A class of multidimensional quadratic BSDEs

Transcription

1 A class of mulidimensional quadraic SDEs Zhongmin Qian, Yimin Yang Shujin Wu March 4, 07 arxiv: v mah.p] Mar 07 Absrac In his paper we sudy a mulidimensional quadraic SDE wih a paricular class of produc generaors bounded erminal values, give a resul of exisence of soluion in a suiable anach space under some consrains on parameers. Key words: Mulidimensional SDEs, quadraic SDEs, MO maringales. Inroducion A mulidimensional quadraic ackward Sochasic Differenial Equaion SDE on 0, T] wih T being he erminal ime, according o he formulaion pu forward by Pardoux Peng 4], is a sochasic inegral equaion wih ˆ T Y = ξ + ˆ T f s,y s,z s ds Z s dw s. where Y is d -valued, Z is d k -valued, erminal value ξ is d -valued F T measurable. The generaor funcion f : 0,T] Ω d d k d is of quadraic growh W is a sard k- dimensional rownian moion defined on Ω,F,F,P wheref is he rownian filraion. SDE wih quadraic growh can be used o solve problems such as uiliy maximizaion wih exponenial uiliy funcion. They were sudied by Kobylanski ] hen exended by ohers for example ]3]4] ec. More precisely in 000, by using he monooniciy mehod adoped from PDE heory, Kobylanski ] solved a class of one dimensional SDEs wih generaor funcion being of quadraic growh in Z. This paricular class of quadraic SDEs wih unbound erminal values were furher sudied by ri Hu ]3] Delbaen, Hu ichou 4]. In 03, arrieu El Karoui 5] adoped a differen approach o prove he exisence under condiions similar o hose of ri Hu ], while ri Elie 6] gave a concise sudy for he case when he erminal value ξ is bounded. The mehod used in he presen paper o ge he main resul was parially inspired by he mehod used in Tevzadze 9] for solving exisence of soluion o a quadraic SDE driven by a coninuous maringale wih bounded erminal values. The case of mulidimensional quadraic SDEs seems significanly more difficul han ha of Lipschiz SDEs, he mehods used in lieraure are ofen quie involved, up unil now he resuls abou quadraic SDEs are mosly only for he one-dimensional case, hey heavily relay on comparison heorems. ecenly, based Exeer College, Turl Sree, Oxford OX 3DP. esearch suppored parially by EC gran ESig ID qianz@mahs.ox.ac.uk Mahemaical Insiue, Universiy of Oxford, Oxford OX 6GG. yimin.yang@mahs.ox.ac.uk School of Finance Saisics, Eas China Normal Universiy, 500 Dongchuan oad, Shanghai sjwu@sa.ecnu.edu.cn

2 on a resul for one-dimensional SDEs in ri Hu ], Hu Tang ] proved he exisence uniqueness of soluion o a mulidimensional SDE wih diagonal quadraic generaor assuming ha each componen f i of he generaor f depends only on he i-h row of he marix variable Z in he SDE.. In his paper we sudy a mulidimensional quadraic SDE wih a paricular class of produc generaors give a resul of exisence of soluion on a suiable anach space under some consrains on parameers. The corresponding PDEs however have significance in fluid dynamics in fac hey are simplified version of fluid equaions. The paper is organized as follows. In Secion, we poin ou he paricular ype of mulidimensional quadraic SDEs ha we sudy lis he assumpions we work under. In Secion 3, we firsly use properies of MO maringales, Girsanov s heorem predicable represenaion propery o show ha he usual ieraion mehod is also well defined for our problem. Then we show ha a conracion map can be found on a suiable complee meric space on a fixed small ime inerval under some consrains on parameers, which gives a resul of exisence of soluion o our SDE on he whole ime inerval by pasing ime ogeher. Definiions assumpions Le us begin wih a few noaions definiions which are neverheless sard in SDE lieraure as follows: y = y T y for y d z = Trzz T for z d k denoe he Euclidean norms. E F ] := E F ] W is a sard k-dimensional rownian moion defined on Ω,F,F,P F 0,T] is he rownian filraion. M d M d k denoe respecively he anach spaces of progressively measurable processes Y Z such ha Y T M = E 0 Y d < Z M = E T 0 Z d <. S d denoes he anach space of bounded progressively measurable processes Y. S C d denoes he anach space of bounded progressively measurable processes Y such ha Y S C C is a non negaive consan. d k denoes he space of progressively measurable processes Z such ha Z s dw s is a MO maringale define Z = Z s dw s MO. Then, is a anach space due o he fac ha he space of MO maringales null a zero is anach he definiion of sochasic inegral. d k = { Z d k : Z } wih being a posiive consan, which denoes a closed subspace of d k. We consider he following SDE: { dy Y T = Z f Y,Z d+ Z dw = ξ. where Y is d -valued, Z is d k -valued, f is k -valued ξ is d -valued F T measurable, which should be inerpreed as a sochasic inegral equaion.. We make he following assumpions:. ξ C for some consan C > 0, i.e. ξ is bounded.

