Chin. Ann. Mah. 25B:3(2004),401 412. ON JENSEN S INEQUALITY FOR g-expectation JIANG Long CHEN Zengjing Absrac Briand e al. gave a conerexample showing ha given g, Jensen s ineqaliy for g-expecaion sally does no hold in general. This paper proves ha Jensen s ineqaliy for g-expecaion holds in general if and only if he generaor g (, z) is sper-homogeneos in z. In pariclar, g is no necessarily convex in z. Keywords Backward sochasic differenial eqaion, Jensen s ineqaliy, g- expecaion, Condiional g-expecaion, Comparison heorem 2000 MR Sbjec Classificaion 60H10 1. Inrodcion I is by now well known ha here exiss a niqe adaped and sqare inegrable solion o a backward sochasic differenial eqaion (BSDE in shor) of ype y = ξ + g(s, y s, z s )ds z s db s, 0 T, (1.1) providing ha he generaor g is Lipschiz in boh variables y and z, and ha ξ and he process g(, 0, 0) are sqare inegrable. We denoe he niqe solion of he BSDE (1.1) by (y ξ (), z ξ ()) [0,T ]. In [1], y ξ (0), denoed by E g [ξ], is called g-expecaion of ξ. The noion of g-expecaion can be considered as a nonlinear exension of he well-known Girsanov ransformaions. The original moivaion for sdying g-expecaion comes from he heory of expeced iliy, which is he fondaion of modern mahemaical economics. Z. Chen and L. Epsein [2] gave an applicaion of g-expecaion o recrsive iliy. Since he noion of g-expecaion was inrodced, many properies of g-expecaion have been sdied in [1, 3 5]. Some properies of classical expecaion are preserved (monooniciy for insance), and some resls on Jensen s ineqaliy for g-expecaion were obained in [3, 5]. B also in [3], he ahors gave a conerexample o indicae ha even for a linear fncion ϕ, which is obviosly convex, Jensen s ineqaliy for g-expecaion sally does no hold. This yields a naral qesion: Wha kind of generaor g can make Jensen s ineqaliy for g-expecaion hold in general? Roghly speaking, for convex fncion ϕ : R R, wha condiions shold be given Manscrip received March 4, 2003. Revised November 9, 2003. School of Mahemaics and Sysem Sciences, Shandong Universiy, Jinan 250100, China. Deparmen of Mahemaics, China Universiy of Mining and Technology, Xzho 221008, Jiangs, China. E-mail: jianglong@mah.sd.ed.cn School of Mahemaics and Sysem Sciences, Shandong Universiy, Jinan 250100, China. Projec sppored by he Naional Naral Science Fondaion of China (No.10131030).
402 JIANG, L. & CHEN, Z. J. o he generaor g sch ha he following ineqaliy E g [ϕ(ξ) F ] ϕ[e g (ξ F )] will hold in general? The objecive of his paper is o invesigae his problem and o prove ha Jensen s ineqaliy for g-expecaion holds in general if and only if g(, z) is sper-homogeneos, and if g is convex, hen Jensen s ineqaliy for g-expecaion holds in general if and only if g(, z) is a posiive-homogeneos generaor; For monoonic convex fncion ϕ, we also ge wo necessary and sfficien condiions. 