BSDES UNDER FILTRATION-CONSISTENT NONLINEAR EXPECTATIONS AND THE CORRESPONDING DECOMPOSITION THEOREM FOR E-SUPERMARTINGALES IN L p

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 43, Number 2, 213 BSDES UNDER FILTRATION-CONSISTENT NONLINEAR EXPECTATIONS AND THE CORRESPONDING DECOMPOSITION THEOREM FOR E-SUPERMARTINGALES IN L p ZHAOJUN ZONG AND FENG HU ABSTRACT. In his paper, we inroduce he noion of F -consisen expecaion defined on L(Ω, F,P) and prove an exisence and uniqueness heorem for soluions and a comparison heorem of BSDE under E µ -dominaed F-expecaions. Furhermore, as an applicaion of his comparison heorem, we obain he decomposiion heorem for E-supermaringales. 1. Inroducion. By [7, we know ha here exiss a unique adaped and square inegrable soluion o a backward sochasic differenial equaion (BSDE for shor) of he ype (1) y = ξ + g(s, y s,z s )ds z s dw s, T, provided he funcion g is Lipschiz in boh variables y and z, andξ and (g(,, )) [,T are square inegrable. The funcion g is said o be he generaor of BSDE (1). We denoe he unique adaped and square inegrable soluion of BSDE (1) by (y (T,g,ξ),z (T,g,ξ) ) [,T. When g also saisfies g(,y,) = for any y R, heny (T,g,ξ), denoed by E g [ξ, is called he g-expecaion of ξ; y (T,g,ξ), denoed by E g [ξ F, is called he condiional g-expecaion of ξ (see [8). The g-expecaion is a kind of nonlinear expecaion, which can be considered o be a nonlinear exension of he well-known Girsanov 21 AMS Mahemaics subjec classificaion. Primary 6H1, 6G48. Keywords and phrases. F -consisen nonlinear expecaion (F-expecaion), backward sochasic differenial equaion (BSDE), generalized g-expecaion, comparison heorem. This work was parly suppored by he Naional Naural Science Foundaion of China (No ), he Docoral Program Foundaion of Minisry of Educaion of China (No ), he Posdocoral Science Foundaion of China (No. 212M52131) and he Naural Science Foundaion of Shandong Province of China (No. ZR212AQ9) Received by he ediors on July 8, 21, and in revised form on Augus 26, 21. DOI:1.1216/RMJ Copyrigh c 213 Rocky Mounain Mahemaics Consorium 677

2 678 ZHAOJUN ZONG AND FENG HU ransformaions. The original moivaion for sudying g-expecaions comes from he heory of expeced uiliy (see [2), which is fundamenally imporan in economics. This heory is seriously challenged by he famous Allais and Ellsberg paradoxes (see [6). The noion of non-addiive probabiliy, or capaciy, is hen inroduced o axiomaize preferences which do no saisfy he von Neumann-Morgensern axioms. Nonlinear expecaions are anoher useful noion in his seing. Since he noion of g-expecaion was inroduced, many researchers have sudied he properies of g-expecaion. In 22, Coque e al. [3 inroduced he noion of F-expecaion defined on L 2 and obained he exisence and uniqueness heorem for soluions of BSDE under E μ - dominaed F-expecaions, comparison heorems and decomposiion heorems for E-supermaringales. Furhermore, by using he decomposiion heorem for E-supermaringales, hey were able o prove he imporan resul ha any E μ -dominaed F-expecaion can be represened as a g-expecaion. Obviously, we know ha, if a random variable ξ is in L 1 (Ω, F,P), he classical mahemaical expecaion of ξ is meaningful. Bu, in he pas several years, researchers have sudied g-expecaions which are confined o L 2 (Ω, F,P). Recenly, [4, 5 have given he exensions of g-expecaions, i.e., he definiional domain of g-expecaion is exended o L(Ω, F, P), which are called generalized g-expecaions and invesigaed wih heir relaed properies. In his paper, inspired by [4, 5, we sudy F -consisen expecaions defined on L(Ω, F,P) and prove an exisence and uniqueness heorem for soluions and a BSDE comparison heorem under E μ -dominaed F- expecaions in L. Furhermore, as an applicaion of his comparison heorem and he monoonic limi heorem of BSDE in [5, we obain a decomposiion heorem for E-supermaringales. These resuls nonrivially generalize he corresponding resuls of [3. This paper is organized as follows. In Secion 2, we presen he noions of generalized g-expecaion and F -consisen nonlinear expecaions defined on L and give heir relaed properies ha are useful in his paper. In Secion 3, we give our main resuls such as he exisence and uniqueness heorem for soluions of BSDE under E μ -dominaed F-expecaions, comparison heorems and decomposiion heorems for E-supermaringales, including he proofs.