3 . f saisfies he Lipschiz condiion has a linear growh: f y,z f y,z C y y +C 3 z z, f y,z C y +C 3 z +C 4, for any y,y d z,z d k, so ha z f y,z has quadraic growh in z. C,C 3,C 4 are non negaive consans. And f Y, Z is progressively measurable when Y, Z is progressively measurable. y a soluion o., we mean a pair of sochasic processes Y,Z on Ω,F,F,P, where Y = Y S d Z =Z d k. Moreover Z d k implies ha Z M d k due o he fac ha T if E F is a MO maringale ˆ T Z M = E Z s ds 0 = E ˆ T 0 ˆ Z s dw s Z s dw s MO.. The following properies abou MO maringales are well known. If Z s dw s is a MO maringale, hen ˆ T ˆ E F Z ds Z s dw s = Z, 0,T] MO Z ds N, for every 0,T] where N is a non negaive consan, hen Z s dw s see Kazamaki 0] for deails. 3 Exisence of soluion Z ˆ = Z s dw s We will use he ieraion mehod. Le S C denoe, for simpliciy, he anach space S C d d k. Suppose ha Y,Z S C f ξ saisfy he above assumpions, hen we have, by he linear growh of f, boundedness of Y properies of he MO maringale Z s dw s, ha f Ys,Z s T dw s is also a MO maringale, which in urn implies ha he sochasic exponenial of MO N, f Y s,z s T dw s is a maringale on 0,T]. Hence we define a probabiliy measureqby ˆ dq dp = E f Y s,z s T dw s FT T 3. define dw Q = dw + f Y,Z d. 3. Then W Q is a sard k-dimensional rownian moion under probabiliy measure Q. The lemma below abou coninuous maringale represenaion is well known is proof may be found in Cohen Ellio 5]. Lemma. Suppose M is a d-dimensional coninuous local maringale under Q wih Q defined as above, hen here exiss a unique predicable process H such ha M M 0 = H s dw Q s. Proof. Since M is a coninuous semi-maringale under P, M M 0 = N+ A where N is a coninuous local maringale null a 0 under P A is a finie variaion process. y he maringale represenaion heorem applying o N, we have ha N = H s dw s for some predicable process H. Thus 3

4 M M 0 H s dws Q = H s f Y s,z s ds+a by equaion 3., which implies ha he coninuous Q-local maringale M M 0 H s dws Q is of finie variaion null a 0. Thus we have ha M M 0 = H s dws Q. Uniqueness can be proved in he usual way. Le δ 0,, consider ime inerval T δt,t]. Le Y,Z SC bu wih duraion T δt,t]. Since ξ C, Ỹ = E F Q ξ] is a coninuous maringale underqont δt,t]. Thus by Lemma, here exiss a unique predicable process Z on T δ T, T] such ha { dỹ = Z f Y,Z d+ Z dw 3.3 Y T = ξ wih Ỹ S C, which means ha Ỹ S C d. We prove he following proposiion. Proposiion. If C C 3 < e where e is jus a universal consan, i does no imply ha i is opimal. Then here is a non negaive consan C 6 depending on C,C,C 3,C 4 δt such ha Z C 6+ Z for any pairsy,z Ỹ, Z ont δt,t] defined by SDE 3.3. Proof. Consider ϕ=e K where K is a posiive consan o be deermined laer. Le η = ϕ Ỹ. Then by Io s formula we have dη d Ỹ = i Y i dy i + Z d = f j Y,Z Y i i, j = Kηd Ỹ + K η j Y i Z i, j d i = Kη Y i Z i, j d+ Z ] d f j Y,Z i, j +Kη Y i Z i, j dw j. i, j Z i, j d+ Z d+ Y i Z i, j dw j, i, j + K η j Se vecor Ũ wih U j = Y i Z i, j, so he previous equaion can be wrien as i i Z i, j d dη = Kη Ũ T f Y,Z+ Z + K Ũ ] d+ KηŨ T dw. Inegraing he equaliy above from o T, we obain η = η T K ˆ T η Ũ T f Y,Z+ Z + K Ũ ] ds K Y i ˆ T ηũ T dw. Take he condiional expecaion wih respec o F o ge ˆ T η = E F η T KE F η Ũ T f Y,Z+ Z + K Ũ ] ds. 4

