc tat (t)(jxj) V (x; t) for all (x; t) R n R + ; (3:5) ten lm jx(t; )j = a:s: (3:6) () If tere are two constants p > R suc tat ( + t) jxj p V (x; t) for all (x; t) R n R + ; (3:7) ten lm sup log jx(t; )j log t? p a:s: (3:8) () If tere are constants p >? < < p log( ) suc tat e t jxj p V (x; t) for all (x; t) R n R + ; (3:9) ten lm sup t log jx(t; )j? p a:s: (3:) Proof. Agan wrte x(t; ) = x(t) for smplcty. () By asserton (c) of Lemma. or.3 condton (3.5), one sees tat Snce K, we must ave lm (jx(t)? G(x(t? ))j) = a:s: lm jx(t)? G(x(t? ))j = a:s: Applyng Lemma 3. wt (so = ) = we obtan te requred asserton (3.6). 7

8 () By asserton (c) of Lemma. or.3 condton (3.7), one sees tat := lm sup ( + t) =p jx(t)? G(x(t? ))j < a:s: To apply Lemma 3., let = (t) = ( + t) =p, ten So Lemma 3. mples = lm sup ( + t) =p ( + t? ) =p = < : lm sup ( + t) =p jx(t)j te requred asserton (3.8) follows mmedately.? a:s: () By asserton (c) of Lemma. or.3 condton (3.9), one sees tat := lm sup e t=p jx(t)? G(x(t? ))j < a:s: To apply Lemma 3., let = (t) = e t=p, ten = lm sup snce < p log( ). Tus, Lemma 3. mples lm sup e t=p jx(t)j e t=p e (t?)=p = e=p <? e =p a:s: te requred asserton (3.) follows mmedately. Te proof s terefore complete. Teorem 3.3 Let all te assumptons of Lemma.3 old. Assume furtermore tat tere are functons K, K 3 K suc tat (jxj) V (x; t) (jxj) 3 (jxj) w(x; t) (3:) for (x; t) R n R +. Ten, for any gven ntal data C b F ([?; ]; R n ), te soluton of equaton (.) wll tend to zero almost surely, tat s lm jx(t; )j = a:s: (3:) Proof. Agan wrte x(t; ) = x(t). Note from (3.) tat? (V (x; t)) jxj ence 3[? (V (x; t))] 3(jxj) w(x; t): (3:3) 8

9 It ten follows from concluson (d) of Lemma.3 tat Z 3 [? (V (x(t)? G(x(t? )); t))]dt < a:s: (3:4) On te oter, concluson (c) of Lemma.3 sows tat Snce 3 [ ()] s contnuous, we ave lm V (x(t)? G(x(t? )); t) < a:s: lm 3[? (V (x(t)? G(x(t? )); t))] = 3? lm V (x(t)? G(x(t? )); t) a:s: Ts, togeter wt (3.4), mples 3? lm V (x(t)? G(x(t? )); t) = a:s: Notng tat 3 [ ()] K, one sees tat Ts, togeter wt (3.), yelds Snce K, we must ave lm V (x(t)? G(x(t? )); t) = a:s: lm (jx(t)? G(x(t? ))j) = a:s: lm jx(t)? G(x(t? ))j = a:s: Fnally, applyng Lemma 3. wt (so = ) = we obtan te requred asserton (3.). Te proof s complete. 4. Asymptotc Moment Propertes In ts secton we sall apply te results obtaned n Secton to sow te asymptotc moment propertes, especally asymptotc moment stablty of te solutons of equaton (.). We need two more lemmas. Lemma 4. If p ; > x; y R n, ten jx + yj p ( + ) p? jxj p +?(p?) jyj p : (4:) Proof. If p =, (4.) olds clearly. If p >, usng Holder's nequalty we derve tat jx + yj p = x + (p?)=p y p ( + ) p? jxj p + (p?)=p?(p?) jyj p 9

10 as requred. Lemma 4. Let (H) old p. Let : R +! (; ) be a contnuous functon z(t), t? be a contnuous R n -valued stocastc process suc tat Ejz(t)j p < for all t?. Assume tat := lm sup (t) (t? ) < p (4:) Ten := lm sup (t)ejz(t)? G(z(t? ))j p < : (4:) lm sup (t)ejz(t)j p? =p p : (4:3) Proof. By condton (4.), we can coose a > for Furtermore, let " > be arbtrarly small for p ( +? ) p? < : (4:4) ( + ") p ( +? ) p? < : (4:5) By (4.) (4.) tere s a constant T = T (") > suc tat (t) (t? ) + " (t)ejz(t)? G(z(t? ))j p + " (4:6) wenever t T. Applyng Lemma 4. we can ten sow tat (t)ejz(t)j p ( + ")( + ) p? + ( + ") p ( +? ) p? (t? )Ejx(t? )j p (4:7) for all t T. From ere we can sow n te same way as n te proof of Lemma 3. tat lm sup (t)ejz(t)j p < : We can terefore take te upper-lmts as t! on bot sdes of (4.7) to obtan Ts, togeter wt (4.5), yelds lm sup (t)ejz(t)j p ( + ")( + ) p? +( + ") p ( +? ) p? lm sup lm sup (t)ejz(t)j p (t)ejz(t)j p : ( + ")( + ) p?? ( + ") p ( +? ) p? :

