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1 Thi aicle wa downloaded by: [Yao, Song] On: 7 Januay 211 Acce deail: Acce Deail: [ubcipion numbe ] Publihe Taylo & Fanci Infoma Ld Regieed in England and Wale Regieed Numbe: Regieed office: Moime Houe, Moime See, London W1T 3JH, UK Sochaic Analyi and Applicaion Publicaion deail, including inucion fo auho and ubcipion infomaion: hp:// On Quadaic g-evaluaion/expecaion and Relaed Analyi Jin Ma a ; Song Yao b a Depamen of Mahemaic, Pudue Univeiy, We Lafayee, Indiana, USA b Depamen of Mahemaic, Univeiy of Michigan, Ann Abo, Michigan, USA Online publicaion dae: 1 June 21 To cie hi Aicle Ma, Jin and Yao, Song(21) 'On Quadaic g-evaluaion/expecaion and Relaed Analyi', Sochaic Analyi and Applicaion, 28: 4, To link o hi Aicle: DOI: 1.18/ URL: hp://dx.doi.og/1.18/ PLEASE SCROLL DOWN FOR ARTICLE Full em and condiion of ue: hp:// Thi aicle may be ued fo eeach, eaching and pivae udy pupoe. Any ubanial o yemaic epoducion, e-diibuion, e-elling, loan o ub-licening, yemaic upply o diibuion in any fom o anyone i expely fobidden. The publihe doe no give any waany expe o implied o make any epeenaion ha he conen will be complee o accuae o up o dae. The accuacy of any inucion, fomulae and dug doe hould be independenly veified wih pimay ouce. The publihe hall no be liable fo any lo, acion, claim, poceeding, demand o co o damage whaoeve o howoeve caued aiing diecly o indiecly in connecion wih o aiing ou of he ue of hi maeial.

2 Sochaic Analyi and Applicaion, 28: , 21 Copyigh Taylo & Fanci Goup, LLC ISSN pin/ online DOI: 1.18/ On Quadaic g-evaluaion/expecaion and Relaed Analyi JIN MA 1 AND SONG YAO 2 Downloaded By: [Yao, Song] A: 12:5 7 Januay Depamen of Mahemaic, Pudue Univeiy, We Lafayee, Indiana, USA and Depamen of Mahemaic, Univeiy of Souhen Califonia, Lo Angele, Califonia, USA 2 Depamen of Mahemaic, Univeiy of Michigan, Ann Abo, Michigan, USA In hi aicle we exend he noion of g-evaluaion, in paicula g-expecaion, of Peng [8, 9] o he cae whee he geneao g i allowed o have a quadaic gowh (in he vaiable z ). We how ha ome impoan popeie of he g-expecaion, including a epeenaion heoem beween he geneao and he coeponding g- expecaion and conequenly he evee compaion heoem of quadaic BSDE a well a he Jenen inequaliy emain ue in he quadaic cae. Ou main eul alo include a Doob Meye ype decompoiion, he opional ampling heoem, and he upcoing inequaliy. The eul of hi aicle ae impoan in he fuhe developmen of he geneal quadaic nonlinea expecaion (cf. [5]). Keywod BMO; Doob Meye decompoiion; Jenen inequaliy; Opional ampling; Quadaic g-evaluaion; Quadaic g-expecaion; Revee compaion heoem; Upcoing inequaliy. Mahemaic Subjec Claificaion Pimay 6G48; Seconday 6H1, 91B3. 1. Inoducion In hi aicle we exend he noion of g-evaluaion, inoduced by Peng [9], o he cae when he geneao g i allowed o have quadaic gowh in he vaiable z. Thi will include he o-called quadaic g-expecaion a a pecial cae, a wa in he linea gowh cae iniiaed in [8]. The noion of g-expecaion, a a nonlinea Received Augu 4, 29; Acceped Decembe 22, 29 We would like o expe ou incee gaiude o he anonymou efeee fo he caeful eading of ou oiginal manucip and many valuable uggeion ha helped u impove he qualiy of he pape ignificanly. J.M. i uppoed in pa by NSF gan # Adde coepondence o Jin Ma, Depamen of Mahemaic, Univeiy of Souhen Califonia, 362 S. Vemon Ave., KAP 18, Lo Angele, CA 989, USA; jinma@uc.edu 711

3 712 Ma and Yao Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 exenion of he well-known Gianov anfomaion and oiginally moivaed by heoy of expeced uiliy, ha been found o have diec elaion wih a faily lage cla of ik meaue in finance. When he nonlinea expecaion i allowed o have poible quadaic gowh, i i expeced ha i will lead o he epeenaion heoem ha chaaceize he geneal convex, bu no neceaily coheen ik meaue in em of a cla of quadaic BSDE. The mo noable example of uch ik meaue i he enopic ik meaue (ee, e.g., [1]), which i known o have a epeenaion a he oluion o a quadaic BSDE bu fall ouide he exiing heoy of he filaion-conien nonlinea expecaion [3], which equie ha he geneao be only of linea gowh. We efe he eade o [2, 3, 8], and he expoioy aicle [9] fo moe deailed accoun fo baic popeie of g- evaluaion and g-expecaion, a well a he elaionhip beween he ik meaue and g-expecaion. A bief eview of he baic popeie of g-evaluaion and g- expecaion will be given in Secion 2 fo eady efeence. The main pupoe of hi aicle i o inoduce he noion of quadaic g- evaluaion and g-expecaion, and pove ome of he impoan popeie ha ae deemed a eenial. In an accompanying aicle [5], we hall fuhe exend he noion of filaion conien nonlinea expecaion o he quadaic cae, and eablih he ulimae elaion beween a convex ik meaue and a BSDE. The main eul in hi aicle include he Doob Meye decompoiion heoem, opional ampling heoem, upcoing inequaliy, and Jenen inequaliy. We alo pove ha he quadaic geneao can be epeened a he limi of he diffeence quoien of he coeponding g-evaluaion, exending he eul in linea gowh cae [2]. Wih he help of hi eul, we can hen pove he o-called eveed compaion heoem, a in he linea cae. Alhough mo of he eul peened in hi aicle look imila o hoe in he linea cae, he echnique involved in he poof ae quie diffeen. We combine he echnique ued in he udy fo quadaic BSDE, iniiaed by Kobylanki [7] and he by now well-known popeie of he BMO maingale. Since many of hee eul ae ineeing in hei own igh, we ofen peen full deail of poof fo fuue efeence. Thi aicle i oganized a follow. In Secion 2 we give he peliminaie, and eview he exiing heoy of g-evaluaion/expecaion and BMO maingale. In Secion 3 we define he quadaic g-evaluaion and dicu i baic popeie. Some fine popeie of g-evaluaion/expecaion ae peened in Secion 4. Thee include a epeenaion of quadaic geneao via quadaic g-evaluaion, a evee compaion heoem of quadaic BSDE, and he Jenen inequaliy. In Secion 5 we pove he main eul of hi pape egading he quadaic g- maingale: a Doob Meye ype decompoiion, he opional ampling heoem, and he upcoing inequaliy. 2. Peliminaie Thoughou hi aicle we conide a fileed, complee pobabiliy pace F PF on which i defined a d-dimenional Bownian moion B. We aume ha he filaion F = F i geneaed by he Bownian moion B, augmened by all P-null e in F, o ha i aifie he uual hypohee (cf. [1]). We denoe o be he pogeively meauable -field on T; and M T o be he e

