Existence, uniqueness and stability results for impulsive stochastic functional differential equations with infinite delay and poisson jumps

Transcription

1 Malaya Journal of Maaik, Vol. 5, No. 4, , 7 hp://doi.org/.6637/mjm54/7 Exinc, uniqun and abiliy rul for ipuliv ochaic funcional diffrnial quaion wih infini dlay and poion jup A.Anguraj * and.banupriya Abrac In hi papr, w udy h xinc and uniqun of ild oluion of ipuliv ochaic funcional diffrnial quaion wih infini dlay and Poion jup undr non-lipchiz condiion wih Lipchiz condiion bing conidrd a a pcial ca by an of h ucciv approxiaion. Furhr, W udy h coninuou dpndnc of oluion on h iniial valu by an of a corollary of h Bihari inqualiy. yword: Sochaic diffrnial quaion, coninuou dpndnc, Poion proc, ipuliv y. AMS Subjc Claificaion 6H5, 34G, 6J65, 6J75 Dparn of Mahaic, PSG Collg of Ar and Scinc, Coibaor-4, Tail Nadu, India. of Mahaic wih CA, PSG Collg of Ar and Scinc, Coibaor-4, Tail Nadu, India. *Corrponding auhor: ı angurajpg@yahoo.co; ı banupriyapg3@gail.co Aricl Hiory: civd 3 Jun 7; Accpd 9 Spbr 7 Dparn Conn Inroducion Prliinari Exinc and niqun Sabiliy Concluion Acknowldgn frnc Inroducion Sochaic diffrnial quaion hav bn invigad a ahaical odl o dcrib h dynaical bhavior of a ral lif phnona. Th hory of ochaic diffrnial quaion ha aracd any rarchr du o i iporanc in any pracical applicaion. nil now, variou udi hav bn carrid ou on ochaic funcional diffrnial quaion(sfde) involving xinc and abiliy rul. ( rfrnc hr in [, 4 6, ]). cnly, In [], h auhor invigad h xinc and uniqun of oluion for SFDE and alo dicud adapiviy and coninuiy, an quar bounddn, and convrgnc of oluion ap fro diffrn iniial daa. c 7 MJM. On h ohr hand, ipuliv diffrnial quaion hriv o b a proiing ara and gaind uch anion aong h rarchr du o i ponial applicaion in variou fild uch a orbial ranfr of alli, doag upply in pharacokinic c. Ipuliv ffc provid a naural dcripion of y and dcrib h phnona which i ubjc o inananou prurbaion and in urn xprinc abrup chang a crain on of i. A larg nubr of rioriou rul abou h SFDE wih ipuliv ffc hav appard in h xiing liraur[3, 6,, ]. I i worh in nioning ha, any ral world y ar ubjcd o ochaic abrup chang and hrfor i i ncary o inviga h uing ipuliv ochaic funcional diffrnial quaion. Fw work hav bn rpord in h udy of SFDE wih ipuliv ffc. For xapl, Anguraj and Vinodkuar[6] ablihd xinc and abiliy rul uing ucciv approxiaion, Sakhivl and Luo [] udid h xinc and aypoical abiliy of ipuliv parial diffrnial quaion. Morovr, any pracical y (uch a uddn pric variaion (jup) du o ark crah, arhquak, hurrican, pidic and o on) ay undrgo o jup yp ochaic prurbaion. Th apl pah of uch y ar no coninuou. Thrfor, i i or appropria o conidr ochaic proc wih jup o dcrib uch odl.

