Title: On mean-square-stability of two-step Maruyama methods for stochastic delay differential equations

Transcription

1 Elsevier Ediorial Sysem(m) for Journal of Compuaional and Applied Mahemaics Manuscrip Draf Manuscrip Number: Tile: On mean-square-sabiliy of wo-sep Maruyama mehods for sochasic delay differenial equaions Aricle Type: Research Paper Secion/Caegory: 65Cxx Keywords: sochasic modeling; uncondiional sabiliy; wo-sep Maruyama mehods Corresponding Auhor: Dr. Wanrong Cao, Dr. Corresponding Auhor's Insiuion: Firs Auhor: Wanrong Cao, Dr. Order of Auhors: Wanrong Cao, Dr.; Zhongqiang Zhang

2 Manuscrip Click here o view linked References On mean-square-sabiliy of wo-sep Maruyama mehods for sochasic delay differenial equaions Wanrong Cao a,b, and Zhongqiang Zhang b, a. Deparmen of Mahemaics, Souheas Universiy, Nanjing 296, P.R.China. b. Division of Applied Mahemaics, Brown Universiy, Providence, RI, US 292. Absrac In his paper we sudy he mean-square-sabiliy of wo-sep Maruyama mehods for sochasic differenial equaions wih ime delay. We propose a family of uncondiionally mean-square sable schemes and prove sabiliy for a model equaion. Numerical resuls for linear and nonlinear equaions show ha his family of wosep Maruyama mehods exhibi beer sabiliy han previous wo-sep Maruyama mehods. Key words: sochasic modeling, uncondiionally sable, wo-sep Maruyama mehods Inroducion Sochasic delay differenial equaions (SDDEs) have been increasingly used o model he effecs of noise and ime delay on various ypes of complex sysems, such as delayed visual feedback sysems [3], conrol problems [, 2], he dynamics of noisy bi-sable sysems wih delay [22], ec. SDDEs are also used in modeling deseases, for example, epidemic diseases[2], neurological diseases [4], ec, and also in finance SDDEs appear in models of sock markes [2]. Corresponding auhor. address: wrcao@seu.edu.cn. This auhor s work was suppored by he NSF of China (No.936). 2 address: Zhongqiang zhang@brown.edu. This auhor s work was suppored by AIRFORCE MURI. Preprin submied o Elsevier 5 July 2

3 Some numerical mehods and heir convergence and sabiliy properies have been esablished [, 4,, 7, 23] recenly, bu mos of hem are on one-sep mehods. Insead of one-sep mehods we here focus on sochasic muli-sep mehods for SDDEs, which have been widely sudied for solving sochasic ordinary differenial equaions (SODEs), i.e. wih no ime delay. To exend he muli-sep mehods for SODEs o hose for SDDEs is a nonrivial ask and hese exensions have no been invesigaed unil recenly. For a review of muli-sep mehods for SODEs, we refer o [3, 9]. Some more recen sudies are as follows. In [7], cerain sochasic linear muli-sep mehods are consruced; and mean-square convergence raes are obained; and consisency condiions in he mean-square-sense are given for wo-sep Maruyama mehods. Ewald and Temam [9] sudied he convergence of a sochasic Adams-Bashforh scheme wih applicaion o geophysical applicaions. Adams-ype mehods for SODEs are also analyzed in [5], where firsorder srong convergence condiions are given. For some special SODEs wih addiive noise, high order muli-sep mehods have been discussed in [8]. In his paper, we follow [6] and sudy wo-sep Maruyama schemes for he scalar equaion dx() = f (,X(),X( τ) ) d+g (,X(),X( τ) ) dw(), J, X() = ξ(), [ τ,], (.) where τ is a posiive fixed delay, J = [,T], W() is a one-dimensional sandard Wiener process and he funcions f : J R R R, g : J R R R.Wenoeha[6]isperhapsheonlyworkonmuli-sepmehodsforSDDEs, wherein muli-sep mehods are proposed for m-dimensional sysems of Iô SDDEs wih d driving Wiener processes and muli-delay, and heir properies are sudied concerning consisency, numerical sabiliy and convergence. Insead of working wih he general SDDE (.), we sudy he lineared es model (.2), which can shed some lighs on he general SDDE (.), dx() = [ax()+bx( τ)]d+[cx()+dx( τ)]dw(),, (.2) X() = ξ(), [ τ,], where a,b,c,d R, τ is a posiive fixed delay, W() is a one-dimensional sandard Wiener process and ξ() is a C([ τ, ]; R)-valued iniial segmen. In his work, we aim o drive mean-square-sable wo-sep Maruyama mehods for he SDDE (.2). 2