5 Nex applying Cauchy-Schwarz inequaliy, he linear growh condiion of f he bound of Y, we deduce ha ˆ T η E F η T KE F η C C +C 3 Z +C 4 Ũ + Z + K Ũ ] ds ˆ T = E F η T +C C +C 4 KE F η Ũ ds +KC 3 E F ˆ T η Z Ũ ds KE F Then by applying he inequaliies wih α, β > 0 we ge ha η I follows ha ˆ T E F ˆ T Ũ α+ Ũ α Z Ũ β Z + Ũ, β E F η T +C C +C 4 KE F KE F KE F η Z ds ˆ T ˆ T ˆ T η Z ds + KC 3 E F η Z + K Ũ ] ds. αηds ˆ T ηβ Z ds η K C ] 3 β C C +C 4 Ũ ds. α K EF η T η +C C +C 4 E F ˆ T +C 3 βe F ˆ T E F η η Z ds ˆ T K C 3 β C C +C 4 α αηds Since Ỹ S C we deduce ha η e KC. We may choose consans such ha ] Ũ ds. 3.4 K C 3 β C C +C α C 3 βe KC =. 3.6 In order o do his, i requires ha K C3 ekc > 0 which means ha Ke C3 ekc > 0 which is possible only if C C 3 < e. When C C 3 < e by considering KC C C 3 ekc > 0 5

6 we can choose K = C lnc C 3 which implies ha K C 3 ekc > 0. Then we can deduce from he previous inequaliy 3.4 ha E F ˆ T Z ds K ekc + αe KC C C +C 4 δt ˆ T +C 3 βe KC E F Z ds for all T δt,t]. Using he properies of MO maringales we may deduce ha Z C 6+ Z where ] C 6 = e KC K + αc C +C 4 δt = C C 3 C lnc C 3 + αc C +C 4 δt ]. 3.7 The above proposiion implies ha we ge a pair Ỹ, Z S C ont δt,t], when C C 3 < e. In his case we define he pair Ỹ, Z = ΦY,Z ont δt,t] Φ : S C S C is well defined. In order o ge a resul abou he global exisence of soluion, we firsly consider he ime inerval T δt,t] for some δ 0, ry o find a conracion map on a closed subspace of S C. This approach is inspired by he mehod used in Tevzadze 9]. Then by working backwards wih respec o ime inervals of lengh δt, we can ge our resul by pasing ime ogeher. Theorem 3. Under he above assumpions on f ξ. If C C 3 < e 34, hen for any erminal ime T, here exiss a posiive consan = δ wih =C 6 some fixed consan δ 0, such ha he SDE { dy = Z f Y,Z d+ Z dw 3.8 = ξ has a pair of soluiony,z S C d d k on0,t]. Y T Proof. Le λ 0, be a posiive consan o be deermined laer. We firsly consider he ime inerval T δt,t] as above assume ha which implies ha C C 3 < e we se consan o be C C 3 < e 4 λ 3.9 =C

