Jensen's Inequality for Generalized Peng's 𝑔-Expectations and Its Applications

Abstract

We study Jensen's inequality for generalized Peng's 𝑔-expectations and give four equivalent conditions on Jensen's inequality for generalized Peng's 𝑔-expectations without the assumption that the generator 𝑔 is continuous with respect to 𝑡. This result includes and extends some existing results. Furthermore, we give some applications of Jensen's inequality for generalized Peng's 𝑔-expectations.

1. Introduction

By Pardoux and Peng [1], we know that there exists a unique adapted and square integrable solution to a backward stochastic differential equation (BSDE for short) of the type 𝑦_𝑡=𝜉+^𝑇_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠−^𝑇_𝑡𝑧_𝑠⋅𝑑𝑊_𝑠,𝑡∈[0,𝑇], (1) provided that the function 𝑔 is Lipschitz in both variables 𝑦 and 𝑧, and 𝜉 and (𝑔(𝑡,0,0))_{𝑡∈[0,𝑇]} are square integrable. 𝑔 is said to be the generator of BSDE (1). We denote the unique adapted and square integrable solution of BSDE (1) by (𝑦_𝑡^{(𝑇,𝑔,𝜉)},𝑧_𝑡^{(𝑇,𝑔,𝜉)})_{𝑡∈[0,𝑇]}.

Based on such a BSDE, Peng [2] introduced the notion of 𝑔-expectation. He proved that the 𝑔-expectation preserves many of properties of the classical mathematical expectation, but not the linearity property, and thus the 𝑔-expectation is a type of nonlinear mathematical expectation. Indeed, 𝑔-expectation is a kind of nonlinear expectation, which can be considered as a nonlinear extension of the well-known Girsanov transformations. The original motivation for studying 𝑔-expectation comes from the theory of expected utility. Since the notion of 𝑔-expectation was introduced, many properties of 𝑔-expectation have been investigated by many researchers. In 1997, Peng [3] introduced the notions of conditional 𝑔-expectation and 𝑔-martingale. Later, Briand et al. [4] studied Jensen's inequality for 𝑔-expectations and gave a counter example and a proposition to indicate that even for a linear function, Jensen's inequality might fail for some 𝑔-expectations. This yields a natural question: under which conditions on 𝑔 in the 𝑔-expectation does Jensen's inequality hold for any convex function? Under the assumptions that 𝑔 does not depend on 𝑦 and is convex, Chen et al. [5, 6] studied Jensen's inequality for 𝑔-expectations and gave a necessary and sufficient condition on 𝑔 under which Jensen's inequality holds for convex functions. Provided that 𝑔 only does not depend on 𝑦, Jiang [7] gave another necessary and sufficient condition on 𝑔 under which Jensen's inequality holds for convex functions. It was an improved result in comparison with the result that Chen et al. yielded. Later, this result was improved by Hu [8] and Jiang [9] showing that, in fact, 𝑔 must be independent of 𝑦. But these results need the assumption that the generator 𝑔 is continuous with respect to 𝑡.

In this paper, without the assumption that the generator 𝑔 is continuous with respect to 𝑡, we study Jensen's inequality for generalized Peng's 𝑔-expectations and give four equivalent conditions on Jensen's inequality for generalized Peng's 𝑔-expectations, which generalize the known results on Jensen's inequality for 𝑔-expectations in Chen et al. [5, 6], Jiang [7, 9], and Hu [8]. Furthermore, we give some applications of Jensen's inequality for generalized Peng's 𝑔-expectations.

This paper is organized as follows: in Section 2, we introduce some notations, assumptions, notions, and lemmas which will be useful in this paper; in Section 3, we give our main results including the proofs and applications.

