Before we give our main results, let us see an example.

Example 12.
Fix T=1 and d=1. Let ξ=f(W1), where f(x)=exp((x2/2p1)-x)1(x≥p1), 1<p1<2.

Obviously, f is an increasing function. We can easily get
(15)E[|ξ|p1]=∫p1∞exp(x22-p1x)12πe-(1/2)x2dx=12πp1e-p12<∞, E[|ξ|p]=∞, ∀p>p1.
Hence, ξ∈ℒ(Ω,ℱ1,P), but ξ∉L2(Ω,ℱ1,P).

Let ξn=ξ∧n, n=1,2,…. Clearly, for each n, ξn∈L2(Ω,ℱ,P). For simplicity, we will write ℰμ[·]≡ℰg[·] for g=μ|z|. From Theorem 1 in Chen and Kulperger’s [12], we know that ℰμ[ξn]=EQ[ξn], where dQ/dP=e-(1/2)μ2+μW1.

By Remark 8(i), we have ℰμ[ξn]→ℰμ[ξ], as n→∞. On the other hand, applying Hölder's inequality and noting that E[e-(1/2)μ2+μW1]=1 and E[e-(1/2)μ2q2+μqW1]=1, we obtain
(16)EQ[ξ]≤(E[|ξ|p1])1/p1(E[(dQdP)q])1/q≤e(1/2)(q-1)μ2(E[|ξ|p1])1/p1<∞,
where (1/p1)+(1/q)=1. It then follows from the monotonic convergence theorem that
(17)EQ[ξn]⟶EQ[ξ], as n⟶∞.
Thus
(18)ℰμ[ξ]=EQ[ξ].

Let φ(x)=(x-k)+, where k∈R. Obviously, φ(x) is a convex and increasing function. From this, we know that φ∘f is an increasing function. In a similar manner of the above, we can deduce that
(19)ℰμ[φ(ξ)]=EQ[φ(ξ)].

From (18), (19), and the classical Jensen's inequality, we have
(20)φ(ℰμ[ξ])=φ(EQ[ξ])≤EQ[φ(ξ)]=ℰμ[φ(ξ)].

This problem yields a natural question: in general, under which conditions on g do generalized Peng's g-expectations satisfy Jensen's inequality for convex functions?

The following theorem will answer this question.

Proof.
(i)⇒(ii) is obvious.

(ii)⇒(iii): let η=ξ+b. By (ii), we have
(22)ℰg[η-b]≥ℰg[η]-b,
That is,
(23)ℰg[ξ]+b≥ℰg[ξ+b].
Thus, for each (ξ,b)∈L2(Ω,ℱT,P)×R,
(24)ℰg[ξ+b]=ℰg[ξ]+b.
For each (X,t,k)∈L2(Ω,ℱT,P) × [0,T] × R, by (24), we know that for each A∈ℱt,
(25)ℰg[1A(X+k)]=ℰg[1AX+1Ak-k]+k=ℰg[1AX+1AC(-k)]+k=ℰg[ℰg[1AX+1AC(-k)∣ℱt]]+k=ℰg[1Aℰg[X∣ℱt]+1AC(-k)]+k=ℰg[1Aℰg[X∣ℱt]+1AC(-k)+k]=ℰg[1A(ℰg[X∣ℱt]+k)].
Thus,
(26)ℰg[X+k∣ℱt]=ℰg[X∣ℱt]+k a.s., ∀t∈[0,T].
On the other hand, for each λ≠0, define
(27)ℰλ[·∣ℱt]=ℰg[λ·∣ℱt]λ, ∀t∈[0,T].
It is easy to check that ℰg[·∣ℱt] and ℰλ[·∣ℱt] are two ℱ-expectations on L2(Ω,ℱT,P) (the notion of ℱ-expectation can be seen in [13]). From (ii), we have if λ>0, for each ξ∈L2(Ω,ℱT,P)(28)ℰλ[ξ]≥ℰg[ξ].
Hence, by Lemma 4.5 in [13], we have
(29)ℰλ[ξ∣ℱt]≥ℰg[ξ∣ℱt] a.s., ∀t∈[0,T].
Similarly, if λ<0, for each ξ∈L2(Ω,ℱT,P)(30)ℰλ[ξ]≤ℰg[ξ].
Hence, by Lemma 4.5 in [13] again, we have
(31)ℰλ[ξ∣ℱt]≤ℰg[ξ∣ℱt] a.s., ∀t∈[0,T].
Thus from (29) and (31), we have ∀(ξ,λ)∈L2(Ω,ℱT,P)×R,
(32)ℰg[λξ∣ℱt]≥λℰg[ξ∣ℱt] a.s., ∀t∈[0,T].
From (26) and (32), we have
(33)∀(ξ,a,b)∈L2(Ω,ℱT,P)×R×R,ℰg[aξ+b∣ℱt]≥aℰg[ξ∣ℱt]+b a.s., ∀t∈[0,T].