7 We may have he following by choosing δ small enough. C C +C 4 δt C δt C3 λ. 3. We can do his because we have condiion 3.9 in equaion 3.7: C 6 = C ] C C 3 lnc C 3 + αc C +C 4 δt which implies ha C3 C 6 = C where α is deermined by inequaliy 3.5: C lnc C 3 + αc C +C 4 δt K C 3 β C C +C 4 0 α which can be achieved when C C 3 < e. Le SC denoe he anach space S C d d k. Since C C 3 < e which is due o condiion 3.9, we have Φ : SC SC as defined above. Then for any pair Y,Z SC we can ge Ỹ, Z = ΦY,Z wih Ỹ, Z SC. y Proposiion we have ha Z C 6+ Z. Togeher wih condiion 3.0 we ge ha Z + Z, which implies ha Ỹ, Z SC. So ha Φ : S C S C is well defined. For any Y,Z, Y,Z SC, we se Y, Z =Φ Y,Z Y, Z =Φ Y,Z. So we have Y, Z, Y, Z S C. Then by seing = Y Y, = Y Y, Λ=Z Z, Λ= Z Z, we ge ha,λ,, Λ S C wih T =0 we also ge Then by Io s formula we have d i = Λ i, j dw j + f j Y,Z Λ i, j d j + j j f j Y,Z f j Y,Z ] Z i, j d. d = ϑ T f Y,Z d+ ρ T f Y,Z f Y,Z ] d + Λ d+ ϑ T dw, 3. where he componens of vecors ϑ ρ are defined as ], ϑ j = i i Λ i, j, ρ j = i i Z i, j, 7

8 ϑ T dw is a maringale. Then by aking condiional expecaion we ge ˆ T + E F Λ ] ˆ T ds = E F ϑ T f Y,Z ] ds ˆ T E F ρ T f Y,Z f Y,Z ] ] ds, which implies ha ˆ T + E F Λ ] ˆ T ds E F ϑ ] f Y,Z ds ˆ T + E F ρ f Y,Z f Y,Z ds ]. Togeher wih he definiion of ϑ ρ, we obain from he inequaliy above ha ˆ T + E F Λ ] ˆ T ds E F Λ f Y,Z ] ds ˆ T + E F Z f Y,Z f Y,Z ] ds for all T δt,t]. Now by using he assumpions on f, we conclude ha ˆ T ˆ T ] + E F Λ s ds ] E F ] C C +C 4 +C 3 Z s s Λ s ds ˆ T + E F C s +C 3 Λ s ] s Z s ds C C +C 4 δt S Λ + C 3 Z S Λ + C δt Z S S + C 3 Z S Λ, from which we deduce ha S C C +C 4 δt Λ + C 3 Z Λ C δt Z S + C 3 Z Λ, Λ C C +C 4 δt S Λ + C 3 Z S Λ C δt Z S S + C 3 Z S Λ. Thus we have ha S C C +C 4 δt Λ + C 3 Λ C δt S + C 3 Λ, Λ C C +C 4 δt S Λ + C 3 S Λ C δt S S + C 3 S Λ. 8

9 Then by condiion 3. we ge ha S λ Λ + S + Λ 3.7 Λ λ S Λ + S + Λ. 3.8 So by subsiuing S in 3.8 wih 3.7 we have Λ 8λ Λ + S + Λ, which implies ha Λ 3λ Λ + S + Λ. y combining wih 3.7 we deduce ha S + Λ 5λ Λ + 4λ S + Λ. If λ < 5 we have ha S + Λ 4λ 5λ S + Λ. 3.9 If we can choose λ < 9 4λ so ha 5λ <. Then Φ : S C S C is a conracion map. y he anach s fixed poin heorem, we deduce ha here exiss a unique pair of soluion Y,Z SC on he ime inerval T δt,t] o he SDE 3.8 resriced on S C. Therefore wha lef o be shown is ha here exiss λ < 9 such ha assumpion 3.9 holds, his 4 can be achieved when C C 3 < e 9 = e 34. We hen consider he ime inerval T δt,t δt] if T δt > 0 0,T δt] oherwise se erminal value ξ a ime T δt o be Y T δt which is he iniial value of he soluion Y solved above on he ime inerval T δt,t]. Then by using he above same mehod, we ge a unique pair of soluion in SC on he ime inerval T δt,t δt] o he SDE 3.8. y repeaing his procedure backwards pasing he soluions on all he ime inervals ogeher we ge a pair of soluion Y,Z SC d d k wih = δ on he ime inerval 0,T] o he SDE 3.8. If δ λ saisfy he condiion ha δ λ < 9, which may be achievable when T C C 3 are small enough. Then he pair of soluion Y,Z S C d d k on he ime inerval 0,T] o he SDE 3.8 is unique. This uniqueness of he soluion can be proved as follows. Suppose here exis wo pairs of soluions Y,Z, Y,Z S C d d k on he ime inerval 0,T] o he SDE 3.8. y seing = Y Y,Λ=Z Z, we ge ha,λ SC d d k wih T =0. Then on he ime inerval T δt,t] by repeaing he procedure saring from equaion 3., we obain an inequaliy which is similar o inequaliy 3.9 as follows: 4 δ S + Λ λ S + Λ. 5 δ λ 9