2. Preliminaries 2.1. Noaions and Assmpions Le (Ω, F, P) be a probabiliy space and (B ) 0 be a d-dimensional sandard Brownian moion on his space sch ha B 0 = 0. Le (F ) 0 be he filraion generaed by his Brownian moion F = σ{b s, s [0, ]} N, [0, T ], where N is he se of all P -nll sbses. Le T >0 be a given real nmber. In his paper, we always work in he space (Ω, F T, P ), and only consider processes indexed by [0, T ]. For any posiive ineger n and z R n, z denoes is Eclidean norm. We define he following sal spaces of processes: S 2 F(0, T ; R) := H 2 F(0, T ; R n ) := { ψ coninos and progressively measrable; E [ sp 0 T { [ ψ progressively measrable; ψ 2 2 = E ψ 2 d 0 ψ 2] } < ; ] } <. We recall he noion of g-expecaion, defined in [1]. We are given a fncion g : Ω [0, T ] R R d R sch ha he process (g(, y, z)) [0,T ] is progressively measrable for each pair (y, z) in R R d, and frhermore, g saisfies some of he following assmpions: (A1) There exiss a consan K 0, sch ha we have [0, T ], y 1, y 2 R, z 1, z 2 R d, g(, y 1, z 1 ) g(, y 2, z 2 ) K( y 1 y 2 + z 1 z 2 ). (A2) The process (g(, 0, 0)) [0,T ] HF 2 (0, T ; R). (A3) (, y) [0, T ] R, g(, y, 0) 0. (A4) (y, z) R R d, g(, y, z) is coninos. Remark 2.1. The assmpion (A3) implies he assmpion (A2). Le g saisfy he assmpions (A1) and (A2). Then for each ξ L 2 (Ω, F T, P ), here exiss a niqe pair (y ξ (), z ξ ()) [0,T ] of adaped processes in S 2 F (0, T ; R) H2 F (0, T ; Rd ) solving he BSDE (1.1) (see [6]). We ofen denoe (y ξ (), z ξ ()) [0,T ] by (y, z ) [0,T ] in shor.
ON JENSEN S INEQUALITY FOR g-expectation 403 2.2. Definiions and Proposiions For he convenience of readers, we recall he noion of g-expecaion and condiional g-expecaion defined in [1]. We also lis some basic properies of BSDEs and g-expecaion. In he following Definiions 2.1 and 2.2, we always assme ha g saisfies (A1) and (A3). Definiion 2.1. The g-expecaion E g [ ] : L 2 (Ω, F T, P ) R is defined by E g [ξ] = y ξ (0). Definiion 2.2. The condiional g-expecaion of ξ wih respec o F is defined by E g [ξ F ] = y ξ (). The following Comparison Theorem is one of he grea achievemens of heory of BS- DEs, readers can see he proof in [7] or [8]. Proposiion 2.1. (cf. [7, 8]) Le g, ḡ saisfy (A1) and (A2), le Y T, Y T L 2 (Ω, F T, P ). Le (y(), z()) [0,T ], (ȳ(), z()) [0,T ] be he solions of he following wo BSDEs: y = Y T + ȳ = Y T + g(s, y s, z s )ds ḡ(s, ȳ s, z s )ds z s db s, 0 T ; z s db s, 0 T. (1) If Y T Y T, g(, ȳ, z ) ḡ(, ȳ, z ), a.s., a.e., hen we have y ȳ, a.e., a.s. (2) In addiion, if we also assme ha P (Y T Y T > 0) > 0, hen P (y ȳ > 0) > 0, in pariclar, y 0 > ȳ 0. Proposiions 2.2 2.5 come from [1], where g is assmed o saisfy (A1) and (A3). Proposiion 2.2. (1) (Preserving of consans) For each consan c, E g [c] = c; (2) (Monooniciy) If X 1 X 2, a.s., hen E g [X 1 ] E g [X 2 ]; (3) (Sric Monooniciy) If X 1 X 2, a.s., and P (X 1 > X 2 ) > 0, hen E g [X 1 ] > E g [X 2 ]. Proposiion 2.3. (1) If X is F -measrable, hen E g [X F ] = X; (2) For all, s [0, T ], E g [E g [X F ] F s ] = E g [X F s ]. ha Proposiion 2.4. E g [X F ] is he niqe random variable η in L 2 (Ω, F, P ), sch E g [X1 A ] = E g [η1 A ] for all A F. Proposiion 2.5. Le g(ω,, y, z) : Ω [0, T ] R R d R be a given fncion saisfying (A1) and (A3). If g does no depend on y, hen we have E g [X + η F ] = E g [X F ] + η, η L 2 (Ω, F, P ), X L 2 (Ω, F T, P ).