3 BSDES Generalized g-expecaions and F -consisen nonlinear expecaion Noaions and assumpions. For a given T (, ), le (W ) [,T be a d-dimensional sandard Brownian moion defined on a compleed probabiliy space (Ω, F,P), and le (F ) [,T be he naural filraion generaed by Brownian moion (W ) [,T,hais, F = σ{w s ; s } N, where N is he se of all P -null subses. For simpliciy, we consider he d = 1 case, bu our mehod is easily exended o higher dimensions. We also make he naural choice of F = F T.Le L p (Ω, F,P)={ξ : ξ is an F-measurable random variable such ha E ξ p < }; L(Ω, F,P)= p>1 L p (Ω, F,P); S p (,T; P ; R) ={V :(V ) [,T is (F ) [,T -adaped process wih E[sup T V p < }; S(,T; P ; R) = p>1 S p (,T; P ; R); L p (,T; P ; R) = {V : (V ) [,T is an (F ) [,T -adaped process wih E[( V s 2 ds) p/2 < }; L(,T; P ; R) = p>1 L p (,T; P ; R); M p (,T; P ; R) ={V :(V ) [,T is an (F ) [,T -adaped process wih E[ V s p ds < }; M(,T; P ; R) = p>1 M p (,T; P ; R). Suppose a funcion g: (2) g(ω,, y, z) :Ω [,T R R R saisfies he following condiions: (A.1) There exiss a Lipschiz consan μ > such ha, for all y 1,y 2 R, z 1,z 2 R, g(, y 1,z 1 ) g(, y 2,z 2 ) μ( y 1 y 2 + z 1 z 2 ), for all [,T; (A.2) g(,y,) =, for any y R.

4 68 ZHAOJUN ZONG AND FENG HU 2.2. Generalized g-expecaions. Lemma 2.1 (see [1). Suppose g saisfies (A.1) and (A.2). Then, for any ξ L p (Ω, F,P), 1 <p<2, hebsde (3) y = ξ + g(s, y s,z s )ds z s dw s, [,T has a unique pair of adaped processes (y (T,g,ξ),z (T,g,ξ) ) [,T S p (,T; P ; R) L p (,T; P ; R). Remark 2.1. From Lemma 2.1, we have: suppose g saisfies (A.1) and (A.2). Then, for each given ξ L(Ω, F,P), BSDE (3) has a unique pair of adaped processes (y (T,g,ξ),z (T,g,ξ) ) [,T S(,T; P ; R) L(,T; P ; R). Now, we inroduce he noions of generalized g-expecaion and generalized condiional g-expecaion. For more deails, see [5. Definiion 2.1 (Generalized g-expecaion). Suppose ha g saisfies (A.1) and (A.2). The generalized g-expecaion E g [ :L(Ω, F,P) R is defined by E g [ξ =y (T,g,ξ). Definiion 2.2 (Generalized condiional g-expecaion). The generalized condiional g-expecaion of ξ wih respec o F is defined by E g [ξ F =y (T,g,ξ). Generalized and generalized condiional g-expecaions are in general nonlinear. However hey mee he following basic properies of usual expecaions (see [5). Proposiion 2.1. Suppose ξ,ξ 1,ξ 2 L(Ω, F,P). Then: (i) If ξ is F -measurable, hen E g [ξ F =ξ; (ii) For all sopping imes τ and σ T, E g [E g [ξ F τ F σ = E g [ξ F τ σ,whereτ σ denoes min(τ,σ);