4 Quadaic g-expecaion 713 Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 of all F-opping ime uch ha T, P-a.., whee T> i ome fixed ime hoizon. In wha follow, we fix a finie ime hoizon T>, and denoe o be a geneic Euclidean pace, whoe inne poduc and nom will be denoed by and, epecively; and denoe o be a geneic Banach pace wih nom. Moeove, he following pace of funcion will be fequenly ued in he equel. Le be a geneic ub--field of F, we denoe fo p, L p o be all -valued, -meauable andom vaiable, wih E p <. In paicula, if p =, hen L denoe he pace of all -valued, -meauable andom vaiable; and if p =, hen L denoe he pace of all -valued, -meauable andom vaiable uch ha = eup < ; p, L p F T o be all -valued, F-adaped pocee, uch ha E p d <. In paicula, p = and fo all -valued, F- adaped pocee; and p = denoe all pocee X L F T uch ha X = eup X < ; F T = X L F T Xha càdlàg pah; F T = X F T Xha coninuou pah; HF 2 T = X L2 F T Xi pedicably meauable. Finally, if d = 1, we hall dop = fom he noaion (e.g., L p F T = L p F T, L F T = L F T, and o on) g-evaluaion and g-expecaion We fi ecall he noion of g-evaluaion inoduced in Peng [9]. Given a ime duaion T, and a geneao g = g y z T d aifying he andad condiion (e.g., i i Lipchiz in all paial vaiable, and i of linea gowh, ec.), conide he following BSDE on, T: Y = + g Y Z d Z db (2.1) whee L 2 F. Denoe he unique oluion by Y Z. The g-evaluaion i defined a he family of opeao { E g L 2 F L 2 F } uch ha fo any T T, E g = Y,. In paicula, fo any L 2 F T, i g-expecaion i defined by E g = Y T, and i condiional g-expecaion i defined by E g F = E g T, fo any T. We hall denoe (2.1) by BSDE g in he equel fo noaional convenience. Remak 2.1. An impoan ingedien in he definiion of g-evaluaion i i domain, namely he ube in L F T on which he opeao i defined (in he cuen cae being naually aken a L 2 F T ). The domain of a g- evaluaion/expecaion may vay a he condiion on he coefficien change, due o he eicion on he well-poedne of he BSDE (2.1). Fo example, owing o he naue of quadaic BSDE, in he e of hi aicle we hall chooe L F T a he domain fo quadaic g-evaluaion. We efe o ou accompanying aicle [5] fo a moe deailed dicuion on he iue of domain fo geneal nonlinea expecaion.

5 714 Ma and Yao By viue of he uniquene of he oluion Y Z, one can how ha he g-evaluaion E g ha he following popeie: (1) (Monooniciy) Fo any[ L 2 F wih, P-a.., E g E, g P-a..; (2) (Time-Coniency) E g E g ] = E, g P-a.., L 2 F, T; (3) (Conan-Peeving) E g =, P-a.., L F, if i hold d dp-a.. ha g y = y (2.2) (4) ( Zeo-One Law ) Fo any L 2 F and any A F,, i hold ha 1 A E g = 1 AE g 1 A P-a. Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 Moeove, if g =, d dp-a.., hen 1 A E g = E1 g A, P-a..; (5) (Tanlaion Invaiance) Aume ha g i independen of y, hen fo any L 2 F and L 2 F, i hold ha E g + = E g +, P-a.. Clealy, if g aifie (2.2), hen one can deduce fom (2) and (3) above ha E g F = E g T = E g [ E g T ] = E g P-a. L2 F T (2.3) and he condiional g-expecaion E g F poee he following popeie ha moe o le juify i name (auming (2.2) fo (2a) and (3a) below): (1a) (Monooniciy) Fo any L 2 F T wih, P-a.., E g F E g F, P-a..; (2a) (Time-Coniency) E g[ E g F ] F = E g F, P-a.., L 2 F T, ; (3a) (Conan-Peeving) E g F =, P-a.., L 2 F ; (4a) (Zeo-One Law) Fo any L 2 F T and A F, i hold ha 1 A E g 1 A F = 1 A E g F, P-a..; Moeove, if g =, d dp-a.., hen 1 A E g F = E g 1 A F, P-a..; (5a) (Tanlaion Invaiance) Aume ha g i independen of y, hen fo any L 2 F T and L 2 F i hold ha E g + F = E g F +, P-a BMO Maingale and BMO Pocee An impoan ool fo udying he quadaic BSDE, whence he quadaic g-expecaion, i he o-called BMO maingale and he elaed ochaic exponenial (ee, e.g., [4]). We efe o he monogaph of Kazamaki [6] fo a complee expoiion of he heoy of coninuou BMO and exponenial maingale. In wha follow, we li ome of he impoan fac ha ae ueful in ou fuue dicuion fo eady efeence. To begin wih, we ecall ha a unifomly inegable maingale M null a zeo i called a BMO maingale on T if fo ome 1 p<, i hold ha M BMOp = up M T EM T M p F 1/p < (2.4) In uch a cae we denoe M BMOp. I i impoan o noe ha M BMOp if and only if M BMO1, and all he BMOp nom ae equivalen (cf. [6]).

