Research & Reviews: Journal of Statistics and Mathematical Sciences

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1 Research & Reviews: Jural f Saisics ad Mahemaical Scieces iuus Depedece f he Slui f A Schasic Differeial Equai Wih Nlcal diis El-Sayed AMA, Abd-El-Rahma RO, El-Gedy M Faculy f Sciece, Alexadria Uiversiy, Egyp Faculy f Sciece, Damahur Uiversiy, Egyp Research Aricle Received dae: //6 Acceped dae: 4/4/6 Published dae: 8/4/6 Fr rrespdece El-Sayed AMA, Faculy f Sciece, Alexadria Uiversiy, Egyp ABSRA I his paper we are ccered wih a lcal prblem f a schasic differeial equai ha cais a Brwia mi he slui cais bh f mea square Riema ad mea square Riema-Seljes iegrals, s we sudy a exisece herem fr uique mea square ciuus slui ad is ciuus depedece f he radm daa X ad he (-radm daa) cefficies f he lcal cdii a Als, a schasic differeial equai wih he iegral cdii will be csidered amasyed@yahcm Keywrds: Iegral cdii, Brwia mi, Uique mea Square slui, iuus depedece, Radm daa, N- Radm daa, Iegral cdii INRODUION May auhrs i he las decades sudied lcal prblems f rdiary differeial equais, he reader is referred [- 7], ad refereces herei Als he hery f schasic differeial equais, radm fixed pi hery, exisece f sluis f schasic differeial equais by usig successive apprximai mehd ad prperies f hese sluis have bee exesively sudied by several auhrs, especially hse cai he Brwia mi as a frmal derivaive f he Gausia whie ise, he Brwia mi W (), R, is defied as a schasic prcess such ha W () ; E(W ()), E(W ()) ad [W ( ) W ( )] is a Gaussia radm variable fr all, R he reader is referred [8,9] ad [-6] ad refereces herei Here we are ccered wih he schasic differeial equai dx() f (, X())d + g()dw (), (, ] () wih he lcal radm iiial cdii ( ), () ( ) +, >, (, ) X ax X a where X is a secd rder radm variable idepede f he Brwia mi W () ad a are psiive real iegers he exisece f a uique mea square slui will be sudied he ciuus depedece he radm daa X ad he radm daa a will be esablished he prblem () wih he iegral cdii will be csidered X + X(s)dv(s) X (3) ( ) RRJSMS Vlume Issue Jue, 6 98

2 INEGRAL REPRESENAION Le (I, L ( Ω)) be he class f all mea square ciuus secd rder schasic prcess wih he rm X sup X sup EX ( ()) c [,] [,] hrughu he paper we assume ha he fllwig assumpis hld (H) he fuci f : [, ] L( ) L( ) (H) here exiss a iegrable fuci [ ] s up (s) ds m [,] Ω Ω is mea square ciuus :, R +, where such ha he fuci f saisfies he mea square Lipschiz cdii (, ( )) (, ( )) ( ) ( ) ( ) f X f X X X (H3) here exiss a psiive real umber m such ha sup f(,) m [,] Nw we have he fllwig lemmas g(s) dw(s) Prf g (s) ds E E g(s)dw( s) lim g( ) W( ) lim g( ) W( ) E lim g ( ) W( ) lim g ( ) ( ) g (s)ds his cmplees he prf Lemma : he slui f he prblem () ad () ca be expressed by he iegral equai m m X ( ) a X a f(s, X (s))ds a g(s)dw(s ) + f (s, X(s))ds +, (4) where a + a Prf Iegraig equai (), we bai ( ) ( ) X X + f (s, X(s))ds + g(s)dw( s), ad ( ) ( ) X X + f (s, X(s))ds +, RRJSMS Vlume Issue Jue, 6 99

