Research Article Existence and Uniqueness of Solutions for a Class of Nonlinear Stochastic Differential Equations

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1 Hindawi Publihing Corporaion Abrac and Applied Analyi Volume 03, Aricle ID 56809, 7 page hp://dx.doi.org/0.55/03/56809 Reearch Aricle Exience and Uniquene of Soluion for a Cla of Nonlinear Sochaic Differenial Equaion Iryna Volodymyrivna Komahynka Mahemaic Deparmen, Univeriy of Jordan, Amman 94, Jordan Correpondence hould be addreed o Iryna Volodymyrivna Komahynka; iryna kom@homail.com Received 8 November 0; Acceped 8 January 03 Academic Edior: Paul Eloe Copyrigh 03 Iryna Volodymyrivna Komahynka. hi i an open acce aricle diribued under he Creaive Common Aribuion Licene, which permi unrericed ue, diribuion, and reproducion in any medium, provided he original work i properly cied. By uing ucceive approximaion, we prove exience and uniquene reul for a cla of nonlinear ochaic differenial equaion. Moreover, i i hown ha he oluion of uch equaion i a diffuion proce and i diffuion coefficien are found.. Inroducion Differenial equaion, which are no olved for he derivaive, have found divere applicaion in many field. Example of equaion of hi ype are Lagrange equaion of claical mechanic or Euler equaion. Conideraion of real objec under he influence of random facor lead o nonlinear ochaic differenial equaion, which are no olved for ochaic differenial. Such equaion were inroduced by Kolmanovkii and Noov in [] for conrucion of ochaic analogue of neural funcional differenial equaion. Work [ 5] were devoed o he problem of exience, uniquene, and properie of oluion of neural ochaic differenial (delay) equaion in finie dimenional pace. Exience of oluion for uch equaioninhilberpacewaudiedinpaper[6 9]. In he paper [9]heauhorconideredaochaicequaionwihou delay. In he monograph of Kolmanovkiĭ and Shaĭkhe [0] condiion were obained for opimaliy in conrol problem for hee equaion. In hi paper we will udy he exience and uniquene of oluion for a cla of nonlinear ochaic differenial equaion, which are no olved for he ochaic differenial. Saemen of he Problem. Le u conider a nonlinear ochaic differenial equaion: d (x G(, x)) =a(, x) d + σ (, x) dw (), () which i no olved for he ochaic differenial. Here x R d, 0, G(, x),and a(, x) are ddimenional vecor funcion; σ(, x) i a d mmarix and W() i an m-dimenional Wiener proce wih independen componen. Le x be a random vecor in R d,uchha E x <. Aume ha x doe no depend on W() W( 0 ), F = σ{x, W() W( 0 ), 0 }. Definiion. An F -dimenional proce x() i aid o be he oluion of ()ifx( 0 )=xand x G(, x ()) =x G( 0,x) + a (, x ()) d + σ (, x ()) dw () 0 0 () hold for every 0 wih probabiliy. Aoluionx() i aid o be unique if for any coninuou oluion x(), y() uch ha x( 0 ) = y( 0 ) = x one ha P{up [0,] x() y() > 0} = 0 for all 0. In hi work we ue he mehod of ucceive approximaion o eablih exience and uniquene (pahwie) of he oluion of (). We udy i probabiliy properie. We prove ha x() i diffuion proce and find coefficien of diffuion.. Main Reul Firly we prove he heorem of exience and uniquene of he oluion.