4 The paper is organized in he following way. In Secion 2 we provide some necessary noaions and preliminaries on SDDEs, including some properies of analyical soluions o Eq.(.2). Also, in his secion he wo-sep Maruyama mehods and heir convergence properies are inroduced. In Secion 3 we derive a series of uncondiionally mean-square-sable wo-sep Maruyama mehods under cerain condiions. Secion 4 illusraes he mean-square-sabiliy of hese wo-sep Maruyama mehods wih numerical examples for he es model (.2) and a nonlinear equaion. 2 Noaions and preliminaries Le (Ω,F,P) be a probabiliy space wih a filraion (F ), which saisfies he usual condiions (increasing and righ-coninuous; each {F }, conains all P-null ses in F). Le W(), in Eq.(.2) be F -adaped and independen of F. Assume ξ(), [ τ,] o be F -measurable and righ coninuous, and E ξ 2 <. Here ξ is defined by ξ = sup τ ξ() and is he Euclidean norm in R. Throughou he paper, Eqs.(.) and (.2) are inerpreed in he Iô sense. Under hese assumpions, Eq.(.2) has a unique srong soluion X() : [ τ,+ ) R, which saisfies Eq.(.2) and X() is a measurable, sampleconinuous and F -adaped process; see [5, 2]. Lemma If a < b ( c + d ) 2, (2.) hen he soluion of Eq.(.2) is mean-square-sable, ha is lim E X() 2 =. (2.2) By Corollary 3.2 in [6], he proof of his lemma is no difficul. Applying he wo-sep Maruyama mehods o Eq.(.) leads o he following j= α j X i j = h + j= j= β j f( i j,x i j,x i m j ) γ j g( i j,x i j,x i m j ) W i j, i = 2,3,,N, (2.3) where α j,β j,γ j,(j {,,}) are parameers; h > is he sepsize in ime which saisfies τ = mh for a posiive ineger m, and n = nh, N = T/h. The incremens W i := W( i+ ) W( i ), are independen N(,h)-disribued 3

5 Gaussian random variables. Suppose ha X i is F i -measurable a he meshpoin i. Then X i is an approximaion o X( i ), where for i, X i are given by he iniial funcion. Definiion 2 The funcion u : R + R R R is said o be uniform Lipschiz coninuous if here exiss a posiive consan L u, such ha he funcion u saisfies u(,x,x 2 ) u(,y,y 2 ) L u ( x y + x 2 y 2 ) (2.4) for every x, x 2, y, y 2 R and, When here exiss a posiive consan K, such ha u(,x,y) K(+x 2 +y 2 ) 2 (2.5) for x, y R and, we say ha u saisfies a linear growh condiion. The characerisic polynomial of (2.3) is given by ρ(λ) = α λ 2 +α λ+α. Definiion 3 The mehod (2.3) is said o fulfill Dahlquis s roo condiion if i) he roos of ρ(λ) lie on or wihin he uni circle; ii) he roos on he uni circle are simple. Here are he consisency and convergence properies of numerical mehods (2.3). Lemma 4 ([6]) Assume ha he coefficiens f and g of he SDDE (.) are Lipschiz coninuous in he sense of (2.4)and have firs-order coninuous parial derivaives wih respec o he firs variable and second-order coninuous parial derivaives wih respec o he second and hird variables; hese parial derivaives saisfy he linear growh condiion (2.5); he coefficiens of he sochasic linear wo-sep Maruyama scheme (2.3) saisfy Dahlquis s roo condiion, and he consisency condiions j= α j =, 2α +α = j= β j, α = γ, α +α = γ. (2.6) Then he global error of he scheme (2.3) applied o (.) saisfies max X( i) X i = O(h /2 ). i=2,,n 4