10 Since λ < so ha 4 9 δ δ λ 5 δ λ <, we deduce ha S = 0 Λ = 0. Thus Y,Z equals Y,Z on he ime inervalt δt,t], in paricular YT δt equals Y T δt. We hen consider he ime inerval T δt,t δt] if T δt > 0 0,T δt] oherwise, erminal values a ime T δt are YT δt Y T δt respecively for he wo soluions. y using he same procedure saring from equaion 3., we deduce ha Y,Z equals Y,Z on he ime inerval T δt,t δt] as well. Thus by repeaing his procedure backwards, we conclude ha Y,Z equals Y,Z on he ime inerval0,t]. emark 4. Theorem 3 says ha, given parameers C,C 3,C 4, if he bound C of he erminal value is small enough hen here exiss a pair of soluion Y,Z S d d k on he ime inerval 0,T] o he SDE 3.8. eferences ] M.Kobylanski. ackward sochasic differenial equaions parial differenial equaions wih quadraic growh. The Annals of Probabiliy, Vol.8, No., ] P.ri Y.Hu. SDE wih quadraic growh unbounded erminal value. Probabiliy Theory elaed Fields, 36, No.4, ] P.ri Y.Hu. Quadraic SDEs wih convex generaors unbounded erminal condiions. Probabiliy Theory elaed Fields, 4, ] F.Delbaen, Y.Hu A.ichou. On he uniqueness of soluions o quadraic SDEs wih convex generaors unbounded erminal condiions. Probabiliés e Saisiques, Vol.47, No., ] P.arrieu N.El Karoui. Monoone sabiliy of quadraic semimaringales wih applicaions o general quadraic SDEs. The Annals of Probabiliy, Vol 4, ] P.ri.Elie. A simple consrucive approach o quadraic SDEs wih or wihou delay. Sochasic Processes heir Applicaions, 38, ] J.P. Lepelier J. San Marin. ackward sochasic differenial equaions wih coninuous coefficiens. Saisics & Probabiliy Leers, 3, ] Mark H.A. Davis. Advances in Conrol, Communicaion Neworks, Transporaion Sysems: In Honor of Pravin Varaiya, chaper Maringale represenaion all ha. Sysems Conrol: Foundaions Applicaions. irkhauser, ].Tevzadze. Solvabiliy of backward sochasic differenial equaions wih quadraic growh. Sochasic Processes heir Applicaions, 83, ] N.Kazamaki. Coninuous exponenial maringales MO, Lecure Noes in Mahemaics, 579. Springer-Verlag, erlin 994. ] G.Liang, A.Lionne Z.Qian. 00. On Girsanov s ransform for backward sochasic differenial equaions. Preprin arxiv: ] Y.Hu S.Tang. Muli-Dimensional ackward Sochasic Differenial Equaions of Diagonally Quadraic Generaors. Sochasic Processes heir Applicaions, 64,

11 3] G.Liang, T.Lyons Z.Qian. ackward sochasic dynamics on a filered probabiliy space. Annals of probabiliy, 394, 4-448, 0. 4] E.Pardoux S.G.Peng. Adaped soluion of a backward sochasic differenial equaion. Sysems & Conrol leers, 4, ] S. N. Cohen. J. Ellio. Sochasic Calculus Applicaions. Springer-Verlag 05.