404 JIANG, L. & CHEN, Z. J. Proposiion 2.6. (cf. [3, 8]) Le ξ L 2 (Ω, F T, P ), and le he assmpions (A1) and (A2) hold. If he process (y, z ) [0,T ] is he solion of BSDE (1.1), hen we have [ E sp s T [ C E e βt ξ 2 + (e βs y s 2 ) + ( where β = 2(K + K 2 ) and C is a niversal consan. e βs z s 2 ds F ] ) 2 ] e (β/2)s F g(s, 0, 0) ds, Proposiion 2.7. (cf. [3]) Sppose g does no depend on y and g saisfies (A1) and (A3). Sppose moreover ha for each [0, T ], z g(, z) is convex. Given ξ L 2 (Ω, F T, P ), le ϕ : R R be a convex fncion sch ha ϕ(ξ) L 2 (Ω, F T, P ). If ϕ[e g (ξ F )] ]0, 1[ c, hen we have ϕ[e g (ξ F )] E g [ϕ(ξ F )]. Proposiion 2.7 can be regarded as an imporan resl on Jensen s ineqaliy for g- expecaion, b if ϕ[e g (ξ F )] ]0, 1[ c =, for example ϕ(x) = x/2, x R, Proposiion 2.7 can no solve his kind of problems. I also can no ell s wha kind of generaor g can make Jensen s ineqaliy hold in general. 3. Jensen s Ineqaliy for Sper-homogeneos Generaor g In he following, we always consider he siaion where he generaor g does no depend on y, ha is, g : Ω [0, T ] R d R. We denoe his kind of generaor g by g(, z). We always assme ha g(, z) saisfies (A1) and (A3). Definiion 3.1. Le g saisfy (A1) and (A3). We say ha g is a sper-homogeneos generaor in z if g also saisfies (, z) [0, T ] R d, λ R : g(, λz) λg(, z). Now we inrodce or main resls on Jensen s ineqaliy for g-expecaion. Theorem 3.1. are eqivalen: Le g saisfy (A1), (A3) and (A4). Then he following wo condiions ( i ) g is a sper-homogeneos generaor; (ii) Jensen s ineqaliy for g-expecaion holds in general, i.e., for each ξ L 2 (Ω, F T, P ) and convex fncion ϕ : R R, if ϕ(ξ) L 2 (Ω, F T, P ), hen for each [0, T ], E g [ϕ(ξ) F ] ϕ[e g (ξ F )]. Proof. (i) (ii). Given ξ L 2 (Ω, F T, P ) and convex fncion ϕ sch ha ϕ(ξ) L 2 (Ω, F T, P ), for each [0, T ], we se η = ϕ [E g (ξ F )]. Then η is F -measrable. Since ϕ is convex, we have Take x = ξ, y = E g (ξ F ). Then we have ϕ(x) ϕ(y) ϕ (y)(x y), x, y R. ϕ(ξ) ϕ[e g (ξ F )] η [ξ E g (ξ F )].