5 BSDES 681 (iii) If ξ 1 ξ 2 almos everywhere, hen E g [ξ 1 F E g [ξ 2 F ; if, moreover, P (ξ 1 >ξ 2 ) >, henp (E g [ξ 1 F > E g [ξ 2 F ) > ; (iv) For each B F, E g [1 B ξ F =1 B E g [ξ F ; (v) E g [ξ F is he unique random variable η in L(Ω, F,P) such ha E g [1 A ξ=e g [1 A η, for all A F ; (vi) For any (ς,η) L(Ω, F,P) L(Ω, F,P), E g [ς + η F = E g [ς F +η if and only if g does no depend on y. The nex proposiion will be useful in his paper. Proposiion 2.2 (see [5). Suppose g saisfies (A.1) and (A.2). (i) For any ξ,η L p (Ω, F,P)(1<p<2), hen E g [ξ F E g [η F e 1/2(p 1)μ2 T +μt (E[ ξ η p F ) 1/p, a.s.; (ii) For any ξ L p (Ω, F,P)(1<p<2), hen E g [ξ F e 1/2(p 1)μ2T (E[ ξ p F ) 1/p, a.s. Definiion 2.3 (Generalized g-maringale). A process (X ) [,T saisfying ha, for each, X L(Ω, F,P) is called a generalized g-maringale (respecively, generalized g-supermaringale, generalized g-submaringale) if, for any, s saisfying s T, E g [X s F =X (respecively X, X ), a.s.. The decomposiion heorem of generalized g-supermaringale obained by [4 will play an imporan role in his paper. Proposiion 2.3 (Decomposiion heorem of generalized g-supermaringale). Assume ha g saisfies (A.1), (A.2) and g is independen of y. Le (X ) be a righ-coninuous generalized g-supermaringale on [,T wih E[sup T X p < (1 <p<2). Then (X ) has he following decomposiion X = M A, for all [,T.

6 682 ZHAOJUN ZONG AND FENG HU Here (M ) is a generalized g-maringale and (A ) is an RCLL increasing process (i.e., a process (A ) is said o be RCLL if i a.s. has sample pahs which are righ coninuous wih lef limi; a process (A ) is said o be increasing if is pahs A : A (ω) are a.s. nondecreasing) wih A =and E(A T ) p <. More specifically, (X ) is he unique soluion of he BSDE X = X T + g(s, Z s )ds +(A T A ) Z s dw s, [,T. For noaional simplificaion, we will henceforh wrie E μ [ F E g [ F forg := μ z and E μ [ F E g [ F forg := μ z F -consisen nonlinear expecaions. Definiion 2.4. A nonlinear expecaion defined on L(Ω, F,P)isa funcional: E[ :L(Ω, F,P) R saisfying he following properies: (i) Sric monooniciy: if X 1 X 2 a.s., E[X 1 E[X 2 ; if X 1 X 2 a.s., E[X 1 =E[X 2 X 1 = X 2, a.s.; (ii) Preservaion of consans: E[c =c, for each consan c. Definiion 2.5. A nonlinear expecaion is called an F -consisen nonlinear expecaion (F-expecaion) defined on L(Ω, F, P) if, for each Y L(Ω, F,P)and [,T, here exiss a random variable η L(Ω, F,P), such ha E[Y 1 A =E[η1 A, for all A F.