6 Quadaic g-expecaion 715 Theefoe, in wha follow we ay ha a maingale M i BMO wihou pecifying he index p; and we hall ue only he BMO2 nom and denoe i imply by BMO. Noe alo ha fo a coninuou maingale M one ha M BMO =M BMO2 = up M T EM T M F 1/2 Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 Fo a given Bownian moion B, we ay ha a poce Z L 2 F T d i a BMO poce, denoed by Z BMO by a ligh abue of noaion, if he ochaic inegal M = Z B = Z db i a BMO maingale. Nex, fo a coninuou maingale M, he Doléan Dade ochaic exponenial of M, denoed cuomaily by EM, i defined a EM = expm 1 M 2,. If M i fuhe a BMO maingale, hen he ochaic exponenial EM i ielf a unifomly inegable maingale (ee [6, Theoem 2.3]). The heoy of BMO wa bough ino he udy of quadaic BSDE fo he following eaon. Conide, fo example, he BSDET g (ee (2.1)) whee he geneao g ha a quadaic gowh. Aume ha hee i ome k> (we may aume wihou lo of genealiy ha k 1 ) uch ha fo d dp-a.. 2 T, g y z k1 +z 2 y z d (2.5) and denoe Y Z F T H 2 F T d be a oluion of he BSDET g. Fo any M T, applying Iô fomula o e 4kY fom o T one ha e 4kY + 8k 2 e 4kY Z 2 d = e 4kY T + 4k e 4kY g Y Z d 4k e 4kY Z db e 4kY T + 4k 2 e ( 4kY 1 +Z 2) d 4k e 4kY Z db I i hen no had o deive, uing ome andad agumen, he following eimae: [ ] E Z 2 d F e 4kY E [ ] e 4k e 4kY F + e 8kY T (2.6) In ohe wod, we conclude ha Z BMO, and ha Z 2 BMO 1 + Te8kY (2.7) 3. Quadaic g-evaluaion on L F T Ou udy of he g-evaluaion/expecaion benefied gealy fom he echnique ued o ea he quadaic BSDE, iniiaed by Kobylanki [7]. We fi li ome eul egading he exience, uniquene, and compaion heoem fo he quadaic BSDE. Thoughou he e of he aicle we aume ha he geneao g in BSDE(T g) (2.1) ake he fom: g y z = g 1 y zy + g 2 y z y z T d

7 716 Ma and Yao and aifie he following Sanding Aumpion: (H1) Boh g 1 and g 2 ae B B d -meauable and boh g 1 and g 2 ae coninuou fo any T ; (H2) Thee exi a conan k> and an inceaing funcion l + +, uch ha fo d dp-a.. T, g 1 y z k and g 2 y z k + lyz 2 y z d (H3) Wih he ame inceaing funcion l, fo d dp-a.. T, g y z z ly1 +z y z d Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 (H4) Fo any >, hee exi a poiive funcion h L 1 T uch ha fo d dp-a.. T, g y y z h + z 2 y z d Unde he aumpion (H1) (H4), i i known (cf. [7, Theoem 2.3 and 2.6]) ha fo any L F T, he BSDE (2.1) admi a unique oluion Y Z F T H 2 F T d. In fac, hi eul can be exended o he following moe geneal fom, which will be ueful in ou fuue dicuion. Popoiion 3.1. Aume ha g aifie (H1) (H4). Fo any L F T and any V F T, he BSDE Y = + g Y Z d + V T V Z db T (3.1) admi a unique oluion Y Z F T H 2 F T d. Poof. We define a new geneao g by g y z = g y V z, y z T d. Then i i eay o ee ha fo any y z T d g 1 y z = g 1 y V z g 2 y z = g 2 y V z g 1 y V zv I can be eaily veified ha g alo aifie (H1) (H4). We can hen conclude (ee, [7]) ha he BSDET + V T g admi a unique oluion ỸZ F T HF 2 T d. Bu hi amoun o aying ha Ỹ V Z i he unique oluion of (3.1), poving he coollay. Popoiion 3.1 indicae ha if g aifie (H1) (H4), hen we can again define a g-evaluaion E g L F L F fo T, a in he peviou ecion. We hall name i a he quadaic g-evaluaion/expecaion fo obviou eaon. Moe geneally, fo any, M T uch ha, P-a.., we can define he

8 Quadaic g-expecaion 717 quadaic g-evaluaion E g L F L F by E g = Y, whee L F, and Y aifie he BSDE: Y = + 1 < g Y Z d Z db T (3.2) wih Z HF 2 T d, and Y = Y and Z = 1 < Z, P-a. In paicula, if = T, we define he quadaic g-expecaion of fo any M T by E g F = E g T. We noe ha, imila o he deeminiic-ime cae, E g ha he following popeie: (1) Time-Coniency: Fo any,, M T wih, P-a.., we have Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 [ E g E g ] = E g P-a. L F (2) Conan-Peeving: Aume (2.2), E g =, P-a.., L F ; (3) Zeo-One Law : Fo any L F and A F, we have 1 A E g 1 A = 1 A E g, P-a.; Moeove, if g =, d dp-a.., hen Eg 1 A = 1 A E g, P-a..; (4) Tanlaion Invaian : If g i independen of y, hen E g + = Eg + P-a. L F L F (5) Sic Monooniciy: Fo any L F wih, P-a.., we have E g E g, P-a..; Moeove, if Eg = Eg, P-a.., hen =, P-a.. We emak ha he la popey (5) above i no compleely obviou. In fac hi will be a conequence of o-called ic compaion heoem fo quadaic BSDE, a enghened veion of he uual compaion heoem (ee, fo example, [7, Theoem 2.6]). Fo compleene we hall peen uch a veion, unde he following condiion ha ae imila o hoe in [7], bu lighly weake han (H1) (H4). (A1) g i B B d -meauable and g i coninuou fo any T ; (A2) Fo any M>, hee exi l L 1 T k L 2 T and C> uch ha fo d dp-a.. T and any y z M M d, g y z l + Cz 2 and g y z z k + Cz (A3) Fo any >, hee exi a poiive funcion h L 1 T uch ha fo d dp-a.. T and any y z d, g y y z h + z 2 Theoem 3.2. Aume (A1) (A3). Le 1 2 L F T and V i, i = 1 2 be wo adaped, inegable, igh-coninuou pocee null a. Le ( Y ) izi F T