ON JENSEN S INEQUALITY FOR g-expectation 405 For each posiive ineger n, we define Ω,n := { E g (ξ F ) + η + ϕ[e g (ξ F )] n}. Becase E g [ξ F ], η, ϕ[e g (ξ F )] are all F -measrable, we see ha Ω,n F. We denoe he indicaor fncion of Ω,n by 1 Ω,n. Se η,n = 1 Ω,n η. Then we have 1 Ω,n [ϕ(ξ) ϕ[e g (ξ F )]] η,n [ξ E g (ξ F )]. (3.1) Since η,n, 1 Ω,n ϕ[e g (ξ F )] are bonded by n and ξ, ϕ(ξ) L 2 (Ω, F T, P ), we dedce ha 1 Ω,n ϕ(ξ), η,n ξ L 2 (Ω, F T, P ), 1 Ω,n ϕ[e g (ξ F )] L 2 (Ω, F, P ), (η,n E g (ξ F s )) s T S 2 F(, T ; R). From he well-known Comparison Theorem we know ha condiional g-expecaion E g [ F ] is nondecreasing. Ths from he ineqaliy (3.1), and by aking condiional g- expecaion, we can ge E g [1 Ω,n [ϕ(ξ) ϕ(e g (ξ F ))] F ] E g [η,n [ξ E g (ξ F )] F ]. Since 1 Ω,n ϕ[e g (ξ F )], η,n E g [ξ F ] L 2 (Ω, F, P ), i follows from Proposiion 2.5 ha E g [1 Ω,n ϕ(ξ) F ] 1 Ω,n ϕ[e g (ξ F )] E g [η,n ξ F ] η,n E g [ξ F ]. (3.2) Le (y, z ) [0,T ] be he solion of he following BSDE (3.3) Then for he given [0, T ], we have y = ξ + g(s, z s )ds z s db s, 0 T. (3.3) η,n y = η,n ξ + η,n g(s, z s )ds η,n z s db s, T. (3.4) We define fncion g 1 (s, z) in his way: for each (s, z) [, T ] R d, { η,n g(s, z/η,n ), if η,n 0; g 1 (s, z) := 0, if η,n = 0. Since η,n is bonded, he following BSDE ȳ = η,n ξ + g 1 (s, z s )ds z s db s, T (3.5) has a niqe solion in S 2 F (, T ; R) H2 F (, T ; Rd ). We denoe i by (ȳ s, z s ) s [,T ]. Also from ha η,n is bonded we know ha (η,n y s, η,n z s ) s [,T ] is in S 2 F (, T ; R) H2 F (, T ; Rd ). From (3.4) and he definiion of g 1, we conclde ha he solion of BSDE (3.5) is js (η,n y s, η,n z s ) s [,T ]. Consider he solions of BSDE (3.5) and he following BSDE (3.6): ỹ = η,n ξ + g(s, z s )ds z s db s, T. (3.6)
406 JIANG, L. & CHEN, Z. J. De o he sper-homogeneiy of g(, z) in z, we can ge ha for each s [, T ], g(s, η,n z s ) η,n g(s, z s ). Combining his wih he definiion of g 1, we have, s [, T ], g(s, z s ) = g(s, η,n z s ) η,n g(s, z s ) = g 1 (s, η,n z s ) = g 1 (s, z s ). Ths from Comparison Theorem, we have, Coming back o (3.2), we can ge E g [η,n ξ F ] = ỹ ȳ = η,n y = η,n E g [ξ F ]. (3.7) E g [1 Ω,n ϕ(ξ) F ] 1 Ω,n ϕ[e g (ξ F )] E g [η,n ξ F ] η,n E g [ξ F ] 0. Applying Lebesge s dominaed convergence heorem o (1 Ω,n ϕ(ξ)) n=1, we can ge easily ha L 2 lim n 1 Ω,n ϕ(ξ) = ϕ(ξ). Since ha ξ E g (ξ F ) is a coninos map from L 2 (F T ) ino L 2 (F ) (see [1, Lemma 36.9]), i follows ha L 2 lim n E g[1 Ω,n ϕ(ξ) F ] = E g [ϕ(ξ) F ]. Ths for he given [0, T ], here exiss a sbseqence (E g [ϕ(ξ)1 Ω,ni F ]) i=1 sch ha, lim E g[ϕ(ξ)1 Ω,ni F ] = E g [ϕ(ξ) F ]. i On