7 BSDES 683 The F-expecaion has he following propery: Proposiion 2.4. For each Y L(Ω, F,P), η is he unique random variable in L(Ω, F,P) such ha (4) E[Y 1 A =E[η1 A, for all A F. The proof is very similar o ha of Lemma 3.1 in [3. We omi i. From Proposiion 2.4, such an η is uniquely defined. We also denoe i by η = E[Y F. E[Y F is called he condiional F-expecaion of Y under F. I is characerized by (5) E[Y 1 A =E[E[Y F 1 A, for all A F. Remark 2.2. Le g saisfy (A.1) and (A.2); hen he relaed generalized g-expecaion E g [ isanf-expecaion. Definiion 2.6 (E μ -dominaion). Given μ>, we say ha an F- expecaion E is dominaed by E μ if (6) E[X + Y E[X E μ [Y, for all X, Y L(Ω, F,P). Remark 2.3. Suppose ha g saisfies (A.1), (A.2) and g does no depend upon y; hen he associaed generalized g-expecaion is dominaed by E μ,whereμ is he Lipschiz consan in (A.1). Lemma 2.2. If E is dominaed by E μ for some μ>, hen E μ [Y E[X + Y E[X E μ [Y, for all X, Y L(Ω, F,P). Proof. A simple consequence of E μ [Y F = E μ [ Y F. The proof of Lemma 2.2 is complee.

8 684 ZHAOJUN ZONG AND FENG HU From now on, we will deal wih he F-expecaion E[ also saisfying he following condiion: (7) E[X + Y F =E[X F +Y, for all X L(Ω, F,P)andforallY L(Ω, F,P). Proposiion 2.5. Le E[, E 1 [ and E 2 [ be F-expecaions saisfying (6) and (7). Then (i) If E 1 [X E 2 [X, for all X L(Ω, F,P), hen E 1 [X F E 2 [X F, a.s., for all [,T. Inparicular,E μ [X F E[X F E μ [X F, a.s., for all [,T. (ii) For a given ζ L(Ω, F,P), he mapping E ζ [ =E[ + ζ E[ζ : L(Ω, F,P) R is also an F-expecaion saisfying (6) and (7). Is condiional expecaion under F is E ζ [ F =E[ + ζ F E[ζ F,a.s. In paricular, E μ [Y F E[X + Y F E[X F E μ [Y F, a.s., for all X, Y L(Ω, F,P). The proof of Proposiion 2.5 is very similar o hose of Lemmas 4.3, 4.4 and 4.5 in [3. We omi i. The following lemma will be very useful laer. Lemma 2.3. If E[ mees (6) and (7), here exiss a consan C p > such ha, for all Y L p (Ω, F,P) (1 <p<2) andforeach T, (8) E[X + Y F E[X F C p (E[ Y p F ) 1/p, a.s. Proof. By Proposiions 2.2 and 2.5, we have E[X + Y F E[X F E μ [Y F E μ [Y F e 1/2(p 1)μ2T (E[ Y p F ) 1/p, a.s., where E μ [Y F E μ [Y F denoes max( E μ [Y F, E μ [Y F ). The proof of Lemma 2.3 is complee.

9 BSDES 685 From now on, we will always assume ha E[ isanf-expecaion saisfying (6) and (7). 3. Main resuls and proofs BSDE under F-expecaions. Le a funcion f :Ω [,T R R saisfying: (B.1) There exiss a Lipschiz consan K>such ha for all y 1, y 2 R, f(, y 1 ) f(, y 2 ) K y 1 y 2, for all [,T; (B.2) f(, ) M(,T; P ; R). For a given erminal daa X L(Ω, F,P), we consider he following ype of equaion: [ (9) Y = E X + f(s, Y s )ds F. Theorem 3.1. We assume (B.1) and (B.2). Then here exiss a unique process Y ( ) M(,T; P ; R) which solves (9). Proof. For a given erminal daa X L(Ω, F, P) and each y( ) M(,T; P ; R), we define a mapping π (y( )) by [ π (y( )) = E X + f(s, y(s)) ds F. Obviously, π (y( )) M(,T; P ; R). Wihou loss of generaliy, suppose y 1 ( ), y 2 ( ) M p (,T; P ; R) (1 <p<2). By Lemma 2.3, we have (1) π (y 1 ( )) π (y 2 ( )) [ T p ) 1/p C p (E [f(s, y 1 (s)) f(s, y 2 (s)) ds F, a.s.