2 Exinc, uniqun and abiliy rul for ipuliv ochaic funcional diffrnial quaion wih infini dlay and poion jup 654/659 Th jup odl ar gnrally bad on h Poion rando aur, and ha h apl pah which ar righ coninuou and hav lf lii. Hnc, hr i a ral nd o dicu SFDE wih Poion Jup. cnly, any rarchr ar focuing hir anion oward h hory and applicaion of SFDE wih Poion Jup. For inanc, Pi and Xu [3] provd h xinc of ild oluion for ochaic voluion quaion wih jup and Tu.S al [3] obaind xinc and uniqun rul of adapd oluion for anicipad backward ochaic diffrnial quaion wih Poion jup undr o wak condiion. To b or prci, xinc and abiliy rul on SFDE wih jup proc can b found in [, 4, 5, ] and h rfrnc hr in. In addiion, non Lipchiz condiion i uch wakr bu ufficin condiion wih wid rang of ponial applicaion. Thrfor, I i nial o conidr SFDE wih non Lipchiz cofficin. Thr ar vral aricl xiing in h liraur ha ar dal wih non Lipchiz cofficin. frnc hr in([3, 7 ]). In [6] Yaada inroducd h hod of ucciv approxiaion undr non Lipchiz cofficin for a nonlinar ochaic diffrnial quaion. Taniguchi[4] ablihd gnralizaion of Yaada hor. Barbu[] and Jakubowki al [8] xndd h rul provd by Taniguch[4] o h infini dinional ca by uing aur of non copacn and fixd poin hor. Cao al[] alo udid Taniguchi yp ucciv approxiaion for a fini dinional SDE wih jup. Thr i uch currn inr in udying SDE wih jup undr non Lipchiz condiion ( [, 3 5]). In [3] and [6], h xinc and abiliy rul wr drivd undr non Lipchiz condiion and undr Lipchiz condiion. To h b of our knowldg, hr i only vry fw aricl in h xiing liraur ha rpor h udy of Ipuliv Sochaic funcional diffrnial quaion wih jup. Th ai of hi papr i o clo hi gap and w inviga h xinc and uniqun rul of ild oluion for Ipuliv SFDE wih infini dlay and Poion jup undr nonlipchiz condiion wih Lipchiz condiion bing conidrd a a pcial ca by an of h ucciv approxiaion. Furhror, w giv h coninuou dpndnc of oluion on h iniial daa by an of a corollary of h Bihari inqualiy. Conidr h following Ipuliv ochaic funcional diffrnial quaion wih jup in h for: dx() = [Ax() f (, x )]d σ (, x )dw () (.) h(, x, u)n(d,, T, 6= k, x(k ) = x(k ) x(k ) = Ik (x(k )), = k, k =,... x() = φ () DbB ((, ], X). fro (, ] o X. Th r of h papr i organizd a follow. In Scion, w giv o baic concp and prliinari. Scion 3 focu on h udy of xinc and uniqun of oluion o ipuliv ochaic funcional diffrnial quaion wih Poion proc by ucciv approxiaion hod. In Scion 4, w ablih h abiliy rul hrough coninuou dpndnc on h iniial valu.. Prliinari L X,Y b ral parabl Hilbr pac and L(Y, X) b h pac of boundd linar opraor apping Y ino X. L (Ω, B, P) b a copl probabiliy pac wih an incraing righ coninuou faily {B } of copl ub σ - algbra of B. L {W () : } dno a Y - valud Winr proc dfind on h probabiliy pac (Ω, B, P) wih covarianc opraor Q, ha i EhW (), xiy hw (), yiy = ( )hqx, yi, for all x, y Y, whr Q i a poiiv lf-adjoin, rac cla opraor on Y. In paricular, w dno W (),, a Y - valud Q- Winr proc wih rpc o {B }. In ordr o dfin ochaic ingral wih rpc o h QWinr proc W (), w inroduc h ubpac Y = Q/ (Y ) of Y which, ndowd wih h innr produc < u, v >Y =< Q / v >Y i a Hilbr pac. W au ha hr xi a copl orhonoral y {i }i in Y, a boundd qunc of nonngaiv ral nubr λi uch ha Qi = λi i, i =,,..., and a qunc {βi }i of indpndn Brownian oion uch ha p hw (), i = λi hi, iβi (), Y, n= B = Bw, and whr Bw i h σ - algbra gnrad by {W () : }. L L = L (Y, X) dno h pac of all Hilbr- Schid opraor fro Y ino X. I urn ou o b a parabl Hilbr pac quippd wih h nor kµkl = r((µq/ )(µq/ ) ) for any µ L. Clarly, for any boundd opraor µ L(Y, X) hi nor rduc o kµkl = r(µqµ ). Th pha pac D((, ], X) i aud o b quippd wih h