3 he ( ) ( ) + + a X a X a f (s, X(s))ds a, ( ) ( ) + + X X a X a f (s, X(s))ds a, ad ( ) + a X () X a f (s, X(s))ds + a, he () ( ) X + a X a f (s, X(s))ds + a Hece X () a X a f (s, X(s))ds a + f (s, X(s))ds + Where a + a Nw defie he mappig FX () a X a f (s, X(s))ds a + f (s, X(s))ds +, he we ca prve he fllwig lemma Lemma 3 F : Prf Le X,, [, ] such ha - < δ, he FX ( ) FX ( ) f (s, X(s))ds + Frm assumpi (ii) we have (, ( )) (,) ) (,) () ( ) f X f f(, X() f X he we have f, X ( ) () X() + f (,) () X c + m S, ( ) FX ( ) FX ( ) f (s, X(s)) ds + usig assumpis ad lemma, we ge FX ( ) FX ( ) X ( ) (s) ds + m + g (s) ds, which prves ha F : EXISENE AND UNIQUENESS Fr he exisece f a uique ciuus slui X f he prblem ()-(), we have he fllwig herem herem 3 Le he assumpis (H) (H3) be saisfied If m <, he he prblem()-() has a uique slui X Prf Le X ad X, he RRJSMS Vlume Issue Jue, 6

4 FX () FX () m X X + a m X X m X X [f (s, X(s)) f (s, X (s)]ds a a [f (s, X(s)) f (s, X (s))]ds f (s, X(s)) f (s, X (s)) ds + a a f (s, X(s)) f (s, X (s)) ds + a a m X X Hece If m <, he F is craci ad here exiss a uique slui X f he lcal schasic prblem ()-(), [] his slui is give by (4) FX F X m X X ONINUOUS DEPENDENE sider he schasic differeial equai () wih he lcal cdii ( ) + X a X ( ) X, (, ) Defiii 4 he slui X f he lcal prblem ()-() is ciuusly depede ( he daa X ) if >, δ > such ha X X δ implies ha X X c Here, we sudy he ciuus depedece ( he radm daa X ) f he slui f he schasic differeial equai () ad () herem 4 Le he assumpis (H) (H3) be saisfied he he slui f he lcal prblem ()-() is ciuusly depede he radm daa X Prf Le X () a X a f (s, X(s))ds a + f (s, X(s))ds + be he slui f he lcal prblem ()-() ad X() a X a f (s, X(s))ds a + f (s, X(s))ds + g(s)dw( s) be he slui f he lcal prblem () ad (6) he X () X() a[ X X ] a a f (s, X(s))ds f (s, X(s))ds + [f (s, X(s) f (s, X (s))]ds Usig ur assumpis, we ge X () X () a X X + a a f (s, X(s)) f (s, X (s)) ds + f (s, X(s) f (s, X (s)) ds aδ + m X X ds he aδ X X c { m} his cmplees he prf RRJSMS Vlume Issue Jue, 6

5 Nw csider he schasic differeial equai () wih he lcal cdii X() + a X( ) X, (, ) Defiii 4 he slui X f he lcal prblem ()-() is ciuusly depede ( he cefficie a f he lcal cdii) if >, δ > such ha a a δ implies ha X X c Here, we sudy he ciuus depedece ( he radm daa X ) f he slui f he schasic differeial equai () ad () herem 43 Le he assumpis (H) (H3) be saisfied he he slui f he lcal prblem ()-() is ciuusly depede he cefficie a f he lcal cdii Prf Le X () a X a f (s, X(s))ds a + f (s, X(s))ds + be he slui f he lcal prblem ()-() ad X () a X a f (s, X(s))ds a + f (s, X(s))ds + be he slui f he lcal prblem () ad (7) he X () X() a[ a a ]X + [f (s, X(s) f (s, X (s))]ds a a a a f (s, X(s))ds + aa f (s, X(s)) ds Nw a a a a a + a ad ( a a) + a + a δ ( a a ) a a f (s, X(s))ds -a a f (s, X(s))ds a + a f (s, X(s))ds -a + a f (s, X(s))ds a f (s, X(s))ds+a f (s, X(s))ds ( ) ( ) a a f (s, X(s))ds+a a f (s, X(s))ds a f (s, X(s))ds+a f (s, X(s))ds [f (s, X(s))ds - f (s, X(s))]ds+[a a ] f (s, X(s))ds a [f (s, X(s))ds - f (s, X(s))]ds+a f (s, X(s))ds a f (s, X(s))ds a f (s, X(s))ds+a f (s, X(s))ds + a f (s, X(s))ds - f (s, X(s))]ds RRJSMS Vlume Issue Jue, 6