10 686 ZHAOJUN ZONG AND FENG HU By Hölder s inequaliy and (B.1), [f(s, y 1 (s)) f(s, y 2 (s)) ds This wih (1) yields p T p 1 f(s, y 1 (s)) f(s, y 2 (s)) p ds T p 1 K p y 1 (s) y 2 (s) p ds, a.s. E[ π (y 1 ( )) π (y 2 ( )) p [ T p 1 (C p K) p E y 1 (s) y 2 (s) p ds. Le C = T p 1 (C p K) p. By muliplying boh sides of he above inequaliy by e 2C and hen inegraing boh sides on [,T wih respec o, i follows ha [ E e 2C π (y 1 ( )) π (y 2 ( )) p d [ CE e 2C [ s = CE y 1 (s) y 2 (s) p y 1 (s) y 2 (s) p ds d e 2C d ds [ =(2C) 1 CE (e 2C 1) y 1 () y 2 () p d 1 [ 2 E e 2C y 1 () y 2 () p d. We observe ha, for any consan α R, he following wo norms are equivalen in M p (,T; P ; R): ( [ 1/p ( [ 1/p E Y d) p E Y p e d) α.

11 BSDES 687 From his, we can obain ha π (y( )) is a conracion mapping on M p (,T; P ; R). I follows ha his mapping has a unique fixed poin Y ( ): [ Y = E X + f(s, Y s )ds F. So he proof of Theorem 3.1 is complee. Theorem 3.2 (Comparison heorem). Le Y be he soluion of (9) and Y he soluion of Y = E [X + [f(s, Y s )+φ(s) ds F, where X L(Ω, F,P) and φ M(,T; P ; R). If (11) X X, a.s., φ(), dp d a.e., hen we have (12) Y Y, a.s. Equaion (12) becomes an equaliy if and only if (11) become equaliies. Proof. The main approach of he following proof derives from Coque e al. [3. Inhecaseofφ() =,dp d almos everywhere, he proof is very similar o ha of Theorem 6.2 in [3, so we omi i. In order o prove he general case φ(), dp d almos everywhere, we define for n =1, 2, 3,..., Y n o be he soluion of Y n [( ) = E X + φ(s)ds + f(s, Y n s )ds F, it/n for [ n i,n i+1 ), n i = it /n, i =, 1, 2,...,n 1. This equaion can be wrien, piece by piece, as Y n [( n = E Y n i+1 ) n i+1 + φ(s)ds + f(s, Y n s n )ds F, i+1 n i [ n i,n i+1 ), Y T n = Y n = n X. n