6 3 Mean-square-sabiliy of he wo-sep Maruyama mehods In his secion, we drive he following schemes in he class of wo-sep Maruyama mehods for Eq. (.2) X i+ +α X i +( α )X i = h(2+α )(ax i+ +bx i m+ ) +(cx i +dx i m ) W i +(+α )(cx i +dx i m ) W i, (3.) where a parameer α <. We will also deermine a series of wo-sep Maruyama schemes, which are uncondiionally sable, see Theorem 5. Applying he wo-sep Maruyama mehods (2.3) o Eq.(.2) gives α j X i j = h β j [ax i j +bx i m j ] j= j= + γ j [cx i j +dx i m j ] W i j, i = 2,3, ; (3.2) j= for i, we have X i = ξ( i ). For beer mean square sabiliy, we compue X using implici Milsein mehod, whichismeansquaresableforevery h = τ/m and has convergence rae O(h) in mean-square-sense. We also suppose ha X is F -measurable a he mesh-poin. By choosing he parameers of he wo-sep Maruyama mehod o saisfy he consisency condiion (2.6) and hen we ge α =, α <, β = β =, (3.3) α = α, β = 2+α, γ =, γ = +α. (3.4) Thus, we obain a family of wo-sep Maruyama schemes (3.) from (3.2). Nex we deermine he condiions on he parameers for mean square sabiliy. I can be checked ha he schemes (3.) saisfy Dahlquis s roo condiion and all assumpions in Lemma 4. Thus, we have he following conclusion on mean-square-asympoic sabiliy. Theorem 5 Assume ha he condiion (2.) holds. If he parameers of he wo-sep Maruyama mehod (3.2) saisfy he resriced condiions (3.3)and (3.4), hen he mehod is uncondiionally mean-square-sable. Tha is 5

7 lim E X n 2 =, n for every sepsize h = τ/m. The proof of his heorem needs he following lemma. Lemma 6 Under condiion (2.), he numerical soluions {X i, i 2} produced by he wo-sep Maruyama mehod (3.) saisfy E(X i X j W k ) =, i,j k, E(X i X j W k W k ) =, i,j k, E(X i X j Wk 2 ) = he(x ix j ), i,j k, E(X i X i W i ) < c he(xi )+ d he( X 2 i X i m ), E(X i X i m W i ) < c he( X i X i m )+ d he(xi m 2 ). (3.5a) (3.5b) (3.5c) (3.5d) (3.5e) Proof. Noe ha E( W k ) =, E [ ( W k ) 2] = h and X j and W j are F i - measurable if j i, hence from properies of condiional expecaion we ge E ( ) [ X i X j W k = E Xi X j E ( )] Fk W k =, i,j k, E ( ) [ X i X j W k W k = E Xi X j W k E ( )] W Fk k =, i,j k, E [ [ X i X j Wk] 2 = E Xi X j E ( )] ( ) Wk 2 F k = he Xi X j, i,j k. Now we prove he inequaliy (3.5d). From he scheme (3.) we have and hen X i+ = X i = [ (2+α )bhx i m+ +(+α )X i α X i ( (2+α )ah) +(cx i +dx i m ) W i +(+α )(cx i +dx i m ) W i ]. [ (2+α )bhx i m +(+α )X i 2 α X i ( (2+α )ah) +(cx i +dx i m ) W i +(+α )(cx i 2 +dx i m 2 ) W i 2 ]. Due o he condiion (2.) and α <, we ge (2+α )ah >. Using (3.5a)-(3.5c), i holds ha 6

8 E(X i X i W i ) [ = (2+α )bhe(x i m X i W i ) ( (2+α )ah) +(+α )E(X i 2 X i W i ) α E(Xi 2 W i )+ce(xi 2 W2 i ) +de(x i m X i Wi )+(+α 2 )ce(x i 2 W i 2 X i W i ) ] +(+α )de(x i m 2 W i 2 X i W i ) ( ) = che(x 2 ( (2+α )ah) i )+dhe(x i m X i ) < c he(xi 2 )+ d he( X i m X i ). Inequaliy (3.5e) can be proved in he same way. This proves he lemma. Proof of Theorem 5. The explici form of he scheme (3.) is ( ) ) (2+α )ah X i+ = (2+α )bhx i m+ + ((+α )X i α X i +(cx i +dx i m ) W i +(+α )(cx i +dx i m ) W i, We square boh sides of he las difference equaion o obain ( (2+α )ah) 2 X 2 i+ = (2+α ) 2 b 2 h 2 X 2 i m+ + ( (+α )X i α X i ) 2 +(cx i +dx i m ) 2 W 2 i +(+α ) 2 (cx i +dx i m ) 2 W 2 i +2(2+α )bhx i m+ ( (+α )X i α X i ) +2(2+α )bhx i m+ (cx i +dx i m ) W i +2(2+α )bhx i m+ (+α )(cx i +dx i m ) W i ) +2 ((+α )X i α X i (cx i +dx i m ) W i ) +2 ((+α )X i α X i (+α )(cx i +dx i m ) W i +2(+α )(cx i +dx i m )(cx i +dx i m ) W i W i. Then 7