9 718 Ma and Yao H 2 F T d, i = 1 2 be oluion o he BSDE: Y i = i + g Y i Zi d + dv i Z i db T i = 1 2 epecively. If 1 2, P-a.. and V 1 V 2 i inceaing, hen i hold P-a.. ha Y 2 T (3.3) Y 1 Moeove, if Y 1 = Y 2 fo ome M T, hen i hold P-a.. ha 1 = 2 and V 1 T V 2 T = V 1 V 2 (3.4) Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 Poof. I i no had o ee ha (3.3) i a mee genealizaion of [7, Theoem 2.6], hu we only need o pove (3.4). Le M =Y 1 +Y 2, and define = 1 2 fo = Y, Z, V, epecively. Then Y aifie: whee dy = ( g Y 1 Z1 2 g Y Z2 ) d dv + Z db 1 ( g = y Y + g ) z Z dd dv + Z db = a Y d dv + Z b d + db (3.5) = Y + Y 2 Z + Z 2, and 1 g 1 a = y d and b g = z d T Noe ha Y + Y 2 M, T, P-a.., by uing ome andad agumen wih he help of aumpion (A1) (A3) a well a he Bukholde Davi Gundy inequaliy we deduce fom (3.5) ha { } E up a d + up b db < (3.6) T T { Define Q = exp a d 1 b 2 2 d + b } db,, and n = inf { T Q >n } T n we ee ha n T, P-a.., and (3.6) indicae ha hee exi a null e uch ha fo each c, T = m fo ome m. On he ohe hand, fo any n, inegaing by pa on n yield ha Q n Y n n = Q Y = n + n Q Z db + n Q Y a d n Y Q a d + n Q dv + Q Z db + n Q Z b d n n Q dv Y Q b db + Y Q b db n Q Z b d

10 Quadaic g-expecaion 719 Taking expecaion on boh ide give: { n } E Q n Y n + Q dv = which implie ha hee exi a null e n uch ha fo any c n {, i hold ha Y n = and V n = V. Theefoe, fo any ( ) } c n n, one ha Y T = and V T = V Thi complee he poof. Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 In mo of he dicuion below, we aume he geneao g aifie (H1) (H4) (hence (A1) (A3)). We fi exend a popey of g-expecaion [2, Popoiion 3.1] o he cae of quadaic g-evaluaion. Popoiion 3.3. Aume (H1) (H4). Aume fuhe ha he geneao g i deeminiic. Fo any T and L F,ifi independen of F fo ome, hen he andom vaiable E g i deeminiic. Poof. Le < T be uch ha L F and ha i i independen of F. I uffice o how ha E g = c, P-a.. fo ome conan c. To ee hi, fo any, we define B = B + B, F = ( B u u ), and F = F. Clealy, B i an F -Bownian moion on. Since F i independen of F, one can eaily deduce ha F. Now we denoe by Y Z he unique oluion o he BSDE: Y = + g + u Y u Z u du Z u db u The imple change of vaiable = v and w = + u yield ha Y v = + gw Y w Z w dw Z w db w = + v v gw Y w Z w dw Z w db w v v v In ohe wod, Y v Z v v i a oluion o BSDE g on. The uniquene of he oluion o BSDE hen lead o ha Y v = Eg v v. In paicula, one ha E g = Y, P-a., which i a conan by he definiion of F and he Blumenhal -1 law, compleing he poof. A we can ee fom he dicuion o fa, o long a he coeponding quadaic BSDE i well-poed, he euling g-evaluaion/expecaion hould behave vey imilaly o hoe wih linea gowh geneao, wih almo idenical poof uing he popeie obained o fa. We heefoe conclude hi ecion by liing ome fuhe popeie of he g-evaluaion/expecion in one popoiion fo eady efeence, and leave he poof o he ineeed eade.

11 72 Ma and Yao Popoiion 3.4. Le g i, i = 1 2, be wo geneao boh aify (H1) (H4). 1) Suppoe ha g i =, i = 1 2, and ha E g 1 = Eg 2 T L F (3.7) hen fo any L F T, i hold P-a.. ha E g 1 T = E g 2 T, T. 2) Suppoe fuhe ha g i, i = 1 2 ae independen of y, Fo any T,ifE g 1 E g 2, L F, hen fo any L F, i hold P-a.. ha E g 1 E g 2,. To end hi ecion, we ae a abiliy eul of quadaic BSDE which i a ligh genealizaion of Theoem 2.8 in [7]. Since hee i no ubanial diffeence in he poof, we omi i. Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 Theoem 3.5. Le g n be a equence of geneao aifying (H1) and (H2) wih he ame conan k> and inceaing funcion l. Denoe, fo each n, Y n Z n F T H F 2 T d o be a oluion of BSDET n g n wih n L F T. Suppoe ha n i a bounded equence in L F T, and convege P-a.. o ome L F T ; and ha fo d dp-a.. T, g n y z convege o g y z locally unifomly in y z d wih g aifying (H1) (H4). Then BSDET g admi a unique oluion Y Z F T H F 2 T d uch ha P-a.. Y n convege o Y unifomly in T and ha Z n convege o Z in HF 2 T d. 4. Some Fine Popeie of Quadaic g-evaluaion In hi ecion we exend ome fine popeie of g-evaluaion o he quadaic cae. Thee popeie have been dicoveed fo diffeen eaon in he linea gowh cae, and hey fom an inegal pa of he heoy of nonlinea expecaion. In he quadaic cae, howeve, he poof need o be adjued, omeime ignificanly. We collec ome of hem hee fo he diinguihed impoance. We begin by a epeenaion heoem fo he geneao via quadaic g- expecaion. Theoem 4.1. Aume (H1) (H4). Le y z T d.ifg aifie (g1) lim y + y g y z= g y z, P-a.. and (g2) Fo ome T and ome >, hee exi an inegable poce h + uch ha fo d dp-a.. +, g y y z h y wih y y hen i hold P-a.. ha 1 ( g y z = lim E g + y + zb + B y ) whee = inf { >B B > 1+z} T.