he oher hand, by he definiion of Ω,n, we can ge, lim 1 Ω,n ϕ[e g (ξ F )] = ϕ[e g (ξ F )]. n Hence we can asser ha (i) implies (ii). Indeed, E g [ϕ(ξ) F ] = lim i E g [1 Ω,ni ϕ(ξ) F ] lim i 1 Ω,ni ϕ[e g (ξ F )] = ϕ[e g (ξ F )]. (ii) (i). Firsly we show ha for each z R d, [0, T [, L 2 lim n n[e g(z (B +1/n B ) F )] = g(, z). (3.8) (3.8) is a special case of [3, Proposiion 2.3]. B for he convenience of readers and he compleeness of or proof, here we give a sraighforward proof. For each given z R d, [0, T [, we choose a large enogh posiive ineger n, sch ha + 1/n T. We denoe by (y s,n, z s,n ) s [,+1/n] he solion of he following BSDE: y s = z (B +1/n B ) + We se Then we have y,n = ȳ,n and ȳ s,n = +1/n s +1/n s g(, z )d +1/n ȳ s,n = y s,n z (B s B ), z s,n = z s,n z. s z db, s + 1/n. (3.9) +1/n g(, z,n + z)d z,n db, s + 1/n. (3.10) s
Since ON JENSEN S INEQUALITY FOR g-expectation 407 [ +1/n E g [z (B +1/n B ) F ] = y,n = ȳ,n = E g(s, z s,n + z)ds F ], by he classical Jensen s ineqaliy and Hölder s ineqaliy, we have E[nE g [z (B + 1 B ) F n ] g(, z)] 2 [ [ + 1 n ]] 2 = E ne (g(s, z s,n + z) g(, z))ds F [ +1/n ] 2 n 2 E (g(s, z s,n + z) g(, z))ds ne 2nE + 2nE +1/n +1/n +1/n g(s, z s,n + z) g(, z) 2 ds g(s, z s,n + z) g(s, z) 2 ds g(s, z) g(, z) 2 ds. (3.11) By (A1), Proposiion 2.6 and (A3), we know ha here exiss a niversal consan C sch ha 2nE +1/n 2nK 2 E +1/n ( +1/n 2nK 2 CE ( +1/n 2nK 2 CE = 2K 4 C z 2 /n, where K is he Lipschiz consan. By (A4), we know ha g(s, z s,n + z) g(s, z) 2 ds z s,n 2 ds ) 2 g(s, z) ds ) 2 K z ds +1/n lim 2n g(s, z) g(, z) 2 ds = 0. n In view of (A3) and (A1), we have 2n +1/n g(s, z) g(, z) 2 ds 2n +1/n I follows from Lebesge s dominaed convergence heorem ha (2K z ) 2 ds = 8K 2 z 2. +1/n lim 2nE g(s, z) g(, z) 2 ds = 0. n
408 JIANG, L. & CHEN, Z. J. Then coming back o (3.11), we can ge Therefore we have lim E[nE g(z (B +1/n B ) F ) g(, z)] 2 n +1/n lim n 2K4 C z 2 /n + lim 2nE n g(s, z) g(, z) 2 ds = 0. L 2 lim n n[e g(z (B +1/n B ) F )] = g(, z). Secondly we prove ha for each riple (, z, λ) [0, T ] R d R, we have g(, λz) λg(, z). (3.12) Given λ R, we define a corresponding convex fncion ϕ λ : R R, sch ha ϕ λ (x) = λx, x R. Given [0, T [, le s pick a large enogh posiive ineger n, sch ha + 1/n T. Then for each z R d, i is obvios ha ϕ λ (z (B +1/n B )) L 2 (Ω, F T, P ). By (ii), we know ha, ha is, E g [ϕ λ (z (B +1/n B )) F ] ϕ λ [E g (z (B +1/n B ) F )]; E g [λz (B +1/n B ) F ] λ[e g (z (B +1/n B ) F )]. (3.13) Becase of (3.8), we know here exiss a sbseqence {n k } k=1 sch ha lim n k[e g (λz (B +1/nk B ) F )] = g(, λz), k lim λn k[e g (z (B +1/nk B ) F )] = λg(, z). k Ths for he given [0, T [, z R d, λ R, by (3.13), we have g(, λz) λg(, z). By (A4), we know ha for each z, he process g(, z) is coninos. Hence we have g(t, λz) = lim g(t ε, λz) lim λg(t ε, z) = λg(t, z). ε 0 + ε 0 + Therefore we can ge (3.12) immediaely. The proof is complee. Remark 3.1. When we prove ha (i) implies (ii), we do no need (A4). Example 3.1. Le g : R R be defined as follows: g(z) = z 4, if z 1 and g(z) = 4 z 3, if z > 1. We can see clearly ha hogh g is convex, g is no sper-homogeneos. Ths for his generaor g, by Theorem 3.1, we know ha Jensen s ineqaliy for g-expecaion does no hold in general. In fac, if we ake T = 1, ξ = B T T and ϕ(x) = x 3, x R, hen we can verify ha (B, 1) [0,T ] is he solion of he following BSDE: y = ξ + g(z s )ds z s db s, 0 T,
and ( B 3 26T + We can calclae ha ON JENSEN S INEQUALITY FOR g-expectation 409 81, 1 3 ) [0,T ] is he solion of he following BSDE: ȳ = ϕ(ξ) + g( z s )ds z s db s, 0 T. E g [ϕ(ξ) F ] ϕ[e g (ξ F )] = 26 ( T ) < 0, when < T. 81 Example 3.1 yields a naral qesion: Wha kind of convex generaor g can make Jensen s ineqaliy for g-expecaion hold in general? The following Theorem 3.2 will answer his qesion. Definiion 3.2. We call a generaor g(, z) is posiive-homogeneos in z if λ 0, [0, T ], z R d, g(, λz) = λg(, z). Theorem 3.2. Sppose g saisfies (A1), (A3) and (A4). Sppose moreover ha for each R, z g(, z) is convex in z. Then he following wo condiions are eqivalen: ( i ) g(, z) is posiive-homogeneos in z; (ii) Jensen s ineqaliy for g-expecaion holds in general. Proof. By Theorem 3.1, i sffices o prove ha if g(, z) is convex in z and g(, 0) 0, hen g(, z) is posiive-homogeneos in z if and only if g(, z) is sper-homogeneos. Sppose g(, z) is posiive-homogeneos in z. We only need o consider he case when λ 0. For each λ 0, (, z) [0, T ] R d, since g is convex and g(, 0) 0, we have, ( 0 = g(, 0) = g, λz 2 + ( λ)z ) g(, λz) g(, λz) g(, λz) + = + 2 2 2 2 Ths we have λ 0, (, z) [0, T ] R d, g(, λz) λg(, z). λg(, z). 2 Hence g(, z) is sper-homogeneos. Sppose g(, z) is sper-homogeneos. For each given riple (, z, λ) [0, T ] R d R +, if 0 λ 1, hen by he convexiy of g and (A3) we have g(, λz) λg(z). Ths by he sper-homogeneiy of g, we have, λ [0, 1], [0, T ], g(, λz) = λg(, z). (3.14) For λ > 1, i follows from (3.14) ha λg(, z) = λg (, 1 ) λ (λz) = λ 1 g(, λz) = g(, λz). λ Ths g(, z) is posiive-homogeneos. This complees he proof. Corollary 3.1. Given µ 0, le he generaor g(, z) = µ z, (, z) [0, T ] R d. Then Jensen s ineqaliy for g-expecaion holds in general. This kind of g-expecaion E g [ ] plays a key role in [4].