2. Preliminaries

Firstly, let us list some notations, assumptions, notions, lemmas, and propositions that are used in this paper. Let (Ω,ℱ,𝑃) be a probability space and let (𝑊_𝑡)_𝑡≥0 be a 𝑑-dimensional standard Brownian motion with respect to filtration (ℱ_𝑡)_𝑡≥0 generated by Brownian motion and all 𝑃-null subsets, that is, ℱ_𝑡=𝜎𝑊_𝑠;𝑠≤𝑡∨𝒩, (2) where 𝒩 is the set of all 𝑃-null subsets. Fix a real number 𝑇>0. For any positive integer 𝑛 and 𝑧∈𝑅^𝑛, |𝑧| denotes its Euclidean norm.

We define the following usual spaces of processes (random variables):(i)Consider 𝐿^𝑝(Ω,ℱ_𝑇,𝑃)={𝜉∶𝜉 is ℱ_𝑇-measurable random variable such that 𝐸[|𝜉|^𝑝]<∞,𝑝≥1};(ii)Consider ℒ(Ω,ℱ_𝑇,𝑃)=⋃_𝑝>1𝐿^𝑝(Ω,ℱ_𝑇,𝑃);(iii)Consider 𝒮^𝑝_ℱ(0,𝑇;𝑃;𝑅)={𝑉∶𝑉 is a continuous process with 𝐸[sup_{0≤𝑡≤𝑇}|𝑉_𝑡|^𝑝]<∞,𝑝≥1};(iv)Consider 𝒮_ℱ(0,𝑇;𝑃;𝑅)=⋃_𝑝>1𝒮^𝑝_ℱ(0,𝑇;𝑃;𝑅);(v)Consider ℒ^𝑝_ℱ(0,𝑇;𝑃;𝑅^𝑛)={𝑉∶𝑉is  a  progressively measurable process with 𝐸[(∫^𝑇₀|𝑉_𝑠|²𝑑𝑠)^𝑝/2]<∞,𝑝≥1};(vi)Consider ℒ_ℱ(0,𝑇;𝑃;𝑅^𝑛)=⋃_𝑝>1𝐿^𝑝_ℱ(0,𝑇;𝑃;𝑅^𝑛).

Suppose the generator 𝑔(𝜔,𝑡,𝑦,𝑧)∶Ω×[0,𝑇]×𝑅×𝑅^𝑑↦𝑅 satisfies the following assumptions:(A.1) there exists a constant 𝜇>0, such that 𝑃-a.s., we have: ∀𝑡∈[0,𝑇], ∀𝑦₁,𝑦₂∈𝑅, 𝑧₁,𝑧₂∈𝑅^𝑑, |𝑔(𝑡,𝑦₁,𝑧₁)−𝑔(𝑡,𝑦₂,𝑧₂)|≤𝜇(|𝑦₁−𝑦₂|+|𝑧₁−𝑧₂|);(A.2) 𝑃-a.s., ∀(𝑡,𝑦)∈[0,𝑇]×𝑅, 𝑔(𝑡,𝑦,0)≡0.

The following lemma is a special case of Theorem 4.2 in Briand et al. [10].

Lemma 1. Suppose 𝑔 satisfies (A.1) and (A.2). Then for each given 𝜉∈𝐿^𝑝(Ω,ℱ_𝑇,𝑃), where 1<𝑝<2, the BSDE (1) has a unique pair of adapted processes (𝑦_𝑡^{(𝑇,𝑔,𝜉)},𝑧_𝑡^{(𝑇,𝑔,𝜉)})_{𝑡∈[0,𝑇]}∈𝒮^𝑝_ℱ(0,𝑇;𝑃;𝑅)×𝑙^𝑝_ℱ(0,𝑇;𝑃;𝑅^𝑑).

From Lemma 1, we have the following.

Remark 2. Suppose 𝑔 satisfies (A.1) and (A.2). Then for each given 𝜉∈ℒ(Ω,ℱ_𝑇,𝑃), the BSDE (1) has a unique pair of adapted processes (𝑦_𝑡^{(𝑇,𝑔,𝜉)},𝑧_𝑡^{(𝑇,𝑔,𝜉)})_{𝑡∈[0,𝑇]}∈𝒮_ℱ(0,𝑇;𝑃;𝑅)×ℒ_ℱ(0,𝑇;𝑃;𝑅^𝑑).