nor kφ k = up <θ < φ (θ ). W alo au DbB ((, ], X) o dno h faily of alo urly boundd, B - aurabl quar ingrabl rando variabl wih valu in X. Furhr, l BT b a Banach pac BT ((, T ], L ), h faily of all BT - adapd proc φ (, w) wih alo urly coninuou in for fixd w Ω wih nor dfind for any φ BT (.) whr A i h infiniial gnraor of an analyic igroup of boundd linar opraor, (S()), dfind on X; f : [, ) D X, σ : [, ) D L(Y, X), h : [, ) X X. Hr D = D((, ], X) dno h faily of all righ picwi coninuou funcion wih lf hand lii φ 654 kφ kbt = ( up E kφ k )/. T L (, E, ν() b a σ -fini aurabl pac. Givn a aionary Poion poin proc (p )>, which i dfind on (Ω, B, P) wih valu in and wih characriic aur ν. W will dno by N(, b h couning aur of p

3 Exinc, uniqun and abiliy rul for ipuliv ochaic funcional diffrnial quaion wih infini dlay and poion jup 655/659 uch ha N (, A)) = E(N(, A)) = ν(a) for A E. Dfin = N(, ν(, h Poion aringal aur N(, gnrad by p. Th ipuliv on j aify < <..., li j k =, x(k ) = x(k ) x(k ), whr x(k ) rprn h jup in h a x a i k wih Ik drining h iz of h jup and x(k ) and x(k ) ar rpcivly h righ and h lf lii of x() a k. L A : D(A) X b h infiniial gnraor of an analyic igroup,(s()), of boundd linar opraor on X. For dail, on can rfr[7] and []. I i wll known ha hr xi M and λ uch ha ks()k Mλ for vry. If (S()) i a uniforly boundd and analyic igroup uch ha ρ(a), whr ρ(a) i h rolvn of A, hn i i poibl o dfin h fracional powr ( A)α for < α, a a clod linar opraor on i doain D( A)α. Furhror, h ubpac D( A)α i dn in X, and khkα = k( A)α hk dfin a nor in D( A)α. If Xα rprn h pac D( A)α ndowd wih h nor k.kα, c(p, T ) > uch ha φ p E up WA T d () T kφ ()k p d. p Morovr, if E kφ ()k d <, hn hr xi a conφ inuou vrion of h proc {WA : }. If (S()) i a conracion igroup, hn h abov rul i ru for p. Dfiniion.5. A igroup S(), i aid o b uniforly boundd if ks()k M for all, whr M i o conan. Dfiniion.6. A ochaic proc {x() BT, (, T ]}, ( < T < ) i calld a ild oluion of h y (.) if, (i) x() X i aurabl and B adapd, (ii) x() ha ca dla g pah alo urly, (iii) La.. ( Bihari inqualiy[8]) L T > and u, u(), v() b a coninuou funcion on [, T ]. L : b a concav coninuou and non dcraing funcion uch ha (r) > for r >. If u() u v()(u())d, for all T, hn u() G G(u )) v()d for all uch [, T ] ha, G(u ) v()d Do(G ), d whr G(r) = r (), r and G i h invr funcion of G. In paricular, if, orovr,u = and u() = for all T. c(p; T ) up ks()k p E x() = S()φ () S( )σ (, x )dw S( ) f (, x )d S( )h(, x, u)n(d, S( k )Ik (x(k )) if [, T ], <k < x() = φ (), =, hn (, ]. (.) 3. Exinc and niqun In ordr o obain h abiliy of oluion, w giv h xndd Bihari inqualiy. La.. ([9])L h aupion of La (.) hold. if u() u T v()(u(())d, for all T, hn u() G G(u ) T v()d for all [, T ] ha In hi cion, w dicu h xinc and uniqun of ild oluion of h y (.). In ordr o prov h rul, w nd h following aupion: (A ) : A i h infiniial gnraor of a rongly coninuou igroup S(), who doain D(A) i dn in H. (A ) : Th funcion f (.), σ (.) and h(.) aify h following condiion G(u ) T v()d Do(G ), r d whr G(r) = (), r and G i h invr funcion of G. (a) For vry [, T ] and x, y H, uch ha k f (, x) f (, y)k kσ (, x) σ (, y)k (kx yk ). Corollary.3. ([9]) L h aupion of La (.) hold and v() for [, T ]. If for all ε >, hr xi d uch ha for u ε, T v()d uε () hold. Thn for vry [, T ], h ia u() ε hold. (b) (i) La.4. ([9]) Suppo ha φ (), i a L -valud φ prdicabl proc and l WA = S( )φ ()dw (), [, T ]. Thn, for any arbirary p > hr xi a conan whr (.) i concav non-dcraing funcion fro du o, () =, (u) >, for u > and (u) =. 655 kh(, x, u) h(, y, u)k ν(d / 4 kh(, x, u) h(, y, u)k ν(d kx yk, / (ii) kh(, x, u)k4 ν(d kxk d.