2 Abrac and Applied Analyi heorem. Aume ha G(, x) i a coninuou funcion; a(, x), σ(, x) aremeaurablefuncionforx R d, 0 and aify he following condiion: () here exi a conan C > 0,uchha G(, x) + a(, x) + σ(, x) C( + x ) for x R d, 0 ; () here exi conan L > 0 and L > 0 uch ha G(, x) G(, x ) L x x and a (, x) a(,x ) + σ (, x) σ(,x ) L x x for x, x R d, 0. If L < (/6) 3/4, hen here exi a unique coninuou oluion x() of ()wihprobabiliy for all 0.Moreover ihaaboundedecondmomenuchhae x() < for all 0. (3) Nex, eimae x () x E up x n+ () x n (), [ 0,] = G (, x) G( 0,x)+ a (, x) d+ σ (, x) dw () 0 0 3[ G (, x) G( 0,x) + + 3[C ( + x ) σ (, x) dw () ] 0 0 a (, x) d Proof. o find a oluion of he inegral equaion (), we ue he mehod of ucceive approximaion. We ar by chooing an iniial approximaion x 0 () = x. A he nex ep, x () =x+g(, x) G( 0,x) Conequenly, + a (, x ()) d + σ (, x ()) dw (). 0 0 x n+ () =x+g(,x n ()) G( 0,x) + a(,x n ())d+ σ(,x n ())dw(). 0 0 (4) (5) herefore, + E up x () x [ 0,] 0 a (, x) d 3(C ( + E x )+E +E up [ 0,] + σ (, x) dw () ]. 0 0 a (, x) d σ (, x ()) dw () ) 0 3(C ( + E x )+( 0 ) C ( + E x )d 0 (7) (8) Conider an inerval [ 0,] uch ha B = 3(L + 5( 0 )L )<.Noehaicanbealwayachievedbychooinga ufficienly mall 0 and by he condiion on L. We prove ha a oluion exi on hi inerval. From (4)i followha E x () x 3(E G (, x) G( 0,x) +E + E σ (, x) dw () ) 0 0 a(, x)d 3(C ( + E x )+( 0 ) C ( + E x ) +( 0 )C ( + E x )), where C i a conan independen of 0,,andx. (6) +4 C ( + E x )d) 0 3(+( 0 ) +4( 0 )) C ( + E x ). he eimaion of he ochaic inegral follow from i properie. Similar o [, p.0], we have E up x n+ () x n () [ 0,] 3(L up x n () x n () [ 0,] +( 0 ) L x n () x n () d 0 + up [ 0,] (σ (, x n ()) σ(,x n ())) dw () ). 0 (9)

3 Abrac and Applied Analyi 3 hu, we have E up x n+ () x n () [ 0,] 3(L E up x n () x n () [ 0,] +( 0 ) L E up x n () x n () [ 0,] of (). Denoe by κ N () he random variable which equal if x() N, y() N and i equal 0 oherwie. hen κ N () (x () y()) =κ N () (G (, x ()) G(,y())) +κ N () [ κ N () [a (, x ()) a(,y())] d 0 + κ N () [σ (, x ()) 0 +4 L E x n () x n () d) 0 3(L +( 0) L +4L ( 0 )) (0) herefore, σ (, y ())]dw()]. (3) E up x n () x n () [ 0,] =BE up x n () x n () [ 0,] B n E up x () x [ 0,] B n 3(+5( 0 )) C ( + E x ). Since B<, i hen follow ha he erie n= P{ up [ 0,] n= x n+ () x n () > n } B n n 4 3(+5( 0 )) C ( + E x ) () E(κ N () x () y() ) 3[E(κ N () L x () y() ) +( 0 ) E(κ N () x () y() )d 0 + E(κ N () L x () y() )d]. 0 he la inequaliy implie he eimaion ( 3L )E(κ N () x () y() ) (+( 0 )) L E(κ N () x () y() )d. 0 (4) (5) By he Gronwall-Bellman inequaliy and he above inequaliy ogeher, we have So, E (κ N () x () y() ) =0. (6) converge. hi implie he uniform convergence wih probabiliy of he erie x+ (x n+ () x n ()), () n=0 on he inerval [ 0,].Iumix(). Sox n () converge o ome random proce. Every x n () i coninuou wih probabiliy. Whence i follow ha he limi x() i coninuou wih probabiliy,oo. hen prove uniquene of hi coninuou oluion. Aume ha here exi a econd coninuou oluion y() P{x() =y()} P{up x () >N} [ 0,] +P{ up y () >N}. [ 0,] (7) Probabiliie a he righ-hand ide of he above inequaliy approach o zero a N,becauex() and y() are coninuou wih probabiliy. herefore x() and y() are ochaic equivalen. We conclude ha P { up x () y() >0} =0. (8) [ 0,] Hence, he exience and uniquene of he oluion are proved on [ 0,].