12 688 ZHAOJUN ZONG AND FENG HU From he firs par of he proof, we have for i = n 1, Y n Y, [ n n 1,T). In paricular, Y Y n n n 1 n 1. An obvious ieraion of his algorihm gives Y n Y, [ n i,n i+1 ), i =,...,n 2. Thus, Y, [,T. Y n In order o prove ha Y Y, i suffices o show he convergence of he sequence (Y n )oy. Wihou loss of generaliy, we assume ha here exiss a 1 <p<2 such ha Y n, Y and φ M p (,T; P ; R). By Lemma 2.3, for fixed [ n i,n i+1 ), [( ) p E[ Y n Y p (C p ) p E φ(s) ds + K Ys n Y s. ds it/n Using Hölder s inequaliy, we have, for each [,T, ( ) p 1 [ T E[ Y n Y p (2C p ) p E φ(s) p ds n [ +(2KC p ) p T p 1 E Ys n Y s p ds. Gronwall s lemma applied o he above inequaliy shows ha E[ Y n Y p, as n. Thus, Y Y. Finally, we invesigae he possible equaliy in (12). From Y = Y, a.s., hen [ [ E X + f(s, Y s )ds = E X + [f(s, Y s )+φ(s) ds. Since X X, a.s., φ(), dp d a.e., i follows from he sric monooniciy of E[ hax = X, a.s., φ() =,dp d a.e. So he proof of Theorem 3.2 is complee F -consisen maringales and decomposiion heorem for E-supermaringales. Definiion 3.1. A process (X ) [,T M(,T; P ; R) is called an E-maringale (respecively E-supermaringale, E-submaringale), if for each s T, X s = E[X F s, (resp. E[X F s, E[X F s ) a.s.

13 BSDES 689 Remark 3.1. An E μ -supermaringale (X ) [,T is boh an E- supermaringale and E μ -supermaringale. An E μ -submaringale (X ) [,T is boh an E-submaringale and E μ -submaringale. An E- maringale (X ) [,T is an E μ -supermaringale and an E μ -submaringale. ThenexlemmashowshaeveryE-maringale admis RCLL pahs. Lemma 3.1. For each X L(Ω, F,P), he process (E[X F ) [,T admis a unique modificaion wih a.s. RCLL pahs. The proof is very similar o ha of Lemma 5.2 in [3. We omi he proof. In he following, we do no disinguish E-maringales and heir RCLL modificaions. Lemma 3.1 has an immediae consequence as follows: Lemma 3.2. Le E[ be an F-expecaion saisfying (6) and (7). Then, for each X L(Ω, F,P) and g M(,T; P ; R), he process (E[X + g(s)ds F ) [,T is RCLL a.s. ThenexlemmashowshaeveryE-maringale admis coninuous pahs. Lemma 3.3. For each X L(Ω, F,P), ley = E[X F. Then here exiss a pair (g(),z ) L(,T; P ; R R) wih (13) g() μ z such ha (14) y = X + g(s)ds z s dw s. Furhermore, ake X L(Ω, F,P), pu y = E[X F and le (g (),z ) L(,T; P ; R R) be he corresponding pair. Then we have (15) g() g () μ z z.

14 69 ZHAOJUN ZONG AND FENG HU Proof. ForeachX L(Ω, F, P), here exiss a 1 < p < 2such ha X L p (Ω, F,P). Since y = E[X F, [,TisanRCLLEmaringale, i is an RCLL E μ -submaringale and E μ -supermaringale. By he dominaion E μ [X F E[X F E μ [X F, we have E[ sup y p <. Thus, by Proposiion 2.3, and in a similar manner T o Lemma 5.3 in [3, we can complee he res proof. So we omi i. Remark 3.2. From Lemma 3.3, he resul of Lemma 3.2 can be improved o: for each X L(Ω, F,P)andg M(,T; P ; R), he process (E[X + g(s)ds F ) [,T is coninuous a.s. Our nex resul generalizes he decomposiion heorem for generalized g-supermaringales proved in [4 o righ coninuous E-supermaringales. The proof mainly uses argumens from [4. Theorem 3.3 (Decomposiion heorem for E-supermaringales). Le E[ be an F-expecaion saisfying (6), (7) and le Y S(,T; P ; R) be a righ-coninuous E-supermaringale. Then here exiss an A S(,T; P ; R) such ha A is RCLL and increasing wih A() = and such ha Y + A is an E-maringale. Proof. For n 1, we define y n o be he soluion of he following BSDE: [ y n = E Y T + n(y s ys n )ds F. We hen have he following: Lemma 3.4. We have, for each and n 1, Y y n, a.s. In a similar manner as Lemma 6.2 in [3, we can prove Lemma 3.4. So we omi i.