4 Exinc, uniqun and abiliy rul for ipuliv ochaic funcional diffrnial quaion wih infini dlay and poion jup 656/659 (A3 ) For all [, T ], hr xi a conan κ > uch ha k f (, )k kσ (, )k kh(,, u)k kik ()k = κ. Thn E kxn k (A4 ) Th funcion Ik C(X, X) and hr xi o conan hk uch ha kik (x) Ik (y)k hk kx yk, for all x, y X and k =,... 5M E kφ ()k T M E M E 5M E x () = φ () for (, ], [ xn xn x () = S()φ () for [, T ]. (3.) κ ]d κ ]d d M hk [E xn κ ] ( xn )d M hk (E xn and, for n=,..., k= Q 5M (T 5)E [ xn κ ]d M E L u now inroduc h ucciv approxiaion o Eqn.(.) a follow [ xn ) k= xn () = φ () for (, ], xn () = S()φ () S( ) f (, xn )d S( )σ (, xn )dw whr Q = 5M (E kφ ()k (T (T ) k= hk )κ ). Givn ha (.) i concav and () =, w can find a pair of poiiv conan a and b uch ha (u) a bu, for all u. S( )h(, xn, u)n(d, S( k )Ik (xn (k )), w obain, for [, (3.) T] E kxn k <k < Q 5M (T 5)bE ), n M hk (E xn wih an arbirary non-ngaiv iniial approxiaion x BT. )d ( xn =,(3.3),... k= Thor 3.. Suppo ha (A ) (A4 ) hold. Thn, h y (.) ha a uniqu ild oluion in BT providd ha whr Q = Q 5M (T 5)Ta. Sinc M hk <. 8 k= E x M E kφ ()k = Q3 <. (3.4) Thu, whr M uch ha ks()k M. E kxn k Q4 <, for all n =,,,.. and [, T ].(3.5) Proof. L x BT b a fixd iniial approxiaion o Eqn.(3.). I i clar ha by (A ) (A4 ), ks()k M for o M Thi prov h bounddn of {xn (), n N}. and all [, T ]. Thn for n, w hav, L u nx how ha {xn ()} i Cauchy in BT. For hi, for n,, w hav E kxn ()k 5M E kφ ()k T M E M E M E [ f (, xn ) f (, ) [ a(, xn ) a(, ) xn () x () k= 8M hk (E xn ka(, )k ]d n () x () up E kx r n x kr Q6 up E kxn x k, ν(d kik ()k ] ). Thu k= up E x M E [ Ik (xn (k )) Ik () ( kxn () x ()k )d 4 4M (T 5)E k f (, )k ]d [ h(, xn, u) h(,, u) kh(,, u)k ]ν(d 5M E h(, xn, u) whr Q5 = 4M (T 5) and Q6 = 8M k= hk. Ingraing boh id of Eqn.(3.6) and applying Jnon 656 d (3.6)

5 Exinc, uniqun and abiliy rul for ipuliv ochaic funcional diffrnial quaion wih infini dlay and poion jup 657/659 alo clai ha for [, T ] inqualiy giv ha up E x n l Q6 r l x kr E dld E up E kxn x kl dld S( ) σ (, xn ) σ (, x ) dw up E kxn x kr r l dld and E S( k ) Ik (xn (k )) Ik (x(k )) x() = S()φ ()! up E kxn x kr dl d r l up E kxn x kl dld S( ) f (, x )d S( )σ (, x )dw S( )h(, x, u)n(d, S( k )Ik (x(k )) <k < Thi crainly donra by h Dfiniion (.) ha x() i a ild oluion o Eqn. (.) on h inrval [, T ]. Nx, w prov h uniqun of h oluion of Eqn.(.). L x, x BT b wo oluion of Eqn.(.) on o inrval (, T ]. Thn, for (, ], w hav Thn,. Hnc, aking lii on boh id of Eqn.(3.), l Ψn, (), <k < up E kxn x kl dld, S( ) h(, xn,u ) h(, x, u) N(d, l,! Q6 n S( ) f (, xn ) f (, x ) d l Q6 up E kx E! x l d (Ψn, ()) d Q6 Ψn, (),(3.7) E kx x k Q6 E kx x k Q5 whr (E kx x k )d. Thu n up l E kx x kl d E kx x k Ψn, () =. Q5 Q6 (E kx x k )d. Thu, Bihari inqualiy yild ha Fro Eqn (3.5), i i ay o ha up E kx x k =, T. [,T ] up Ψn, () <. Thu x () = x (), for all T. Thrfor, for all < T, x () = x (). Thi prov our dird rul. n, So ling Ψ() = li