4 4 Abrac and Applied Analyi Nex, we how boundedne of a econd momen of he oluion. Denoe again by κ N () he indicaor of he e Ue he Gronwall-Bellman inequaliy o find he eimaion Eκ N () x () x D(+E x ), (3) hen {ω : up x () x N}. (9) [ 0,] κ N ()(x () x) where D i a conan independen of N. Applying he Faou lemma o he la inequaliy and auming ha N,we have E x () x D(+E x ). (4) Similarly, we have =κ N () (G (, x ()) G( 0,x)) E(κ N ()(x () x)) +κ N () κ N () a (, x ()) d 0 +κ N () κ N () σ (, x ()) dw (). 0 3 [Eκ N () (L x () x (0) Since E x, hen i follow from [8]haE x < for every [ 0,]. hu exience, uniquene, and boundedne of he econd momen of he oluion x() are proved on [ 0,]. Since he conan B i dependen only on 0 and E x <,andx() i independen of W() W() for,hen by imilar manner we can prove he exience and uniquene of he oluion of he IVP wih iniial condiion (, x()) on he inerval [, ],where i choen uch ha he inequaliy 3(L +5( )L )<. hi procedure can be repeaed in order o exend he oluion of () oheenireemiaxi 0.heheoremi proved. So, + G (, x) + G( 0,x) ) +( 0 ) E(κ N () ( a (, x ()) a(, x) 0 + a (, x ()) ) )d + E(κ N () ( σ (, x ()) σ(, x) 0 + σ (, x ()) ) )d] 9L E(κ N () x () x ) + 8C ( + E x ) +6( 0 ) E(κ N () L x () x )d 0 +6( 0 ) C ( + E x ) +6 E(κ N () L x () x )d 0 +6( 0 ) C ( + E x ). ( 9L )E(κ N () x () x ) 8C ( + E x )+( 0 )C ( + E x ) () + (6L ( 0 )+6L ) E(κ N () x () x )d. 0 () Noe. he exience and uniquene of he oluion can be obained a corollary from work [], where hi reul wa proved for an SDE of neural ype by replacing a condiion for Lipchiz conan L < (/6) 3/4 wih more weak condiion L <. Bu by uing our mehod, pah of obained oluion are coninuou wih probabiliy.oherwie,inhepoined work only he meaurabiliy of he oluion and boundedne of i econd momen were aed. Now, we ae ome probabiliy properie of he oluion obained in heorem. We prove ha under aumpion of heorem, he oluion of () iamarkovrandomproce. Moreover, if he coefficien are coninuou hen i i a diffuion proce. We will find i diffuion coefficien. heorem 3. Under condiion ()-() of heorem he oluion x() of () i a Markov proce wih a raniion probabiliy defined by P (, x,, A) =P{x,x () A}, (5) where x,x () i a oluion of ha equaion x,x () =G(,x,x ())+x G(, x) + a(u,x,x (u))du where 0,x R d. + σ(u,x,x (u))dw(u), (6) Proof. A in heorem, weolve() byhemehodof ucceive approximaion. I can be hown ha x,x () i compleely defined by he nonrandom iniial value x and he

5 Abrac and Applied Analyi 5 proce W() W() for >, which are independen of F. Since x() i a oluion of (), i i F meaurable. Hence x,x () i independen of x() and even from F.Noeha,fromhe uniquene of he oluion x() for >,ifollowhaiia unique oluion of he equaion x () =G(, x ()) +x() G(, x ()) + a (u, x (u)) du + σ (u, x (u)) dw (u). (7) Since he proce x,x() () i alo a oluion of hi equaion, hen x() = x,x() () wih probabiliy. A for he re, he proof i he ame a he proof of he heorem for ordinary ochaic equaion []. he heorem i proved. We