(iii)⇒(iv): Firstly, we prove that g is independent of y. From (iii), we can obtain that for each (ξ,y)∈L2(Ω,ℱT,P)×R,
(34)ℰg[ξ-y∣ℱt]=ℰg[ξ∣ℱt]-y, a.s., ∀t∈[0,T].
For each (t,y,z)∈[0,T]×R×Rd, let Y·t,y,z be the solution of the following SDE defined on [t,T]:
(35)Yst,y,z=y-∫tsg(r,Yrt,y,z,z)dr+z·(Ws-Wt).
From (34), we have
(36)Yrt,y,z-y=ℰg[Yst,y,z∣ℱr]-y=ℰg[Yst,y,z-y∣ℱr], t≤r≤s≤T.
Let Ys=Yst,y,z-y, s∈[t,T] and Z be the corresponding part of Itô's integrand. It then follows that
(37)Ys=-∫tsg(r,Yrt,y,z,z)dr+∫tsz·dWr=-∫tsg(r,Yr,Zr)dr+∫tsZr·dWr.
Thus, Zr≡z and
(38)g(r,Yr,z)=g(r,Yrt,y,z-y,z)=g(r,Yrt,y,z,z).
Then, we can apply Lemma 4.4 in Peng [14] to obtain that for each (y,z)∈R×Rd,
(39)g(t,y,z)=g(t,0,z), dP×dt a.s.
Namely, g is independent of y.

Now we prove that g is superhomogeneous with respect to z. From (iii), we can obtain that for each (ξ,λ)∈L2(Ω,ℱT,P)×R,
(40)λℰg[ξ∣ℱt]≤ℰg[λξ∣ℱt], a.s., ∀t∈[0,T].
For each (t,z)∈[0,T]×Rd, let Y·t,z be the solution of the following SDE defined on [t,T]:
(41)Yst,z=-∫tsg(r,z)dr+z·(Ws-Wt).
From (40), we have
(42)ℰg[λYst,z∣ℱr]≥λℰg[Yst,z∣ℱr]=λYrt,z, t≤r≤s≤T.
Thus, (λYst,z)s∈[t,T] is an ℰg-submartingale. From the decomposition theorem of ℰg-supermartingale (see [15]), it follows that there exists an increasing process (As)s∈[t,T] such that
(43)λYst,z=-∫tsg(r,Zr)dr+As-At+∫tsZr·dWr, s∈[t,T].
This with λYst,z=-∫tsλg(r,z)dr+∫tsλz·dWr yields Zr≡λz and
(44)λg(r,z)≤g(r,λz), dP×dt a.s.

At last, we prove that g is positively homogeneous with respect to z. From (iii), we can obtain that for each fixed λ>0 and ξ∈L2(Ω,ℱT,P),
(45)1λℰg[λξ∣ℱt]≤ℰg[ξ∣ℱt], a.s., ∀t∈[0,T],
that is,
(46)ℰg[λξ∣ℱt]≤λℰg[ξ∣ℱt], a.s., ∀t∈[0,T].
Thus, we have
(47)ℰg[λξ∣ℱt]=λℰg[ξ∣ℱt], a.s., ∀t∈[0,T].
Obviously, if λ=0, (47) still holds. Thus, for each λ≥0,
(48)ℰg[λξ∣ℱt]=λℰg[ξ∣ℱt], a.s., ∀t∈[0,T].
For each (t,z)∈[0,T]×Rd, let Y·t,z be the solution of SDE (34). From (48), for each λ≥0, we have
(49)ℰg[λYst,z∣ℱr]=λℰg[Yst,z∣ℱr]=λYrt,z, t≤r≤s≤T.
This implies that there exists a process Z·t,z,λ such that
(50)λYst,z=-∫tsg(r,Zrt,z,λ)dr+∫tsZrt,z,λ·dWr, s∈[t,T].
Comparing this with λYst,z=-∫tsλg(r,z)dr+∫tsλz·dWr, it follows that Zrt,z,λ≡λz and
(51)λg(r,z)=g(r,λz), dP×dt a.s.