9 ( (2+α )ah) 2 Xi+ 2 =αx 2 i 2 +c 2 Wi 2 Xi 2 +(+α ) 2 Xi 2 +(+α ) 2 c 2 Wi X 2 i 2 +(2+α ) 2 h 2 b 2 Xi m+ 2 +d2 Wi 2 X2 i m +(+α ) 2 d 2 Wi 2 X2 i m 2α (+α )X i X i 2α (+α )c W i X i X i 2α (2+α )bhx i X i m+ 2α (+α )d W i X i X i m +2(+α )(2+α )bhx i X i m+ +2cd Wi 2 X i X i m +2(+α ) 2 cd Wi X 2 i X i m +O i ( W i, W i ), (3.6) where we define O i ( W i, W i ) =2(+α ) 2 (cx i +dx i m )X i W i +2(2+α )bhx i m+ (cx i +dx i m ) W i +2(2+α )bhx i m+ (+α )(cx i +dx i m ) W i ) +2 ((+α )X i α X i (cx i +dx i m ) W i +2(+α )(cx i +dx i m )(cx i +dx i m ) W i W i. By he lineariy of expecaion and (3.5a)-(??) in Lemma 6, we ge E(O i ( W i, W i )) =. Le Y i = E(X 2 i), i =,,2,. Taking he expecaion over boh sides of (3.6), and using he inequaliy 2ab a 2 +b 2 and (3.5c)-(3.5e) in Lemma 6, i follows ha where P Y i+ P Y i +P 2 Y i +P 3 Y i m+ +P 4 Y i m +P 5 Y i m, (3.7) i = 2,3,, P =( (2+α )ah) 2, P =α 2 +c2 h α (+α ) α (2+α ) b h+ cd h, P 2 =(+α ) 2 c 2 h+(+α ) 2 α (+α ) 2α (+α )c 2 h 2α (+α ) cd h+(+α )(2+α ) b h+(+α ) 2 cd h, P 3 =(2+α ) 2 b 2 h 2 α (2+α ) b h+(+α )(2+α ) b h, P 4 =d 2 h+ cd h, P 5 =(+α ) 2 (d 2 + cd )h 2α (+α )d 2 h 2α (+α ) cd h. Le P = P(a,b,c,d,α,h) = ( P +P 2 +P 3 +P 4 +P 5 ) /P. I is obvious ha 8

10 Y i+ P max { Y i,y i,y i m+,y i m,y i m }. i = 2,3,. (3.8) We now claim ha, for any sepsize h = τ/m, Y i+ max{p i+,p i,,p [i m m+2 ]+ }E ξ 2. (3.9) Thus, if P <, hen we can ge lim i Y i =, as E ξ 2 <. In fac, we have Y i+ P max{y i,y i,y i m+,y i m,y i m } P 2 max{y i,y i 2,,Y i 2m 3 } P [i m m+2 ]+ max{y i [ i m ],,Y,E ξ 2 } m+2 P i+ max{e ξ 2, P E ξ 2,, } m+2 ] E ξ 2 P i [i m max{p i+,p i,,p [i m m+2 ]+ }E ξ 2. Here we noice ha in he process of ieraion, Y i m (i m+) will be he firs erm down o Y = E ξ 2 and i calls for is previous [ i m ]+ seps, m+2 where [z] means he larges ineger no more han a real number z. This proves he claim (3.9). I only remains o verify ha P < i.e. P +P 2 +P 3 +P 4 +P 5 < P. Recall ha α < and hus 2 α 2 2(2+α ), hen from (3.7), we have P +P 2 +P 3 +P 4 +P 5 P =+ (2(2+α ) b +(2 α 2)( c + d ) ) 2 h +(2+α ) 2 b 2 h 2 ( (2+α )ah) 2 =(2+α ) 2 (b 2 a 2 )h 2 +2 ((2+α )a+(2+α )b+ ) 2 (2 α2 )( c + d )2 h ) (2+α ) 2 (b 2 a 2 )h 2 +2(2+α ) (a+ b +( c + d ) 2 h. Based on condiion (2.), we obain ha P +P 2 +P 3 +P 4 +P 5 P < holds for each sepsize h = τ/m. This proves he heorem. 9