11 Lettng "! we ave sown tat lm sup (t)ejz(t)j p ( + ) p?? p ( +? ) p? (4:8) olds for any > satsfyng (4.4). If =, we can let! n (4.8) to obtan lm sup (t)ejz(t)j p ; tat s, (4.3) olds n ts case. On te oter, f >, coosng = =p? =p wc satses (4.4), we can easly obtan te desred concluson (4.3) from (4.8). Te proof s complete. Teorem 4.3 Let p. Under te assumptons of eter Lemma. or Lemma.3, for every gven ntal data C b F ([?; ]; R n ), te soluton of equaton (.) as te followng propertes: () If tere s a functon H a convex functon K suc tat (t)(jxj p ) V (x; t) for all (x; t) R n R + ; ten lm Ejx(t; )jp = : () If (3.7) olds wt R, ten lm sup log(ejx(t; )j p ) log t?: () If (3.9) olds wt? < < p log( ), ten lm sup t log(ejx(t; )jp )?: Ts teorem can be proved n te same way as Teorem 3. usng Lemma 4. nstead of Lemma 3.. Te detals are left to te reader. 5. Furter Results on Exponental Stablty Te purpose of ts secton s to compare our new results wt tose of te autor's earler works Mao [{4]. We sall sow tat te results obtaned ere are better tan te autors' earler works even n te respect of exponental stablty wtout mentonng te oter new asymptotc estmates.

12 As ponted out n Secton, te equatons dscussed n Mao [{4] are of form (.). If we dene G(') = G('(?)); f('; t) = f('(); '(?); t); g('; t) = g('(); '(?); t) (5:) for ('; t) C([?; ]; R n ) R +, ten equaton (.) can be wrtten as equaton (.). Tat s, equaton (.) s more general tan equaton (.). However, let us rst explan wy te teory establsed by Mao [] for equaton (.) cannot be appled to equaton (.). To be more precse, recall tat G s a functonal from C([?; ]; R n ) to R n. A key condton mposed n Mao [] s tat Z jg(')j (s)j'(s)j ds; ' C([?; ]; R n ) (5:)? for some number (; ) some Borel measurable bounded nonnegatve functon () dened on [?; ] wt property tat Z? (s)ds = : Clearly f G s dened by (5.), ten condton (5.) cannot be satsed. Hence results obtaned by Mao [] cannot be appled to te neutral delay equaton (.). It was n ts sprt Mao [3, 4] developed te Razumkn tecnque to deal wt neutral stocastc functonal derental equatons, te teory establsed n Mao [3, 4] can be appled to equaton (.). In fact, Mao [3, 4] obtaned te followng results. Corollary 5. (Corollary 6. of Mao [3]) Let (H) (H) old. Assume tat tere are two postve constants suc tat (x? G(y)) T f(x; y; t) + jg(x; y; t)j? jxj + jyj (5:3) for all x; y R n t. If < < > (? ) ; (5:4) ten te trval soluton of equaton (.) s exponentally stable n mean square. Corollary 5. (Corollary of Mao [4]) Under te same condtons of Corollary 5., te trval soluton of equaton (.) s also almost surely exponentally stable. To sow te generalty usefulness of our results obtaned n te prevous sectons, we furter establs some sucent crtera for te exponental stablty n order to compare wt te above exstng results. Teorem 5.3 Let (H) (H) old. Let V C ; (R n R + ; R + ) p; ; ; 3, c ; c ; c 3 be all postve constants. Assume tat p, >, c jxj p V (x; t) c jxj p (5:5)