Now, we introduce the notions of generalized Peng's 𝑔-expectation and generalized conditional Peng's 𝑔-expectation.

Definition 3 (generalized Peng's 𝑔-expectation [11]). Suppose 𝑔 satisfies (A.1) and (A.2). For any 𝜉∈ℒ(Ω,ℱ_𝑇,𝑃), let (𝑦_𝑡^{(𝑇,𝑔,𝜉)},𝑧_𝑡^{(𝑇,𝑔,𝜉)})_{𝑡∈[0,𝑇]} be the solution of BSDE (1). Consider the mapping ℰ_𝑔[⋅]∶ℒ(Ω,ℱ_𝑇,𝑃)↦𝑅, denoted by ℰ_𝑔[𝜉]=𝑦₀^{(𝑇,𝑔,𝜉)}. One calls ℰ_𝑔[𝜉] the generalized Peng's 𝑔-expectation of 𝜉.

Definition 4 (generalized Peng's conditional 𝑔-expectation [11]). Suppose 𝑔 satisfies (A.1) and (A.2). The generalized Peng's conditional 𝑔-expectation of 𝜉 with respect to ℱ_𝑡 is defined by ℰ_𝑔𝜉∣ℱ_𝑡=𝑦_𝑡^{(𝑇,𝑔,𝜉)},𝑡∈[0,𝑇]. (3)

Then, let us list some basic properties of generalized Peng's 𝑔-expectation.

Proposition 5 (see [11]). Considerℰ_𝑔[𝜉∣ℱ_𝑡] is the unique random variable 𝜂 in ℒ(Ω,ℱ_𝑡,𝑃) such that ℰ_𝑔1_𝐴𝜉=ℰ_𝑔1_𝐴𝜂,∀𝐴∈ℱ_𝑡. (4)

Proposition 6 (see [11]). Suppose 𝑔 satisfies (A.1) and (A.2). If 𝑔 does not depend on 𝑦, that is, 𝑔(𝜔,𝑡,𝑧)∶Ω×[0,𝑇]×𝑅^𝑑↦𝑅, then ℰ_𝑔𝜉+𝜂∣ℱ_𝑡=ℰ_𝑔𝜉∣ℱ_𝑡+𝜂,∀𝜉∈ℒΩ,ℱ_𝑇,𝑃,∀𝜂∈ℒΩ,ℱ_𝑡,𝑃. (5)

Proposition 7 (see [11]). Suppose 𝑔 satisfies (A.1) and (A.2). For 𝜉,𝜂_𝑛∈𝐿^𝑝(Ω,ℱ_𝑇,𝑃), where 𝑛=1,2,… and 𝑝>1, if 𝐸[|𝜉−𝜂_𝑛|^𝑝∣ℱ_𝑡]→0,a.s.,𝑡∈[0,𝑇], then lim_𝑛→∞ℰ_𝑔𝜂_𝑛∣ℱ_𝑡=ℰ_𝑔𝜉∣ℱ_𝑡,𝑎.𝑠.,𝑡∈[0,𝑇]. (6) Applying Proposition 7, one can immediately obtain the following.

Remark 8. (i) For any 𝜉∈ℒ(Ω,ℱ_𝑇,𝑃), let 𝜉^𝑛=(𝜉∧𝑛)∨(−𝑛), 𝑛=1,2,…, then lim_𝑛→∞ℰ_𝑔[𝜉^𝑛∣ℱ_𝑡]=ℰ_𝑔[𝜉∣ℱ_𝑡], a.s., ∀𝑡∈[0,𝑇].
(ii) For any 𝜉_𝑛∈ℒ(Ω,ℱ_𝑇,𝑃), if lim_𝑛→∞𝜉_𝑛=𝜉 a.s. and |𝜉_𝑛|≤𝜂 a.s. with 𝜂∈ℒ(Ω,ℱ_𝑇,𝑃), then lim_𝑛→∞ℰ_𝑔[𝜉^𝑛∣ℱ_𝑡]=ℰ_𝑔[𝜉∣ℱ_𝑡], a.s., ∀𝑡∈[0,𝑇].