upn, Ψn, () and aking ino accoun h Faou la, w yild ha b Ψ() = Q b= (Ψ()) d, whr Q 4. Sabiliy In hi cion, w udy h abiliy of h y (.) hrough h coninuou dpndnc of oluion on iniial condiion. Q5. Q6 Dfiniion 4.. A ild oluion x() of h y (.) wih iniial valu φ i aid o b abl in h an quar if for all ε >, hr xi δ > uch ha E kx() x ()k ε whn E φ φ < δ, for all [, T ], whr x i anohr ild oluion of h y (.) wih iniial valu φ. Now, applying h La(.) idialy rval Ψ() = for any [, T ]. Thi furhr an{xn (), n N} i a Cauchy qunc in BT. So hr i an x BT uch ha T li up E kxn xk d =. n E kxk In addiion, by Eqn(3.5), i i ay o follow ha Q4. Thu w clai ha x() i a ild oluion o Eqn.(.). On h ohr hand, by aupion (A ) and ling n, w can Thor 4.. L x() and y() b ild oluion of h y (.) wih iniial valu φ and φ rpcivly. If h aupion (A ) (A4 ) ar aifid hn h ild oluion of h y (.) i abl in h quadraic an. 657

6 Exinc, uniqun and abiliy rul for ipuliv ochaic funcional diffrnial quaion wih infini dlay and poion jup 658/659 ark 4.4. If h y (4.) aifi h ark 4., hn by Thor 4., h ild oluion of h y (4.) i abl in h an quar. Proof. By h aupion, x() and y() ar wo ild oluion of h y (.) wih iniial valu φ and φ rpcivly, hn for T x() y() = S() [φ () φ ()] In hi papr, w hav udid h xinc and abiliy rul for ipuliv ochaic funcional diffrnial quaion wih infini dlay and Poion jup undr non-lipchiz condiion wih Lipchiz condiion bing conidrd a a pcial ca by an of h ucciv approxiaion. Manwhil, W ablih h coninuou dpndnc of oluion on h iniial valu by an of a corollary of h Bihari inqualiy. S( ) [σ (, x ) σ (, y )] dw () 5. Concluion S( ) [ f (, x ) f (, y )] d S( )h(, x, u) h(, y, u)n(d, S( k ) [Ik (x(k )) Ik (y(k ))]. <k < Acknowldgn So, iaing a bfor, w g E kx yk 5M E kφ φ k 5M (T ) Thu, Th auhor would lik o xpr incr graiud o h rviwr for hi/hr valuabl uggion. Th cond auhor (E kx yk )d 5M hk E kx yk wih. o acknowldg h GC, India k= (F MP-58/5(SEO/GC)) for upporing h prn work. E kx yk 5M E kφ φ k 5M k= hk 5M (T ) 5M k= hk frnc [] (E kx yk )d. [] 5M (T ) L (u) = 5M h (u), whr i a concav incrak= k ing funcion fro o uch ha () =, (u) > for du u > and (u) =. So, (u) i obviouly, a concav funcion fro o uch ha () =, (u) (u), for u and du(u) =. Now for any ε >, ε = ε, [3] w hav li ε du(u) =. So, hr i a poiiv conan [4] δ < ε, uch ha δε du(u) T. L u = 5M 5M k= hk E kφ φ k [5] u() = E kx yk, v() =, whn u δ ε. Fro corollary.3 w hav ε u du(u) ε δ du(u) T = T v()d. So, for any [, T ], h ia u() ε hold. Thi copl h proof. [6] ark 4.3. If = in h y(.), hn h y bhav a ochaic parial funcional diffrnial quaion wih infini dlay of h for [7] dx() = [Ax() f (, x )]d σ (, x )dw () (4.) [8] h(, x, u)n(d,, T, 6= k, x() = φ () DbB ((, ], X). [9] (4.) By applying Thor 3. undr h hypoh (A ) (A3 ), h y (4.) guaran h xinc and uniqun of h ild oluion. 