(iv)⇒(iii): By comparison theorem (for example, we can see [3]), it is easy to obtain (iii).

(iii)⇒(i): Suppose (iii) holds. From (iii) and by Remark 8 (i), we have
(52)∀(X,k)∈ℒ(Ω,ℱT,P)×R, ℰg[X+k∣ℱt]=ℰg[X∣ℱt]+k a.s.,(53)∀(X,λ)∈ℒ(Ω,ℱT,P)×R,ℰg[λX∣ℱt]≥λℰg[X∣ℱt] a.s.
From (53), we can deduce that for each bounded variable ζ∈ℱt,
(54)∀X∈ℒ(Ω,ℱT,P), ℰg[ζX∣ℱt]≥ζℰg[X∣ℱt] a.s.
In fact, let {Ai}i=1m be a ℱt-measurable partition of Ω and let λi∈R (i=1,2,…,m). By (53), we have
(55)ℰg[∑i=1mλi1AiX∣ℱt]=∑i=1m1Aiℰg[λiX∣ℱt]≥∑i=1m1Aiλiℰg[X∣ℱt] a.s.
In other words, for each X∈ℒ(Ω,ℱT,P) and each simple function ζ∈ℒ(Ω,ℱt,P),
(56)ℰg[ζX∣ℱt]≥ζℰg[X∣ℱt] a.s.
Thus, thanks to Remark 8(ii), it follows that (54) is true.

The main idea of the following proof is derived from [7]. Given ξ∈ℒ(Ω,ℱT,P) and convex function φ such that φ(ξ)∈ℒ(Ω,ℱT,P), we set ηt=φ-′(ℰg[ξ∣ℱt]). Then ηt is ℱt-measurable. Since φ is convex, we have
(57)φ(x)-φ(y)≥φ-′(y)(x-y), ∀x,y∈R.
Take x=ξ, y=ℰg[ξ∣ℱt]. Then we have
(58)φ(ξ)-φ(ℰg[ξ∣ℱt])≥ηt(ξ-ℰg[ξ∣ℱt]) a.s.
For each n∈N, we define
(59)Ωt,n:={|ℰg[ξ∣ℱt]|+|ηt|+|φ(ℰg[ξ∣ℱt])|≤n},
so we have
(60)ℰg[1Ωt,nφ(ξ)∣ℱt] ≥ℰg[1Ωt,nφ(ℰg[ξ∣ℱt])-1Ωt,nηtℰg[ξ∣ℱt] +1Ωt,nηtξ∣ℱt] a.s.
By the definition of 1Ωt,n, we know
(61)1Ωt,nφ(ℰg[ξ∣ℱt])-1Ωt,nηtℰg[ξ∣ℱt]∈ℒ(Ω,ℱt,P).
Thus, in view of (52) and from Proposition 10, we can get
(62)ℰg[1Ωt,nφ(ξ)∣ℱt]≥1Ωt,nφ(ℰg[ξ∣ℱt])-1Ωt,nηtℰg[ξ∣ℱt]+ℰg[1Ωt,nηtξ∣ℱt] a.s.
Moreover, from (54), considering that 1Ωt,nηt∈ℱt and is bounded by n, we can get(63)ℰg[1Ωt,nηtξ∣ℱt]≥1Ωt,nηtℰg[ξ∣ℱt] a.s.
Hence, we can deduce that for each n∈N,
(64)ℰg[1Ωt,nφ(ξ)∣ℱt]≥1Ωt,nφ(ℰg[ξ∣ℱt]) a.s.
Finally, thanks to Remark 8 (ii) again, we can get
(65)ℰg[φ(ξ)∣ℱt]≥φ(ℰg[ξ∣ℱt]) a.s.
Hence, Jensen's inequality for ℰg[·∣ℱt] holds in general. The proof of Theorem 13 is complete.