13 L V (x; y; t)? jxj p + jyj p + c 3 e? 3t (5:6) for all x; y R n t. equaton (.) as te propertes tat Ten for every C b F ([?; ]; R n ), te soluton of lm t log jx(t; )j? p a:s: (5:7) were > s dened by lm t log(ejx(t; )jp )?; (5:8) := max n" (; 3 ^ (p=) log(=) ] : e "? + "c + e "=p o p : (5:9) Proof. Let " (; ) be arbtrary. Dene Clearly V C ; (R n R + ; R + ). Compute V (x; t) = e"t c V (x; t) for (x; t) R n R + : LV (x; y; t) = e"t c " V (x? G(y); t) + L V (x; y; t) e"t c "c jx? G(y)j p? jxj p + jyj p + c 3 e? 3t : (5:) By Lemma 4. ypotess (H) we ave, for any >, jx? G(y)j p ( + ) p? jxj p + p jyjp : p? Coosng = e "=p, wc mnmzes ( + ) p? [ + p e " = p? ] on >, we obtan jx? G(y)j p? + e "=p p? jxj p + e?"(p?)=p jyj p : Substtutng ts nto (5.) yelds LV (x; y; t) c 3 c e?(?")t? e"t c? "c? + e "=p p? jxj p + e"t c + "c? + e "=p p? e?"(p?)=p jyj p : (5:) To apply Lemma., we dene, for (x; t) R n R +, w (x; t) = e"t c? "c? + e "=p p? jxj p 3

14 wle w (x; t) = e"t c + "c? + e "=p p? e?"(p?)=p jxj p : Notng from (5.9) " (; ) tat we observe? "c? + e "=p p? " " + "c? + e "=p p? e?"(p?)=p ; w (x; t) w (x; t + ): Hence te assumptons of Lemma. are all satsed. Moreover, we clearly ave e "t jxj p V (x; t) for all (x; t) R n R + : Applyng Teorems we ten get lm t log jx(t; )j? " p a:s: lm t log(ejx(t; )jp )?": Fnally te requred assertons (5.7) (5.8) follows by lettng "!. Te proof s complete. Corollary 5.4 Let (H), (H) (5.3) old. If >, ten te trval soluton of equaton (.) s exponentally stable n mean square s also almost surely exponental stable. Ts corollary follows from Teorem 5.3 drectly by coosng V (x; t) = jxj, p = c 3 =. Clearly, Corollary 5.4 s an mprovement of Corollares snce > s muc weaker tan condton (5.4). In oter words, te above result as already mproved te autor's earler work even n te case of p =. Teorem 5.5 Let all te assumptons of Teorem 5.3 old except tat p (; ) nstead of p. Ten for every C b F ([?; ]; R n ), te soluton of equaton (.) as te property tat were > s dened by lm t log jx(t; )j? p a:s: (5:) := max " (; 3 ^ (p=) log(=) ] : e " + p "c? + p e " : (5:3) Proof. We use te same notatons as n te proof of Teorem 5.. Note tat for p (; ), jx? G(y)j p (jxj + jg(y)j) p [(jxj _ jg(y)j)] p p (jxj p + p jyj p ): 4

15 Substtutng ts nto (5.6) yelds LV (x; y; t) c 3 c e?(?")t? e"t c (? p "c )jxj p + e"t c ( + p p "c )jyj p : (5:4) Te remer of te proof s te same as before te proof s terefore complete. 6. Examples In ts secton we sall dscuss some examples to llustrate our teory. Example 6. We rst sow tat under ypoteses (H) (H), for any ntal data C b F ([?; ]; R n ), te soluton of equaton (.) as te propertes tat To sow tese, we dene c( + + c) lm log jx(t; )j a:s: (6:) t (? ) lm t log(ejx(t; )j ) V (x; t) = e?t jxj for (x; t) R n R + ; were > s a constant to be determned. Ten 4c( + + c) (? ) : (6:) LV (x; y; t) = e?t?jx? G(y)j + (x? G(y)) T f(x; y; t) + jg(x; y; t)j : (6:3) Compute jx? G(y)j jxj? jxjjyj + jyj (? )jxj? (? )jyj : (6:4): Moreover, x " > arbtrarly compute (x? G(y)) T f(x; y; t) c(jxj + jyj)( + jxj + jyj) c jxj + jyj + jxj + ( + )jxjjyj + jyj c "? + "jxj + "? + "jyj + jxj + ( + )(jxj + jyj ) + jyj = 4c"? + c(3 + + ")jxj + c( ")jyj (6:5) jg(x; y; t)j c + p " p "jxj + p" p "jyj c ( + "? )( + "jxj + "jyj ) = c ( + "? ) + c ( + ")(jxj + jyj ): (6:6) 5