658 [] B.Boufoui and S.Hajji, Succiv approxiaion of nural funcional ochaic diffrnial quaion wih jup J. Sai. Probab. L.8() Bo Du, Succiv approxiaion of nural funcional ochaic diffrnial quaion wih variabl dlay, Appl.Mah.Copu. 68(5) A.Vinodkuar, Exinc, niqun and abiliy rul of ipuliv ochaic ilinar funcional diffrnial quaion wih infini dlay, J. Nonlinar Sci. Appl. 4(), no. 4, T.Taniguchi, Succiv approxiaion o oluion of ochaic diffrnial quaion, J. Diffr. Equ. 96(99)5-69. V.B.olanovkii and A.Myhki, 99, Applid hory of funcional diffrnial quaion, luwr Acadic publihr. A.Anguraj and A.Vinodkuar, Exinc,niqun and abiliy rul of ipuliv ochaic ilinar nural funcional diffrnial quaion wih infini dlay, Elcron.J.Qual.Thory Diffr.Eqn.67(9)-3. A.Goldin and Jro, Sigroup of linar opraor and applicaion.in: Oxford Mahaical Monograph. Th Clarndon Pr, Oxford nivriy pr, Nw York (985). I.Bihari, A gnralizaion of a la of Blan and i applicaion o uniqun probl of diffrnial quaion, Aca.Mah.Acad.Sci.Hungar.(956) 7,7-94. Da.Prao and J.abczyk, Sochaic quaion in infini dinion, Cabridg nivriy pr, 4..Sakhivl and J.Luo, Aypoic abiliy of nonlinar ipuliv ochaic parial diffrnial quaion wih infin dlay,j.mah.anal.appl. 356(9)-6.

7 Exinc, uniqun and abiliy rul for ipuliv ochaic funcional diffrnial quaion wih infini dlay and poion jup 659/659 [] [] [3] [4] [5] [6] [7] [8] [9] [] [] [] [3] A.Pazy, Sigroup of Linar Opraor and Applicaion o Parial Diffrnial Equaion In: Applid Mahaical Scinc.(983) vol. 44. Springr-Vrlag, Nw York. Y.ao,Q.hu and W.Qi Exponnial abiliy and inabiliy of ipuliv ochaic funcional diffrnial quaion wih Markovian wiching, Appl.Mah.Copu.7(5) B.Pi and Y.Xu Mild oluion of local non Lipchiz ochaic voluion quaion wih jup, Appl.Mah.Copu.5(6) J.Cui and L.Yan, Succiv approxiaion of nural ochaic voluion quaion wih infini dlay and Poion jup, Appl.Mah.Copu. 8 () C.Yu, Nural ochaic funcional diffrnial quaion wih infini dlay and Poion jup in h Cg pac, Appl.Mah.Copu. 37 (4) T.Yaada, On h ucciv approxiaion of oluion of ochaic diffrnial quaion, J. Mah. yoo niv. (98):5-55. A.Jakubowki, M.anki, and P.aynaud d Fi, Exinc of wak oluion o ochaic voluion incluion, Soch. Anal. Appl. (5) 3(4): A.E.odkina, On xinc and uniqun of oluion of ochaic diffrnial quaion wih hrdiy, Sochaic Monograph (984):87-. Y.n and N.Xia, Exinc, uniqun and abiliy of h oluion o nural ochaic funcional diffrnial quaion wih infini dlay, Appl. Mah. Copu. (9), D.Barbu, Local and global xinc for ild oluion of ochaic diffrnial quaion, Porugal. Mah.(998) 55(4):4-44. G.Cao,. H and X. hang, Succiv approxiaion of infini dinional SDE wih jup, Soch. Dyna. Vol. 5, No. 4 (5) F.Wu, G.Yin and H.Mi, Sochaic funcional diffrnial quaion wih infini dlay: Exinc and uniqun of oluion, oluion ap, Markov propri, and rgodiciy, J. Diffr. qu. (6), hp://dx.doi.org/.6/j.jd.6..6 (Aricl in pr). Tu. S, Hao.W and Chn.J, Th adapd oluion and copariion hor for anicipad backward ochaic diffrnial quaion wih Poion jup undr h wak condiion, Sa. Probabil. L. (7), hp://dx.doi.org/.6/j.pl.7.. (Aricl in pr).????????? ISSN(P): Malaya Journal of Maaik ISSN(O):????????? 659