Proof.
Let g(t,z)= ess supv∈Dvt·z. Obviously, g(t,z) is superhomogeneous and positively homogeneous with respect to z. and satisfies (A.1) and (A.2).

From El Karoui and Quenez [16], we have ess supv∈DEQv[ξ∣ℱt]=ℰg[ξ∣ℱt], a.s., ∀ξ∈L2(Ω,ℱT,P). Now we prove ess supv∈DEQv[ξ∣ℱt]=ℰg[ξ∣ℱt], a.s., ∀ξ∈ℒ(Ω,ℱT,P). Indeed, for any ξ∈ℒ(Ω,ℱT,P), there exists 1<p<2 such that ξ∈Lp(Ω,ℱT,P). Let ξn=(ξ∧n)∨(-n), n=1,2,…. Clearly, for each n, ξn∈L2(Ω,ℱT,P), then
(68)ess supv∈DEQv[ξn∣ℱt]=ℰg[ξn∣ℱt], a.s.

Since
(69)ess supv∈D EQv[ξn∣ℱt] =ess supv∈D(EQv[ξn-ξ∣ℱt]+EQv[ξ∣ℱt]) ≤ess supv∈DEQv[ξn-ξ∣ℱt]+ess supv∈DEQv[ξ∣ℱt],
we have
(70)ℰg[ξn∣ℱt]-ess supv∈DEQv[ξ∣ℱt] ≤ess supv∈DEQv[ξn-ξ∣ℱt].
With an approach similar to the one above, we can get easily that
(71)ℰg[ξn∣ℱt]-ess supv∈D EQv[ξ∣ℱt]≥ess infv∈D EQv[ξn-ξ∣ℱt].
Combining (42) with (43), we have
(72)|ℰg[ξn∣ℱt]-ess supv∈DEQv[ξ∣ℱt]| ≤({|ess supv∈DEQv[ξn-ξ∣ℱt]|}|ess infv∈D EQv[ξn-ξ∣ℱt]| ∨|ess supv∈DEQv[ξn-ξ∣ℱt]|) ≤ess supv∈D EQv[|ξn-ξ|∣ℱt].
By Hölder's inequality and noting that (e-(1/2)∫0t|vs|2ds+∫0tvs·dWs)t∈[0,T] and (e-(1/2)∫0t|qvs|2ds+∫0tqvs·dWs)t∈[0,T] are both martingales with respect to (ℱt)t∈[0,T], we can obtain
(73)EQv[|ξn-ξ|∣ℱt] =E[|ξn-ξ|(dQv/dP)∣ℱt]E[(dQv/dP)∣ℱt] ≤(E[|ξn-ξ|p∣ℱt])1/p(E[(dQv/dP)q∣ℱt])1/qE[(dQv/dP)∣ℱt] ≤e(1/2)(q-1)μ2T(E[|ξn-ξ|p∣ℱt])1/p,
where (1/p)+(1/q)=1. It then follows from Lebesgue's dominated convergence theorem that
(74)ess supv∈D EQv[|ξn-ξ|∣ℱt]⟶0, as n⟶∞.
Hence,
(75)|ℰg[ξn∣ℱt]-ess supv∈D EQv[ξ∣ℱt]|⟶0, as n⟶∞.

On the other hand, from Remark 8 (i), we have
(76)ℰg[ξn∣ℱt]⟶ℰg[ξ∣ℱt], as n⟶∞.
Thus,
(77)ℰg[ξ∣ℱt]=ess supv∈D EQv[ξ∣ℱt], a.s.,∀ξ∈ℒ(Ω,ℱT,P).

Applying Theorem 13, we have
(78)φ(ess supv∈DEQv[ξ∣ℱt])≤ess supv∈DEQv[φ(ξ)∣ℱt], a.s.