have a corollary from hi heorem. Corollary 4. Suppoe ha condiion of heorem are aified. hen () if funcion G, a, and σ are independen of, hen he oluion x() i a homogeneou Markov proce; () if funcion G, a, and σ are periodic funcion wih period θ, hen a raniion probabiliy i periodic funcion; ha i, P(+θ,x,+θ,A)=P(,x,,A). Now, we inveigae condiion for which he oluion of () i a diffuion proce. For hi we mu find an addiional eimae. Lemma 5. Le x,x () be a oluion of () uch ha x,x () = x, x R d. If condiion of heorem are aified and G (, x) i coninuou for 0, x R d,henaninequaliy E x,x () x 4 H( ) (+ x 4 ) (8) hold. Here H>0i a conan dependen only on C, L,L, and G (, x). Proof. Denoe by κ N () he indicaor of he e {ω : up x,x (u) x N}. (9) u [,] Similarly o finding eimaion (4), we have E(κ N () x,x () x 4 ) 7E (κ N () (G (, x,x ()) G(, x) +G (, x) G(, x) ) 4 ) + 7E + 7E 4 κ N (u) a(u,x, (u))du 4 κ N (u) σ(u,x,x (u))dw(u) 6 3 L 4 E(κ N () x,x () x 4 ) +6 3 G4 (, x) ( )4 +o x ( ) 4 +7( ) 3 Eκ N (u) a(u,x,x (u)) 4 du + (7)(36)( ) Eκ N (u) σ(u,x,x (u)) 4 du, (30) where o x poin dependence on x. he la eimae follow from Hölder inequaliy and properie of ochaic inegral. From inequaliy (30) and condiion for L,wehave ( 6 3 L 4 )E(κ N () x,x () x 4 ) 6 3 G (, x) 4 ( ) 4 +o x ( ) 4 + (7)(8)( ) 3 L 4 E(κ N (u) x,x (u) x 4 )du + (7)(8)( ) 3 a (u, x) 4 du + (7)(36)(8)( ) σ (u, x) 4 du + (7)(36)(8)( ) L 4 E(κ N (u) x,x (u) x 4 )du 6 3 G (, x) 4 ( ) 4 +o x ( ) ( ) 4 C ( + x 4 )+6 5 ( ) C ( + x 4 ) +6 5 ( ) L 4 E(κ N (u) x, (u) x 4 )du, (3) where C depend only on C. Furher,herigh-handideof(3)canbeeimaedby ( + x 4 )R( ) +R E(κ N (u) x,x (u) x 4 )du, (3) where R=6 3 G (, x) C +6 5 C +and R =6 5 L 4, +. Uing he lemma from [,p.38],weobain Eκ N (u) x,x (u) x 4 H( ) ( + x 4 ), (33) where H>0depend only on C, L,andG (, x). Now, aume ha N and obain eimae (8). hi complee he proof of he Lemma.

6 6 Abrac and Applied Analyi heorem 6. If condiion ()-() of heorem are aified and funcion a(, x), σ(, x), G(, x), G (, x), G xi (, x), and G xi x j (, x), i, j =, d, areconinuoufor 0, x R d. Funcion G (, x), G xi (, x), andg xi x j (, x) aify Lipchiz condiion wih repec o x in neighborhood of every poin (, x). hen he oluion of () i diffuion proce wih a (, x) =(I G x (, x)) [G (, x)+a(, x)+ G xx (, x) (I G x (, x)) and diffuion marix wih σ (, x) σ (, x) ((I G x (, x)) ) ] (34) σ (, x)=(i G x (, x)) σ (, x) σ (, x) ((I G x (, x)) ). (35) Proof. ake any poin (, x) from a region 0, x R d. Conider a ochaic IO equaion in a cloed neighborhood of hi poin dy = (I G x (, y)) [G (, y) + a (, y) + G xx (, y) (I G x (, y)) σ(,y)σ (, y) ((I G x (, y)) ) ]d +(I G x (, y)) σ(,y)dw(). (36) From he condiion of he heorem, i follow ha here exi a cloed neighborhood of he poin (, x), uchha coefficien of he equaion are Lipchiz wih repec o x. Fix hi neighborhood. Exend hee coefficien on he all region 0, x R d uch ha hey remain coninuou by boh variable, Lipchiz and linear wih repec o x. hen he equaion dy = a (, y) d + σ (, y) dw () (37) wih hee exended coefficien ha a unique oluion of he IVP y() = x for >. From IO formula and coincidence of coefficien of (36) and (37) in hi neighborhood of he poin (, x), one can how ha procee x,x and y() coincide in hi neighborhood. I i known ha under hee aumpion he proce y() i diffuion. I diffuion coefficien in he poin (, x) are defined by coefficien of (36). Nex, how ha x() ha he ame diffuion coefficien. o do hi i i ufficien o eimae he following limi and ue eimaion (8) lim (y x)p(,x,,dy), y x ε 0 (38) lim (z, y x) P(,x,,dy), y x ε 0 where ε 0 i choen uch ha he poin (, y) i laying in hi neighborhood. We ge lim (y x) P (, x,, dy) y x ε 0 = lim (x,x () x)p(dw) {w: x,x () x ε 0 } = lim (y () x)p(dw) {w: y() x ε 0 } =(I G x (, x)) [G (, x) +a(, x) + G xx (, x) (I G x (, x)) σ (, x) σ (, x) ((I G x (, x)) ) ]. (39) Similarly, obain he econd limi in (38): lim (z, y x) P(,x,,dy) y x ε 0 =((I G x (, x)) σ (, x) σ (, x) ((I G x (, x)) ) z, z). (40) So, he exience of limi in (38) i proved for every poin (, x) and fixed ε 0, choen for hi poin. I i clear ha from (8), hee limi exi for every ε > 0 and hey are independen of ε. From he above and he definiion of diffuion proce [, p.67]we can finih he proof.he heorem i proved. Reference [] V. B. Kolmanovkii and V. R. Noov, Sabiliy and Periodic Mode ofconrolsyemwihafereffec, Nauka, Mocow, Ruia, 98. [] V.Kolmanovkii,N.Koroleva,.Maizenberg,X.Mao,andA. Maaov, Neural ochaic differenial delay equaion wih Markovian wiching, Sochaic Analyi and Applicaion,vol., no. 4, pp , 003. [3] X. Mao, Sochaic Differenial Equaion and heir Applicaion, Horwood,Chicheer,UK,997. [4] X.X.LiaoandX.Mao, Exponenialabiliyinmeanquareof neural ochaic differenial difference equaion, Dynamic of Coninuou, Dicree and Impulive Syem,vol.6,no.4,pp , 999.

7 Abrac and Applied Analyi 7 [5] X. Mao, Aympoic properie of neural ochaic differenial delay equaion, Sochaic and Sochaic Repor,vol.68,no. 3-4, pp , 000. [6]A.M.Samoilenko,N.I.Mahmudov,andO.M.Sanzhykii, Exience, uniquene, and conrollabiliy reul for neural FSDES in Hilber pace, Dynamic Syem and Applicaion, vol.7,no.,pp.53 70,008. [7]. E. Govindan, Sabiliy of mild oluion of ochaic evoluion equaion wih variable delay, Sochaic Analyi and Applicaion,vol.,no.5,pp ,003. [8] Y. Boukfaoui and M. Erraoui, Remark on he exience and approximaion for emilinear ochaic differenial equaion in Hilber pace, Sochaic Analyi and Applicaion,vol.0, no. 3, pp , 00. [9] N. I. Mahmudov, Exience and uniquene reul for neural SDE in Hilber pace, Sochaic Analyi and Applicaion, vol.4,no.,pp.79 95,006. [0] V. B. Kolmanovkiĭ and L. E. Shaĭkhe, ConrolofSyem wih Afereffec,vol.57,AmericanMahemaicalSociey,Providence, RI, USA, 996. [] I. I. Gikhman and A. V. Skorokhod, Sochaic Differenial Equaion,NaukovaDumka,Kyiv,Ukrania,968. [] I. I. Gikhman and A. V. Skorokhod, Inroducion in heory of Random Procee,NaukovaDumka,Kyiv,Ukrania,977.

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