12 Quadaic g-expecaion 721 Poof. We e M = 1 +y+ z, and M = km + 2l4Mz 2. By educing 1+z,we may aume ha M e k 1. 1+z 4l4M Fix ln 2 k. Since zb + B z, hee exi a unique 1+z oluion { Y Z } + F + H F 2 + d o he following BSDE: Y = y + zb + B < g Y Z d Z db + We know fom Coollay 2.2 of [7] ha Y y+ z 1+z + kek 2M. Now le Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 Ỹ = Y y zb B Z = Z 1 <z + I i eay o check ha Ỹ Z + i a oluion of he BSDE: + + Ỹ = g Ỹ Z d Z db + (4.1) wih g y z = y 1 < g ( y + y + zb B z + z ), y z + d whee 1 i an abiay C 1 funcion ha equal o 1 inide 3M 3M, vanihe ouide 3M 1 3M + 1 and aifie up 3M<x<3M+1 x 2. Fo any y z + d, we ee ha g y z = g 1 y z y + g 2 y z wih ( g 1 y z = y 1 < g 1 y + y + zb B z + z ) ( g 2 y z = y 1 < g 1 y + y + zb B z + z ) ( y + zb B ) + y 1 < g 2 ( y + y + zb B z + z ) One can eaily deduce fom (H2) and (H3) ha fo d dp-a.. +, i hold fo any y z d ha g 1 y z k (4.2) g 2 y z km + 2l4M ( z 2 +z 2) = M + 2l4Mz 2 (4.3) and g z y z l4m( 1 +z +z ) (4.4)

13 722 Ma and Yao Coollay 2.2 of [7] once again how ha Ỹ Me k M e k 1. Applying Iô fomula o Ỹ 4l4M 2 we obain ha 1+z Ỹ Z 2 d = 2 + Ỹ + g Ỹ Z d 2 Ỹ Z db + (4.5) Uing (4.2) (4.4) and ome andad manipulaion one deive eaily ha Downloaded By: [Yao, Song] A: 12:5 7 Januay = 2 2 Ỹ g Ỹ Z d Ỹ C g Ỹ d + 2 Ỹ ( kỹ + M ) d + 2l4M ( Ỹ Z 1 + g Ỹ z Z )d d Ỹ ( 2kỸ +2 M + l4m1 +z 2) d + 2l4M + Z 2 d + Ỹ Z ( 1 +z+ 1 2 Z )d + Ỹ Z 2 d whee C i a geneic conan depending on y z k and l4m, which may vay fom line o line. Taking he condiional expecaion E F on boh ide of (4.5) we have { + E Z 2 d } F C 2 + (4.6) Now, aking he condiional expecaion in he BSDE (4.1) we have 1 g = 1 { + Ỹ E ( g Ỹ Z g ) d } F = 1 { + [ 1 E Z g Ỹ z Z d + Ỹ 1 g Ỹ y ]d d + g g F } We know fom (g2) and (H4) ha fo d dp-a.. +, h g y y + y + zb B z h 1 +z 2 hold fo any y wih y. I follow ha fo d dp-a z, g y y = y 1 < g y + y + zb B z

14 Quadaic g-expecaion g y 1 < y y + y + zb B z 2k1 + 4M + ( 1 + 2l4M ) z 2 + h +h 1 = h Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 hold fo any y wih y. Clealy, h 1+z + i an inegable poce. Then applying (4.4), (4.6) and he Hölde Inequaliy we have 1 g Ỹ 1 { + [ ) ] } E l4m (1 +z Z +12 Z 2 +Ỹ h d F { E g g } d F C ( + ) [ + ] + Me k E h d F { E g g d F } (4.7) A lim + 1 < = 1 and lim +B B =, P-a.., one can deduce fom (g1) ha lim g = lim g y + zb + + B z = g y z = g which implie ha 1 lim + g g d = P-a. P-a. Since g M fo d dp-a.. +, Lebegue Convegence Theoem implie ha he igh hand ide of (4.7) convege P-a.. o a +. Theefoe, 1 g y z = g = lim + = lim Ỹ + 1 Y y 1 ( = lim E g + y + zb + B y ) P-a. whee (3.2) wa ued in he la equaliy. The poof i now complee. A imple applicaion of he heoem above give ie o a evee o he Compaion Theoem of quadaic BSDE: Theoem 4.2. Aume ha g i, i = 1 2 aify (H1) (H4) and (2.2). Le T.If E g 1 F E g 2 F, P-a.. fo any L F T, and if boh g i aify (g1) and (g2) fo any y z d, hen i hold P-a.. ha g 1 y z g 2 y z y z d We alo have he following coollay of Theoem 4.1.

15 724 Ma and Yao Popoiion 4.3. Aume ha g aifie (H1) (H4) and (2.2). We alo aume ha P- a.., g yz i coninuou fo any y z d.ifg aifie (g1) and (g2) fo any y z T d, hen g i independen of y if and only if E g + c = E g + c L F T c Poof:. : A imply applicaion of anlaion invaiance of quadaic g- expecaion. Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 : Fo any c, we define a new geneao g c y z = g y c z, y z T d. I i eay o check ha g c aifie (H1) (H4) a well a he ohe aumpion on g in hi popoiion. Fo any L F T, le Y Z denoe he unique oluion o BSDET g. Seing Ỹ = Y + c, T one obain ha Thu, i hold P-a.. ha Ỹ = + c + g c Ỹ Z d Z db E gc + cf = Ỹ = Y + c = E g F + c T T In paicula, aking = give ha E gc = E g fo any L F T. Since g aifie (2.2), i eay o ee ha he condiion (3.7) i aified fo g 1 = g and g 2 = g c. Hence, Popoiion 3.4 implie ha fo any L F T, i hold P-a.. ha E g F = E gc F, T. Applying Theoem 4.1 we ee ha fo any z T d, i hold P-a.. ha g c z = g c c z = g z. Then (H1) implie ha fo any T, i hold P-a.. ha g y z = g z, y z d. Evenually, by ou aumpion, i hold P-a.. ha g y z = g z, y z T d. Thi pove he popoiion. To end hi ecion we exend anohe impoan feaue of he g-expecaion o he quadaic cae: The Jenen inequaliy. We begin by ecalling ome baic fac fo convex funcion, and we efe o Rockafella [11] fo all he noion o appea below. Recall ha if F n i a convex funcion, hen by conideing he convex eal funcion f = Fx ( Fx + 1 F ),, wih f = f1 =, i i eay o check ha fo any x n, i hold ha { Fx Fx + 1 F if 1 Fx Fx + 1 F if 1 c (4.8) Nex, if F i a convex (eal) funcion, hen we denoe by F he ubdiffeenial of F (ee [11]). In paicula, fo any x, Fx i imply an ineval F x F + x, whee F and F + ae lef-, and igh-deivaive of F, epecively. The following eul i an exenion of he linea gowh cae (cf. [2, Popoiion 5.2]).