6 ad a he a a a a + a a + a [ a a ] [ ] [ ] aa aa a a [ a a ], X () X() δ X + f (s, X(s) f (s, X(s)) ds + δ δ [ m X + m ] + a f (s, X(s))- f (s, X(s)) Usig ur assumpis we ge δ δ δ X X X + m X X + g (s)d(s) + [ m X + m ] + am X X, he X X X + m X + m + g (s)d(s) + ( + a) m X Hece δ X δ X + m X + m + g (s)d + (s) m X X δ X + m X + m + g (s)d(s) X X { m} his cmplees he prf ds NON LOAL INEGRAL ONDIION Le a v( ) v( ), (, ), where ( < < < 3 < < ) he, he lcal cdii () will be i he frm + X() X( )( v( ) v( ) ) X Frm he mea square ciuiy f he slui f he lcal prblem ()-(), we bai frm [5] lim X ( )( v( ) v( )) X(s)dv( s), ha is, he lcal cdiis () is rasfrmed he mea square Riema-Seljes iegral cdii X () + X(s)dv(s) X, Nw, we have he fllwig herem herem 54 Le he assumpis (H)-(H3) be saisfied, he he schasic differeial equai () wih he lcal iegral cdii (3) has a uique mea square ciuus slui represeed i he frm RRJSMS Vlume Issue Jue, 6 3

7 X a X f( X )d dv(s) g( )dw( )dv(s) f( X )d g( )dw( s s ( ), ( ), ( ) ), θ θ θ θ θ + θ θ θ + θ θ Prf aig he limi f equai (4) we ge he prf REFERENES Bucherif AA Firs-rder differeial iclusis wih lcal iiial cdiis Appl Mah Le ;5:49-44 Bucherif A ad Precup R O he lcal iiial value prblem fr firs rder differeial equais Fixed Pi hery 3;4:5-3 Byszewsi L ad Lashmiaham V herem abu he exisece ad uiqueess f a slui f a lcal absrac cauchy prblem i a baach space Applicable Aal99;4:-9 4 El-Sayed AMA, e al Uifrmly sable slui f a lcal prblem f cupled sysem f differeial equais Differ Equ Appl 3;5: El-Sayed AMA, e al Exisece f slui f a cupled sysem f differeial equai wih lcal cdiis Malaya J Mah 4;: El-Sayed AMA ad Amee I iuai f a parameerized impulsive differeial equai a ieral lcal cauchy prblem Alexadria J Mah 7 El-Sayed AMA ad Bi-ahir EO A arbirary fracial rder differeial equai wih ieral lcal ad iegral cdiis Advaces i Pure Mahemaics ;: Admia G Schasic Sysem, Academic Press, Bharucha-eid A Fixed pi herems i prbabilisic aalysis Bull Amer Mah Sc 976;8: El-awil MA O he applicai f mea square calculus fr slvig radm differeial equais Elecric J Mah Aal Appl 3;:- El-awil MA ad Shaly MA Mea square umerical mehds fr iiial value radm differeial equais Ope J Discree Mah ;:66-84 Ih S Radm fixed pi herems wih a applicai radm differeial equais i baach spaces J Mah Aal Appl 979;6: Philipse AP Nes Brwia mi Urech Uiversiy Debye Isiue Va H Labrary, 4 Plae E A irduci umerical mehds fr schasic differeial equais, Aca Numerica 999;8: Sg Radm differeial equais i sciece ad egieerig Mah Sci Eg 3:973 6 Zhu W, e al Expeial sabiliy f schasic differeial equai wih mixed delay J Appl Mah 4;4:- RRJSMS Vlume Issue Jue, 6 4