15 BSDES 691 Lemma 3.4 wih Theorem 3.2 implies ha y n monoonically converges o some Y Y. Indeed, wriing φ() =Y y n+1 showsha(y n ) is an increasing sequence of funcions. Observe hen ha y n + n(y s ys n )ds is an E-maringale. By Lemma 3.3, here exiss a (g n (),z n ) L(,T; P ; R R) wih (16) g n () μ z n, n =1, 2,..., such ha y n + n(y s ys n )ds = yn T + n(y s ys n )ds + g n (s)ds zs n dw s; hence, as y n T = Y T, (17) y n = Y T + Equaion (15) also ells us ha [g n (s)+n(y s ys n ) ds zs n dw s. (18) g n () g m () μ z n zm, n,m =1, 2,.... Le us denoe, for each n =1, 2,..., A n () = n(y s ys n )ds; hen A n is a coninuous increasing process such ha A n () =. Wihou loss of generaliy, assume Y S p (,T; P ; R) (1<p<2). We have he following: Lemma 3.5. There exiss a posiive consan C which is independen of n such ha: (19) (1) E ( z n s 2 ds) p/2 C, (2) E(A n (T )) p C.

16 692 ZHAOJUN ZONG AND FENG HU Proof. From (16) and (17), we have (A n (T )) p = yn yn T g n (s)ds + zs n dw p s ( p 4 ( y p n p + yt n p + μ zs n ds) + z n s dw s p). For he hird erm on he righ side, applying Schwarz s inequaliy, we obain ( ) p ( p/2 μ zs n ds m p zs ds) n 2, where m p is a posiive consan independen of n. For he las erm, i follows by he BDG (Burkholder-Davis-Gundy) inequaliy ha p ( p/2 E zs n dw s n p E zs n ds) 2, where he posiive consan n p is independen of n. Observeha y n is dominaed by y 1 + Y. Thus, here exiss a posiive consan C independen of n, such ha E [ sup y n p C. T I follows ha here exis wo posiive consans C 1 and C 2, independen of n, such ha ( p/2 (2) E(A n (T )) p C 1 + C 2 E zs ds) n 2. On he oher hand, we apply Io s formula o y n 2 : Then (21) z n s 2 ds y n T 2 +2 ( y n s gn (s)ds+2 ) p/2 ( ( zs n 2 ds d p yt n p + ( + p/2 ys n da (s)) n + y n s dan (s) 2 ) p/2 ys n gn (s) ds y n s zn s dw s. y n s z n s dw s p/2),

17 BSDES 693 where d p is a posiive consan independen of n. For he second erm on he righ side of (21), we ge ( p/2 ( ) p/2 E ys n gn (s) ds) E ys n μ zn s ds e p E [ sup y n p T + 1 ( p/2 E zs ds) n 4d 2, p where e p is a posiive consan independen of n. For he hird erm, ( p/2 [ E ys n da (s)) n E ( [ E sup y n p/2 (A n (T )) p/2 T ) 1/2 sup y n p (E(A n (T )) p ) 1/2 T [ 1 4C 2 d p E(A n (T )) p + C 2 d p E sup y n. p T For he las erm, i follows from he BDG inequaliy ha p/2 ( E ys n zn s dw T ) p/4 s h p E ys n zn s 2 ds [ ( ) p/4 h p E sup y n p/2 zs n 2 ds T ( 1 4d p E ) p/2 zs n 2 ds + d p h 2 p E [ sup T where h p is a posiive consan independen of n. Consequenly, ( ) p/2 (22) E zs n 2 ds C E(A n (T )) p, 2C 2 y n p, where C 3 is a posiive consan independen of n. This wih (2) implies ha (1) and (2) hold. The proof of Lemma 3.5 is complee. Wih he help of Lemma 3.5, we can end he proof of Theorem 3.3.