16 Quadaic g-expecaion 725 Theoem 4.4. Aume ha g i independen of y and aifie (H1) (H4) and (2.2). Le T.Ifg z i convex in z fo d dp-a.. T, hen F ( E g F ) E g FF P-a. fo any L F T wih F ( E g F ) 1 c, P-a.. Poof. Since boh F x and F + x ae nondeceaing funcion, we can define anohe non-deceaing funcion: x = 1 F x F x + 1 F x> F + x x Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 Thu, = ( E g F ) i an F -meauable andom vaiable. Since x 1 c fo any x wih Fx 1 c, i follow ha One can deduce fom he convexiy of F ha 1 c P-a. (4.9) ( E g F ) F F ( E g F ) (4.1) Since L F T, i i clea ha F, E g F, F ( E g F ) ( a well a E g F ) ae all of L F T. Taking E g F on boh ide of (4.1), and uing Tanlaion Invaiance of quadaic g-expecaion we have E g F E g F = E g[ ( E g F ) ] F E g[ F F ( E g F ) ] F = E g FF F ( E g F ) P-a. Hence, i uffice o how ha E g F E g F, P-a.. To ee hi, le Y = E g F, T.A F, one ha Y = + g Z d Z db T Since g i convex and aifie (2.2), uing (4.8) and (4.9) we obain Y + g Z d Z db = E g F T In paicula, we have E g F E g F, P-a., poving he heoem. 5. Main Reul In hi ecion we pove he main eul of hi pape egading he quadaic g- maingale. To begin wih, we give he following definiion. Recall ha E g, T denoe he g-evaluaion.

17 726 Ma and Yao Definiion 5.1. An X L F T i called a g-ubmaingale (ep. g- upemaingale) if fo any T, i hold ha E g X ep. X P-a. X i called a g-maingale if i i boh a g-ubmaingale and a g-upemaingale. Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 We hould noe hee ha, in he above he maingale i defined in em of quadaic g-evaluaion, inead of quadaic g-expecaion a we have uually een. Thi ligh elaxaion i meely fo convenience in applicaion. I i clea, howeve, ha if g aifie (2.2), hen he quadaic g-maingale defined above hould be he ame a he one defined via quadaic g-expecaion, hank o (2.3). We hall exend hee main eul fo g-expecaion o he quadaic cae: he Doob Meye decompoiion, he opional ampling heoem, and he upcoing heoem. Alhough he eul look imila o he exiing one in he g-expecaion lieaue, he poof ae moe involved due o he pecial naue of he quadaic BSDE. We peen hee eul epaaely. We begin by poving a Doob Meye ype decompoiion heoem fo g- maingale. Theoem 5.2 (Doob Meye Decompoiion Theoem). Aume (H1) (H4). Le Y be any g-ubmaingale (ep. g-upemaingale) ha ha igh-coninuou pah. Then hee exi a càdlàg inceaing (deceaing) poce A null a and a poce Z HF 2 T d uch ha Y = Y T + g Y Z d A T + A Z db T Poof. We fi aume ha Y i a g-ubmaingale. Se M = Y + kte kt and K = lm + 1, wele 1 be any C 2 funcion ha equal o 1 inide [ e 2KM e 2KM] and vanihe ouide ( e 2KM+1 e 2KM+1). Le u conuc a new geneao: Fo any y z T d, [ ( g y z = ln y y 2Ky g 2K z 2Ky ) ] z2 2y One can deduce fom (H2) ha fo d dp-a.. T, g y z 2M + 2kKyy y z d Since 2M + 2kKyy i Lipchiz coninuou in y, we can conuc (cf. [7]) a deceaing equence g n y z of geneao unifomly Lipichiz in y z uch ha P-a.. g n y z g y z y z T d Now fix T, fo any L F wih Y, we define y = E g,. I follow fom [7, Coollay 2.2] ha y Y + kte kt = M.

18 Quadaic g-expecaion 727 Applying Iô fomula we ee ha ỹ = e 2Ky, ogehe wih a poce z H 2 F d i a oluion of he following BSDE: ỹ = e 2K + g ỹ z d z db Since g n i Lipchiz, a andad compaion heoem implie ha e 2KEg =ỹ E g n e 2K P-a. In paicula, aking = Y how ha e 2KY e 2KEg Y E g n e 2KY P-a. Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 Namely, Ỹ = e 2KY i a igh-coninuou g n -ubmaingale in he ene of g n - evaluaion fo any n. Applying he known g-ubmaingale decompoiion heoem fo he Lipchiz cae (ee [9, Theoem 3.9]), we can find a càdlàg inceaing poce A n null a and a poce Z n H 2 F T d uch ha Ỹ = Ỹ T + g n Ỹ Z n d An T + An Z n db T (5.1) fom which we ee ha Ỹ, whence Y i càdlàg. Noe ha, in he epeenaion (5.1), he maingale pa mu coincide fo any m and n. In ohe wod, one mu have Z m = Z n a he elemen in H 2 F T d. Thu, fo any n, (5.1) can be ewien a Ỹ = Ỹ T + g n Ỹ Z d A n T + An Z db T Since g n g, he Lebegue Convegence Theoem implie ha Conequenly, i hold P-a.. ha [ gn Ỹ Z g Ỹ Z ] d P-a. A n à = Ỹ Ỹ + g Ỹ Z d Z db T I i eay o check ha à i alo a càdlàg inceaing poce null a. Now le u define a new C 2 funcion by y = y ln y, y. Applying Iô fomula o 2K Ỹ fom o T one ha Y = Y T ] [ g Ỹ Z d dã Z db 2KỸ Z 2 2KỸ 2 d T { Y Ỹ 2KỸ }