410 JIANG, L. & CHEN, Z. J. 4. Jensen s Ineqaliy for Monoonic Convex Fncion ϕ In his secion, we will consider he following problem: If g is independen of y, ϕ is a monoonic convex fncion, hen wha condiions shold be given o he generaor g, sch ha Jensen s ineqaliy for g-expecaion holds for ϕ? We will give wo necessary and sfficien condiions o solve his problem, one condiion is for increasing convex fncion ϕ, he oher condiion is for decreasing convex fncion ϕ. Theorem 4.1. Le g saisfy (A1), (A3) and (A4). Then he following wo condiions are eqivalen: ( i ) (, z, λ) [0, T ] R d R +, g(, λz) λg(, z); (ii) Jensen s ineqaliy for g-expecaion holds for increasing convex fncion, i.e., for each ξ L 2 (Ω, F T, P ) and increasing convex fncion ϕ : R R, if ϕ(ξ) L 2 (Ω, F T, P ), hen for each [0, T ], E g [ϕ(ξ) F ] ϕ[e g (ξ F )]. Proof. (i) (ii). Given ξ L 2 (Ω, F T, P ) and increasing convex fncion ϕ sch ha ϕ(ξ) L 2 (Ω, F T, P ). For each [0, T ] and posiive ineger n, js as in he proof of Theorem 3.1, we se or define η = ϕ [E g (ξ F )], Ω,n := { E g [ξ F ] + η + ϕ[e g (ξ F )] n}, η,n = 1 Ω,n η. We already know ha Ω,n F, η,n, 1 Ω,n are F -measrable; η,n, 1 Ω,n ϕ[e g (ξ F )] are bonded by n; 1 Ω,n ϕ(ξ), η,n ξ L 2 (Ω, F T, P ), 1 Ω,n ϕ[e g (ξ F )] L 2 (Ω, F, P ); (η,n E g (ξ F s )) s [,T ] SF 2 (, T ; R). Moreover, we also know ha E g [1 Ω,n ϕ(ξ) F ] 1 Ω,n ϕ[e g (ξ F )] E g [η,n ξ F ] η,n E g [ξ F ]. (4.1) Le (y, z ) [0,T ] be he niqe sqare inegrable solion of he following BSDE: Then for he given [0, T ], we have y = ξ + g(s, z s )ds z s db s, 0 T. (4.2) η,n y = η,n ξ + η,n g(s, z s )ds η,n z s db s, T. (4.3) For he given, again we define fncion g 1 (s, z) in his way: for each (s, z) [, T ] R d, { η,n g(s, z/η,n ), if η,n 0; g 1 (s, z) := 0, if η,n = 0. Consider he solions of he following BSDE (4.4) and BSDE (4.5): ȳ = η,n ξ + ỹ = η,n ξ + g 1 (s, z s )ds g(s, z s )ds z s db s, T, (4.4) z s db s, T. (4.5)
ON JENSEN S INEQUALITY FOR g-expectation 411 Analogos o he proof of Theorem 3.1, from (4.3) we dedce ha (η,n y s, η,n z s ) s [,T ] is he niqe solion of BSDE (4.4). For he given [0, T ] and ϕ, since ϕ is increasing, we have η = ϕ [E g (ξ F )] 0, η,n = 1 Ω,n η 0. In view of (i), for each s [, T ], we have Therefore, for each s [, T ], we can ge, g(s, η,n z s ) η,n g(s, z s ). (4.6) g(s, z s ) = g(s, η,n z s ) η,n g(s, z s ) = g 1 (s, η,n z s ) = g 1 (s, z s ). Ths from Comparison Theorem we have This wih (4.1), i follows ha E g [η,n ξ F ] = ỹ ȳ = η,n y = η,n E g [ξ F ]. (4.7) E g [1 Ω,n ϕ(ξ) F ] 1 Ω,n ϕ[e g (ξ F )] E g [η,n ξ F ] η,n E g [ξ F ] 0. Applying Lebesge s dominaed heorem o (1 Ω,n ϕ(ξ)) n=1, we can ge easily ha L 2 lim n 1 Ω,n ϕ(ξ) = ϕ(ξ). Similarly o he proof of Theorem 3.1, we can ge Hence for each [0, T ], P -a,s., we have L 2 lim n E g[1 Ω,n ϕ(ξ) F ] = E g [ϕ(ξ) F ]. E g [ϕ(ξ) F ] ϕ[e g (ξ F )]. (ii) (i). Given λ 0, we define a corresponding increasing convex fncion ϕ λ : R R, sch ha ϕ λ (x) = λx, x R. For each [0, T [, z R d, le s pick a large enogh posiive ineger n, sch ha + 1/n T. I is obvios ha ϕ λ (z (B +1/n B )) L 2 (Ω, F T, P ). By (ii), we know ha Jensen s ineqaliy holds for he increasing fncion ϕ λ. Ths we have, ha is, E g [ϕ λ (z (B +1/n B )) F ] ϕ λ [E g (z (B +1/n B ) F )]; E g [λz (B +1/n B ) F ] λ[e g (z (B +1/n B ) F )]. (4.8) By (3.8), we know ha here exiss a sbseqence {n k } k=1 sch ha lim n k[e g (λz (B +1/nk B ) F )] = g(, λz), k lim λn k[e g (z (B +1/nk B ) F ] = λg(, z). k Ths for each [0, T [, z R d, λ 0, i follows from (4.8) ha (A4) and (4.9) imply ha g(, λz) λg(, z). (4.9) g(t, λz) = lim g(t ε, λz) lim λg(t ε, z) = λg(t, z). ε 0 + ε 0 + Hence (ii) implies (i). The proof is complee.