Lemma 9. Suppose 𝑔 satisfies (A.1) and (A.2). Let {𝐴_𝑖}^𝑚_𝑖=1 be a ℱ_𝑡-measurable partition of Ω (i.e., 𝐴_𝑖∈ℱ_𝑡, 𝐴_𝑖⋂𝐴_𝑗=∅ if 𝑖≠𝑗 and ⋃^𝑚_𝑖=1𝐴_𝑖=Ω), where 𝑡≤𝑇. Then for each 𝑋_𝑖∈ℒ(Ω,ℱ_𝑇,𝑃), 𝑖=1,…,𝑚, one has ^𝑚_𝑖=11_𝐴𝑖ℰ_𝑔𝑋_𝑖∣ℱ_𝑡=ℰ_𝑔^𝑚_𝑖=11_𝐴𝑖𝑋_𝑖∣ℱ_𝑡𝑎.𝑠. (7)

Proof. We consider the following BSDEs: ℰ_𝑔𝑋_𝑖∣ℱ_𝑡=𝑋_𝑖+^𝑇_𝑡𝑔𝑠,ℰ_𝑔𝑋_𝑖∣ℱ_𝑠,𝑧^{(𝑇,𝑔,𝑋𝑖)}_𝑠−𝑑𝑠_𝑇𝑡𝑧^{(𝑇,𝑔,𝑋𝑖)}_𝑠⋅𝑑𝑊_𝑠,𝑖=1,…,𝑚,ℰ_𝑔^𝑚_𝑖=11_𝐴𝑖𝑋_𝑖∣ℱ_𝑡=^𝑚_𝑖=11_𝐴𝑖𝑋_𝑖+^𝑇_𝑡𝑔𝑠,ℰ_𝑔^𝑚_𝑖=11_𝐴𝑖𝑋_𝑖∣ℱ_𝑠,𝑧^{(𝑇,𝑔,∑𝑚𝑖=11𝐴𝑖𝑋𝑖)}_𝑠−𝑑𝑠_𝑇𝑡𝑧^{(𝑇,𝑔,∑𝑚𝑖=11𝐴𝑖𝑋𝑖)}_𝑠⋅𝑑𝑊_𝑠. (8)(9) By the fact that ∑^𝑚_𝑖=11_𝐴𝑖𝑔(𝑠,ℰ_𝑔[𝑋_𝑖∣ℱ_𝑠],𝑧^{(𝑇,𝑔,𝑋𝑖)}_𝑠) = 𝑔(𝑠,∑^𝑚_𝑖=11_𝐴𝑖ℰ_𝑔[𝑋_𝑖∣ℱ_𝑠],∑^𝑚_𝑖=11_𝐴𝑖𝑧^{(𝑇,𝑔,𝑋𝑖)}_𝑠),𝑡≤𝑠≤𝑇 and from (8), we have ^𝑚_𝑖=11_𝐴𝑖ℰ_𝑔𝑋_𝑖∣ℱ_𝑡=^𝑚_𝑖=11_𝐴𝑖𝑋_𝑖+^𝑇_𝑡𝑔𝑠,^𝑚_𝑖=11_𝐴𝑖ℰ_𝑔𝑋_𝑖∣ℱ_𝑠,^𝑚_𝑖=11_𝐴𝑖𝑧^{(𝑇,𝑔,𝑋𝑖)}_𝑠−𝑑𝑠_{𝑇𝑡𝑚𝑖=1}1_𝐴𝑖𝑧^{(𝑇,𝑔,𝑋𝑖)}_𝑠⋅𝑑𝑊_𝑠. (10) Comparing this with (9), it follows that ∑^𝑚_𝑖=11_𝐴𝑖ℰ_𝑔[𝑋_𝑖∣ℱ_𝑡]=ℰ_𝑔[∑^𝑚_𝑖=11_𝐴𝑖𝑋_𝑖∣ℱ_𝑡] a.s. The proof of Lemma 9 is complete.