19 728 Ma and Yao = Y T + = Y T + g 1 [ g Ỹ Z d dã c 2KỸ ] 1 Z db + 2 ( ) Z Y 1 d dã c 2KỸ 2KỸ Z 2 d Y 2KỸ 2 T Z 2KỸ db T Y whee he econd equaliy i due o he fac ha Ỹ = à > and à c denoe he coninuou pa of Ã. Clealy, A = 1 dã c 2KỸ + Y i a càdlàg inceaing poce null a, finally we ge Y = Y T + g Y Z d A T + A Z db T Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 On he ohe hand, if Y i a g-upemaingale, hen one can eaily check ha Y i coepondingly a g -ubmaingale wih g y z = g y z y z T d (5.2) Clealy, g alo aifie (H1) (H4), hu hee exi a càdlàg inceaing poce A null a and a poce Z H 2 F T d uch ha Y = Y T + g Y Z d A T + A Z db We can ewie hi BSDE a: Y = Y T + g Y Z d A T + A Z db T T The poof i now complee. We now un ou aenion o he opional ampling heoem. We begin by peening a lemma ha will play an impoan ole in he poof of he opional ampling heoem. Lemma 5.3. Le M T be finie valued in a e = < 1 < < n = T. If i < i+1 fo ome i 1n 1, hen fo any F E g = 1 i + 1 i+1 E g P-a. (5.3) Poof. Fo any F, le Y Z be he unique oluion o he BSDE (3.2) wih =. Then we have E g = Y = + = + 1 u< gu Y u Z u du 1 u< gu Y u Z u du 1 u< Z u db u 1 u< Z u db u

20 Quadaic g-expecaion 729 Fo any, ince i = i+1 c F i F, one can deduce ha 1 i Y = 1 i + 1 i 1 u< gu Y u Z u du 1 i 1 u< Z u db u = 1 i (5.4) and ha 1 i+1 Y = 1 i+1 + = 1 i i+1 1 u< gu Y u Z u du 1 i+1 gu Y u Z u du 1 i+1 1 u< Z u db u 1 i+1 Z u db u (5.5) Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 On he ohe hand, we le Y = Eg,. Then fo any, by he definiion of quadaic g-evaluaion, one ha 1 i Y = 1 i + 1 i gu Y u Z u du 1 i Z u db u (5.6) Adding (5.6) o (5.5) how ha Ỹ = 1 i+1 Y + 1 i Y and Z = 1 i+1 Z + 1 i Z olve he following BSDE Ỹ = + gu Ỹ u Z u du Z u db u Then i i no had o check ha Ŷ = 1 Ỹ + 1 < E g Ỹ i he unique oluion of BSDE g. Hence we can ewie Ŷ = E g,. In paicula, i hold P-a.. ha 1 i+1 Y = 1 i+1 Ỹ = 1 i+1 Ŷ = 1 i+1 E g (5.7) Leing = in (5.4) and hen adding i o (5.7), he lemma follow. We ae now eady o pove he opional ampling heoem. Theoem 5.4. Aume (H1) (H4). Fo any g-ubmaingale X (ep., g- upemaingale, g-maingale) uch ha eup up T X <, and fo any, M T wih, P-a.. Aume eihe ha and ae finiely valued o ha X i igh-coninuou, hen E g X ep. =X P-a. Poof. We hall conide only he g-ubmaingale cae, a he ohe cae can be deduced eaily by andad agumen. To begin wih, we aume ha ake value in a finie e = < 1 < < n = T. Noe ha if n, hen i i clea ha E X g = E g X = X, P-a.. We can hen ague inducively ha fo any T, E g X X P-a. (5.8)

21 73 Ma and Yao In fac, aume ha fo ome i 1n, (5.8) hold fo any i. Then fo any i 1 i, he ime-conience and he monooniciy of quadaic g-evaluaion a well a (5.3) imply ha E g X [ = E g i E g i X ] E g i X i = 1 i 1 X i + 1 i E g i X i = 1 i 1 X + 1 i E g i X i P-a. Since i = i 1 c F, he zeo-one law of quadaic g-evaluaion how ha Downloaded By: [Yao, Song] A: 12:5 7 Januay i E g i X i = 1 i E g i 1 i X i = 1 i E g i 1 i X i = 1 i E g i X i 1 i X = 1 i X P-a. Hence, (5.8) hold fo any i 1, hi complee he inducive ep. If i alo finiely valued, fo example in he e = < 1 < < m = T, hen i hold P-a.. E g X = E g X = m 1 =j E g j X j= m 1 =j X j = X = X (5.9) j= Fo a geneal M T, we define wo equence n and n of finie valued opping ime uch ha P-a.. n n and n n n Fix n and le Y n Z n be he unique oluion o he BSDE (3.2) wih = X n and = n. We know fom (5.9) ha P-a.. Y n m = E g m n X n X m m n In ligh of he igh-coninuiy of X and Y n, leing m give ha Y n X P-a. Now le Y Z be he unique oluion o he BSDE (3.2) wih = X. I i eay o ee ha fo d dp-a.. T 1 n g y z convege o 1 g y z unifomly in y z d. Theoem 3.5 hen implie ha P-a.. Y n convege o Y unifomly in T. Thu, we have E g X = Y = lim n Y n X P-a. poving he heoem. Finally, we udy he o-called upcoing inequaliy fo quadaic g- ubmaingale, which would be eenial fo he udy of pah egulaiy of g- ubmaingale.