11 4 Numerical examples In all our numerical examples, E(Xn) 2 = 2 X n (ω i ) 2, 2 are he sampled average over 2 rajecories in Malab. Table liss a number of wo-sep Maruyama schemes wih differen parameers under es here. Noe ha only he schemes in bold (TS3 and TS5) saisfy he required condiions in Theorem 5 and hence are uncondiionally mean-square-sable. However, he oher Maruyama schemes TS, TS2, TS4, TS6 do no saisfy he condiions in Theorem 5 and hus may be only condiionally sable and even no sable as we show laer on. i= Table parameers for differen wo-sep Maruyama schemes scheme α β β wo-sep mehod (TS) -/2 /4 5/4 wo-sep mehod 2 (TS2) -3/2 /2 wo-sep mehod 3 (TS3) -/2 3/2 wo-sep mehod 4 (TS4) -/3 /3 4/3 wo-sep mehod 5 (TS5) -2/3 4/3 wo-sep mehod 6 (TS6) -4/3 2/3 Example. We consider he linear es model dx() = [ax()+bx( τ)]d+[cx()+dx( τ)]dw(),, (4.) X() = +τ, [ τ,] o illusrae he mean-square-sabiliy of he wo-sep Maruyama schemes in Table. We choose he parameers as a = 4, b = 2, c =.5, d =.5 and τ =, which ensures ha he exac soluion of he equaion (4.) is mean-squaresable by Lemma. From Fig., we observe ha TS is no mean-squaresable for boh large sepsize h = /4 and small sepsize h = /64. From Figs. 2 and 3, we see ha TS3 and TS5 are mean-square-sable even for a large sepsize h = /4; Fig. 2 illusraes ha TS4 is condiionally mean-

12 5 (a) (b) (c) 5 5 (d) Fig.. Simulaions wih TS and TS3. (a): TS, h = /4; (b): TS3, h = /4; (c): TS, h = /64; (d): TS3, h = /64. square-sable; TS4 is mean square sable if he sepsize h is small enough (like h = /64 here). In Fig. 3, we fix he sepsize h = /8 and show ha o grea exen he mean square sabiliy of he implici scheme TS3 is beer han he explici wo-sep scheme TS6. If we es TS6 for a raher long ime inerval, hen he numerical soluion will oscillae and will finally diverge. From Figs., 2 and 3, we observe ha he numerical soluion from TS6 blows up earlier han any oher unsable implici mehods shown in Fig and 2. On he oher hand, numerical soluions of boh TS3 and TS5 converse o zero very fas. In Fig. 4 we choose differen parameers for he equaion (4.): a = 3,b =,c =.5, d =.5 and τ = when (4.) is mean square sable. Here we use a very large sepsize h = /2 for TS, TS2 and TS3. The resuls show ha he scheme TS3 mainains he mean-square-sabiliy even wih large sepsize h. Example 2. We es he proposed wo-sep Maruyama mehods for he following nonlinear SDDE (Example 5.2., [8]): dx()= a + X()+ b + X()sin(X( τ))dw(),, (4.2) X()=+τ, [ τ,].

13 8 (a) (b) (c) 5 5 (d) Fig. 2. Simulaions wih TS4 and TS5. (a): TS4, h = /4; (b): TS5, h = /4; (c): TS4, h = /64; (d): TS5, h = /64. 4 (a) (b) Fig. 3. Simulaions wih TS3 and he explici mehod TS6. (a): TS6; (b): TS3. h = /8. 2

14 TS3 TS2 TS Fig. 4. Simulaions wih TS, TS2 and TS3 for fixed sepsize h = /2. The soluion of Eq.(4.2) is mean square sable if 2a b 2 and b [8]. We ake a =, b = and τ =. I is shown from Figs. 5 and 6 ha he schemes TS3 and TS5 mainain heir mean-square-sabiliy for nonlinear SDDE (4.2); TS and TS4 are condiionally mean-square-sable. I is ineresing o menion ha in some range of sepsize h, smaller h leads o a greaer insabiliy, comparing o (a),(c) in Fig. 5 and (a),(c) in Fig Conclusion We have sudied he mean-square-sabiliy of wo-sep Maruyama mehods and proposed a family of uncondiionally mean-square-sable wo-sep Maruyama schemes. Numerical ess ha his family of numerical mehods exhibis of mean-square-sabiliy for boh linear and some paricular nonlinear SDDE. The numerical resuls sugges ha our proposed scheme can be adoped for general nonlinear SDDEs, bu furher numerical sudies are required. 3

15 5 x 5 (a) (b) x (c) (d) (e) (f) Fig. 5. Simulaions wih TS and TS3. (a): TS, h = /2; (b): TS3, h = /2; (c): TS, h = /6; (d): TS3, h = /6; (e): TS, h = /64; (f): TS3, h = /64 5 x 4 (a) (b) x (c) (d) x (e) (f) Fig. 6. Simulaions wih TS4 and TS5. (a): TS4, h = /2; (b): TS5, h = /2; (c): TS4, h = /6; (d): TS5, h = /6; (e): TS4, h = /64; (f): TS5, h = /64. 4

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