412 JIANG, L. & CHEN, Z. J. Corollary 4.1. Given µ 0, le he generaor g(, z) = µ z, (, z) [0, T ] R d. Then Jensen s ineqaliy for g-expecaion holds for increasing convex fncion ϕ. Similarly we can ge he following Theorem 4.2. Le g saisfy (A1), (A3) and (A4). Then he following wo condiions are eqivalen: ( i ) λ 0, (, z) [0, T ] R d, g(, λz) λg(, z); (ii) Jensen s ineqaliy for g-expecaion holds for decreasing convex fncion, i.e., for each ξ L 2 (Ω, F T, P ) and decreasing convex fncion ϕ : R R, if ϕ(ξ) L 2 (Ω, F T, P ), hen for each [0, T ], E g [ϕ(ξ) F ] ϕ[e g (ξ F )]. Proof. The proof of Theorem 4.2 is similar o ha of Theorem 4.1. We omi i. By Theorem 4.2, we can obain he following corollary immediaely. Corollary 4.2. Le g saisfy (A1) and (A3). If (, z) [0, T ] R d, g(, z) 0, hen Jensen s ineqaliy for g-expecaion holds for decreasing convex fncion ϕ. Acknowledgemen. The ahors hank Professor S. Peng and Professor J. Mémin for heir commens and help, and also hank he referee for his sggesions. References [ 1 ] Peng, S., BSDE and relaed g-expecaions, Piman Research Noes in Mahemaics Series, 364, 1997, 141 159. [ 2 ] Chen, Z. & Epsein, L., Ambigiy, risk and asse rerns in coninos ime, Economerica, 70(2002), 1403 1443. [ 3 ] Briand, P., Coqe, F., H, Y., Mémin, J. & Peng, S., A converse comparison heorem for BSDEs and relaed properies of g-expecaion, Elecon. Comm. Probab., 5(2000), 101 117. [ 4 ] Coqe, F., H, Y., Mémin, J. & Peng, S., Filraion consisen nonlinear expecaions and relaed g-expecaion, Probab. Theory Relaed Fields, 123(2002), 1 27. [ 5 ] Chen, Z. & Peng, S., A general downcrossing ineqaliy for g-maringales, Saisics and Probabiliy Leers, 46(2000), 169 175. [ 6 ] Pardox, E. & Peng, S., Adaped solion of a backward sochasic differenial eqaion, Sysems Conrol Leers, 14(1990), 55 61. [ 7 ] Peng, S., A generalized dynamic programming principle and Hamilon-Jacobi-Bellman eqaion, Sochasics, 38:2(1992), 119 134. [ 8 ] El Karoi, N., Peng, S. & Qenez, M. C., Backward sochasic differenial eqaions in finance, Mah. Finance, 7:1(1997), 1 71.