22 Quadaic g-expecaion 731 Theoem 5.5. Given a g-ubmaingale X, we e J = ( X + kt ) e kt and denoe X = X + kj + 1, T. A uual, fo any finie e = < 1 < < n T, weleua b X denoe he numbe of upcoing of he ineval a b by X ove. Then hee i a BMO poce { } uch ha n [ ( E U b a X n exp db 1 n )] 2 2 d X + kj + 1T +a b a and ha E n 2 d C, a conan independen of he choice of. Poof. Fo any j 1n we conide he following BSDE: Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 j Y j = X j + g Y j Zj d j Z j db Applying Coollay 2.2 of [7] one ha j 1 j Y j ( X j + k j j 1 ) e k j j 1 J (5.1) Now le u define a d-dimenional poce = 1 d, n by l = n j=1 1 j 1 j 1 g z l ( Y j Zj1 Z jl ) d l 1d I i eay o ee fom Mean Value Theoem ha fo any j 1 j, g Y j Zj d = = l=1 d l=1 j g Y { g ( Y j Zj1 Z jl ) g ( Y j Zj1 Z jl 1 )} Z jl l =Zj (5.11) Moeove, (H3) implie ha l n lj 1 j 1 j 1 +Z j n l 1 d (5.12) j=1 We ee fom (2.7) ha each Z j i a BMO poce, hu o i. In fac, fo any M n, one can deduce fom (5.12) ha [ n E 2 d F ] C n + C [ n j E Z j 2 d ] F j=1 j 1 j { [ n j CT + C 1 j 1 E Z j 2 d ] F j 1 j=1 j 1 [ j + 1 j 1 < j E Z j 2 d ]} F j 1 j j 1 j

23 732 Ma and Yao { [ n j CT + C 1 j 1 E E Z j 2 d F j 1 ] F j 1 j=1 j 1 } + 1 j 1 < j Z j 2 BMO 2 CT + C n Z j 2 BMO 2 (5.13) j=1 whee C = 2dlJ 2. Thu, { E ( B ) } i a unifomly inegable maingale. n By Gianov heoem we can find an equivalen pobabiliy Q uch ha dq /dp = E ( B ). Then (5.11) and (H2) how ha fo any j 1n and n any j 1 j, Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 j Y j = X j + = X j + j [ g Y j +Zj ] j d Z j db g Y j d j X j + kj + 1 j j Z j db Z j db whee B denoe he Bownian Moion unde Q. Taking he condiional expecaion E Q F on boh ide of he above inequaliy one can obain ha E g j X j = Y j E Q X j F + kj + 1 j P-a. j 1 j In paiculaly, aking = j 1 we have X j 1 E g j 1 j X j E Q X j F j 1 + kj + 1 j j 1 P-a. Hence, X j n j= i a Q -ubmaingale. Applying he claical upcoing heoem one ha [ E Q U b a X ] E [ Q X n a +] X + kj + 1T +a b a b a Fuhemoe, we denoe C> o be a geneic conan depending only on d T J k X, and i allowed o vay fom line o line. Leing = in (5.13) one can deduce ha n E 2 d C + C C + C C + C n j=1 n j=1 j E Z j 2 d j 1 {e 4 KJ E [ e 4 KY j j e 4 KY j ] } j 1 + e 8 KJ j j 1 n E [ e 4 KX j e 4 KX j 1 ] [ ] = C + CE e 4 KX n e 4 KX C j=1

24 Quadaic g-expecaion 733 whee we applied (2.6) and (5.1) wih K = 1 kj + 1 lj o deive he econd 2 inequaliy and he hid inequaliy i due o he fac ha Y j j 1 = E g j 1 j X j X j 1. The poof i now complee. Wih he above upcoing inequaliy, we can dicu he coninuiy of he quadaic g-ub(upe)maingale. Coollay 5.6. If X i a g-ubmaingale (ep. g-upemaingale), hen fo any denumeable dene ube of T, i hold P-a.. ha lim X exi fo any T and lim X exi fo any T Downloaded By: [Yao, Song] A: 12:5 7 Januay 211 Poof. If X i a g-upemaingale, hen X i coepondingly a g -ubmaingale wih g defined in (5.2). Hence, i uffice o aume ha X i a g-ubmaingale. Le n n be an inceaing equence of finie ube of uch ha n n =. Fo any wo eal numbe a<b, Theoem 5.5 and Jenen Inequaliy imply ha: C = 1 + X + kj + 1T +a b a [ { 1 + E U b a X n n exp db 1 n 2 [ (1 = E + U b a X n ) { n exp db 1 n 2 { [ exp E ln ( 1 + U b a X n ) n + db 1 2 fom which one can deduce ha }] 2 d }] 2 d n E [ ln ( 1 + U b a X n )] ln C L 2 F n d C ]} 2 d whee C i a conan independen of he choice of. Since Ua b X n Ua ( X ) b a n, Monoone Convegence Theoem implie ha ln ( 1 + Ua ( X )) b i inegable, hu Ua ( X ) b <, P-a. Then a claical agumen yield he concluion fo X, hu fo X. The poof i now complee. Refeence 1. Baieu, P., and El Kaoui, N. 24. Opimal Deivaive Deign Unde Dynamic Rik Meaue, Conempoay Mahemaic. Vol Ame. Mah. Soc., Povidence, RI. 2. Biand, P., Coque, F., Hu, Y., Mémin, J., and Peng, S. 2. A convee compaion heoem fo BSDE and elaed popeie of g-expecaion. Elecon. Comm. Pobab. 5: (eleconic) 3. Coque, F., Hu, Y., Mémin, J., and Peng, S. 22. Filaion-conien nonlinea expecaion and elaed g-expecaion. Pobab. Theoy Relaed Field 123(1): Hu, Y., Imkelle, P., and Mülle, M. 25. Uiliy maximizaion in incomplee make. Ann. Appl. Pobab. 15(3): Hu, Y., Ma, J., Peng, S., and Yao, S. 28. Repeenaion heoem fo quadaic Fconien nonlinea expecaion. Sochaic Poce. Appl. 118(9):

25 Downloaded By: [Yao, Song] A: 12:5 7 Januay Ma and Yao 6. Kazamaki, N Coninuou Exponenial Maingale and BMO, Lecue Noe in Mahemaic, Vol Spinge-Velag, Belin. 7. Kobylanki, M. 2. Backwad ochaic diffeenial equaion and paial diffeenial equaion wih quadaic gowh. Ann. Pobab. 28(2): Peng, S Backwad SDE and Relaed g-expecaion, Piman Re. Noe Mah. Se., Vol Longman, Halow. 9. Peng, S. 24. Nonlinea Expecaion, Nonlinea Evaluaion and Rik Meaue, Lecue Noe in Mah., Vol Spinge, Belin. 1. Poe, P Sochaic Inegaion and Diffeenial Equaion, Applicaion of Mahemaic (New Yok), Vol. 21. Spinge-Velag, Belin. 11. Tyell Rockafella, R Convex Analyi. Pinceon Mahemaical Seie, No. 28. Pinceon Univeiy Pe, Pinceon, NJ.