18 694 ZHAOJUN ZONG AND FENG HU Equaion (19) (1) wih (16) imply ha E( gn (s) 2 ds) p/2 μ p C. Bu equaion (19) (2) implies ha y n Y. From Theorem 3.1 in [4, i follows ha we can wrie Y in he form Y = Y T + g(s)ds + A(T ) A() Z s dw s, for some (g, Z) L p (,T; P ; R R) and an increasing process A wih A() = and E(A(T )) p <. From he same heorem we have ha, moreover, z n Z, weakly in L p (,T; P ; R); g n () g(), weakly in L p (,T; P ; R); A n () A(), weakly in L p (Ω, F,P). Since y n = E[Y T + A n (T ) A n () F, by Lemma 2.3, i follows ha Y = E[Y T + A(T ) A() F. Thus, Y + A() =E[Y T + A(T ) F isan E-maringale. The proof of Theorem 3.3 is complee. A las, we will prove an imporan resul: an F-expecaion can be idenified as a generalized g-expecaion, proving ha (6) and (7) hold. Theorem 3.4. We assume ha an F-expecaion E[ saisfies (6) and (7) for some μ >. Then here exiss a unique funcion g = g(, z) :Ω [,T R saisfying (A.1) and (A.2) such ha E[X F =E g [X F, a.s., for all X L(Ω, F,P). Proof. For any X L(Ω, F,P), le X n =(X n) ( n), n =1, 2,... Then, for each n, X n L 2 (Ω, F,P). By Theorem 7.1 in [3, hen here exiss a unique funcion g = g(, z) :Ω [,T R saisfying (A.1) and (A.2) such ha E[X =Eg[X, for all X L 2 (Ω, F,P). So E[X n =E g [X n, n =1, 2,.... By Lemma 2.3, choosing =, E[X n E[X, as n. On he oher hand, by Proposiion 2.2, choosing =,E g [X n E g [X, as n. Thus, E[X =E g [X, for all X L(Ω, F,P).

19 BSDES 695 For all A F, E[E[X F 1 A =E[X1 A =E g [X1 A =E g [E g [X F 1 A =E[E g [X F 1 A. By he uniqueness of he condiional F-expecaion of X under F,we have E[X F =E g [X F, a.s., for all X L(Ω, F,P). So we have compleed he proof of Theorem 3.4. Acknowledgmens. We would like o hank he anonymous referee who gave us many consrucive suggesions and commens on he previous version of his paper. REFERENCES 1. P. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Soica, L p soluions of backward sochasic differenial equaions, Soch. Proc. Appl. 18 (23), Z. Chen and L. Epsein, Ambiguiy, risk and asse reurns in coninuous ime, Econom. 7 (22), F. Coque, Y. Hu, J. Mémin and S. Peng, Filraion-consisen nonlinear expecaions and relaed g-expecaions, Probab. Theory Rela. Fields 123 (22), F. Hu, Monoonic limi heorem of BSDE and nonlinear decomposiion heorem of Doob-Meyer s ype in L p, Chinese J. Appl. Probab. Sais., submied. 5. F. Hu and Z. Chen, Generalized g-expecaions and relaed properies, Sais. Probab. Le. 8 (21), I. Levi, The paradoxes of Allais and Ellsberg, Econ. Philos. 2 (1986), E. Pardoux and S. Peng, Adaped soluion of a backward sochasic differenial equaion, Sysems Conrol Le. 14 (199), S. Peng, Backward SDE and relaed g-expecaion, in Backward sochasic differenial equaions, N. El Karoui and L. Mazliak, eds., Piman Res. Noes 364, Longman, Harlow, School of Mahemaical Sciences, Qufu Normal Universiy, Qufu, Shandong , China address: zongzj1@163.com School of Mahemaical Sciences, Qufu Normal Universiy, Qufu, Shandong , China and School of Mahemaics, Shandong Universiy, Jinan 251, China address: hufengqf@163.com