16 Substtutng (6.4){(6.6) nto (6.3) gves LV (x; y; t) = e?t 4c"? + c ( + "? )? (? )? c(3 + + ")? c ( + ") jxj + (? ) + c( ") + c ( + ") jyj : (6:7) Coosng we ave = (? ) 4c( + + ") + c (4 + ") ; (6:8) (? )? c(3 + + ")? c ( + ") = (? ) + c( ") + c ( + ") > : We can ten apply Teorems to obtan tat lm t log jx(t; )j a:s: lm t log(ejx(t; )j ) : Terefore te desred assertons (6.) (6.) follow by lettng "!. Example 6. Let (H) (H) old. Assume tat tere are sx constants { 6 suc tat for x; y R n t. If x T f(x; ; t)? jxj ;?G(y) T f(; y; t) 3 jyj ; jf(x; y; t)? f(x; ; t)j jyj; jf(x; y; t)? f(; y; t)j 4 jxj; jg(x; y; t)j 3 jxj + 4 jyj?? 4? 5 > ; (6:9) ten for every CF b ([?; ]; R n ), te soluton of equaton (.) as te propertes tat lm t log jx(t; )j? a:s: (6:) were > s dened by n = max " (; (=) log(=) ] : lm t log(ejx(t; )j )?; (6:) ( )e " + "? + e "=?? 4? 5 o: (6:) 6

17 Usng te condtons, we compute (x? G(y)) T f(x; y; t) + jg(x; y; t)j x T f(x; ; t) + jxjjf(x; y; t)? f(x; ; t)j? G(y) T f(; y; t) + jg(y)jjf(x; y; t)? f(; y; t)j + jg(x; y; t)j? jxj + jxjjyj + 3 jyj + 4 jxjjyj + 5 jxj + 6 jyj?(?? 4? 5 )jxj + ( )jyj : So te conclusons follow from Corollary 5.4. Example 6.3 Consder a one-dmensonal neutral stocastc derental delay equaton d[x(t)? G(x(t? ))] =? + t x(t)dt + db(t) (6:3) + t on t wt ntal data C b F ([?; ]; R), were B(t) s a scalar Brownan moton G : R! R satses ypotess (H). To apply Teorems , let " > be arbtrarly small dene Compute, for (x; y; t) R R R +, Denng we ave V (x; t) = ( + t)?" x for (x; t) R R + : LV (x; y; t) =? " ( + t) " (x? G(y))? = = Clearly we also ave ( + t) " (x? G(y))x + ( + t) "?( + ")x + "xg(y) + (? ")G (y) ( + t) +" + ( + t) "?( + ")x + "(x + y ) + (? ") y + ( + t) " (?x + y ) + w (x; t) = ( + t) +" : x ( + t) " w (x; t) = x ( + t) " LV (x; y; t)?w (x; t) + w (y; t) + w (x; t) w (x; t + ): ( + t) +" : ( + t) +" ( + t) +" Applyng Teorems we see tat te soluton of equaton (6.3) as te propertes tat log jx(t; )j lm sup?? " a:s: log t lm sup log(ejx(t; )j ) log t 7?(? ")

18 Lettng "! we nally obtan tat lm sup log jx(t; )j log t? a:s: lm sup log(ejx(t; )j ) log t?: Acknowledgements Te autor wses to tank te referee for s/er very elpful comments suggestons. He would also lke to tank te Royal Socety te EPSRC/BBSRC for ter nancal supports. REFERENCES [] Arnold, L., Stocastc Derental Equatons: Teory Applcatons, Jon Wley & Sons, 97. [] Elworty, K.D., Stocastc Derental Equatons on Manfolds, Cambrdge Unversty Press, 98. [3] Fredman, A., Stocastc Derental Equatons Ter Applcatons, Vol., Academc Press, 976. [4] Hale, J.K. Lunel, S.M.V., Introducton to Functonal Derental Equatons, Sprnger, 993. [5] Hale, J.K. Meyer, K.R., A class of functonal equatons of neutral type, Mem. Amer. Mat. Soc. 76 (967), {65. [6] Has'mnsk, R.Z., Stocastc Stablty of Derental Equatons, Sjto Noordo, 98. [7] Kolmanovsk, V.B. Mysks, A., Appled Teory of Functonal Derental Equatons, Kluwer Academc Publsers, 99. [8] Kolmanovsk, V.B. Nosov, V.R., Stablty of Functonal Derental Equatons, Academc Press, 986. [9] Lptser, R. S. Sryayev, A. N., Teory of Martngales, Kluwer Academc Publsers, 989 (translaton of te Russan edton, Nauka, Moscow, 986). [] Mao, X., Stablty of Stocastc Derental Equatons wt Respect to Semmartngales, Longman Scentc Tecncal, 99 [] Mao, X., Exponental Stablty of Stocastc Derental Equatons, Marcel Dekker, 994. [] Mao, X., Exponental stablty n mean square of neutral stocastc derental functonal equatons, Systems Control Letters 6 (995), 45{5. [3] Mao, X., Razumkn type teorems on exponental stablty of neutral stocastc functonal derental equatons, SIAM J. Mat. Anal. 8() (997), 389{4. [4] Mao, X., Stocastc Derental Equatons Applcatons, Horwood, 997. [5] Moammed, S.-E.A., Stocastc Functonal Derental Equatons, Longman Scentc Tecncal,