Proposition 10. Suppose 𝑔 satisfies (A.1) and (A.2). Then the following two statements are equivalent:(i)consider ∀(𝑋,𝑘)∈ℒ(Ω,ℱ_𝑇,𝑃)×𝑅, ℰ_𝑔[𝑋+𝑘∣ℱ_𝑡]=ℰ_𝑔[𝑋∣ℱ_𝑡]+𝑘a.s.,(ii)consider ∀(𝑋,𝜂)∈ℒ(Ω,ℱ_𝑇,𝑃)×ℒ(Ω,ℱ_𝑡,𝑃), ℰ_𝑔[𝑋+𝜂∣ℱ_𝑡]=ℰ_𝑔[𝑋∣ℱ_𝑡]+𝜂a.s.

Proof. It is obvious that (ii) implies (i). We only need to prove that (i) implies (ii). Suppose (i) holds. Let {𝐴_𝑖}^𝑚_𝑖=1 be a ℱ_𝑡-measurable partition of Ω and let 𝜆_𝑖∈𝑅 (𝑖=1,2,…,𝑚). From Lemma 9 and (i), we deduce that for each 𝑋∈ℒ(Ω,ℱ_𝑇,𝑃), ℰ_𝑔𝑋+^𝑚_𝑖=1𝜆_𝑖1_𝐴𝑖∣ℱ_𝑡=ℰ_𝑔^𝑚_𝑖=11_𝐴𝑖𝑋+𝜆_𝑖∣ℱ_𝑡=^𝑚_𝑖=11_𝐴𝑖ℰ_𝑔𝑋+𝜆_𝑖∣ℱ_𝑡=^𝑚_𝑖=11_𝐴𝑖ℰ_𝑔𝑋∣ℱ_𝑡+𝜆_𝑖=ℰ_𝑔𝑋∣ℱ_𝑡+^𝑚_𝑖=1𝜆_𝑖1_𝐴𝑖a.s. (11) In other words, for any 𝑋∈ℒ(Ω,ℱ_𝑇,𝑃) and any simple function 𝜂∈ℒ(Ω,ℱ_𝑡,𝑃), ℰ_𝑔𝑋+𝜂∣ℱ_𝑡=ℰ_𝑔𝑋∣ℱ_𝑡+𝜂a.s. (12) Let 𝜂_𝑛∶=^{𝑛2𝑛−1}_𝑖=0𝑖2^𝑛1_{(𝑖/2^𝑛_{)≤𝜂<((𝑖+1)/2}^𝑛_)}+𝑛1_{{𝜂≥𝑛}}+^{𝑛2𝑛−1}_𝑖=0−𝑖2^𝑛1_{{−((𝑖+1)/2}^𝑛_{)≤𝜂<−(𝑖/2}^𝑛_)}+(−𝑛)1_{{𝜂<−𝑛}},𝑛=1,2,…. (13) Obviously, for each 𝑛, 𝜂_𝑛 is a simple function in ℒ(Ω,ℱ_𝑡,𝑃). From (12), we have ℰ_𝑔𝑋+𝜂_𝑛∣ℱ_𝑡=ℰ_𝑔𝑋∣ℱ_𝑡+𝜂_𝑛a.s. (14) On the other hand, lim_𝑛→∞(𝑋+𝜂_𝑛)=𝑋+𝜂, |𝑋+𝜂_𝑛|≤|𝑋|+|𝜂|. Thus, from Remark 8 (ii), it follows that (ii) is true. The proof of Proposition 10 is complete.

3. Main Results and Applications

Definition 11. Let 𝑔: Ω×[0,𝑇]×𝑅×𝑅^𝑑↦𝑅. The function 𝑔 is said to be superhomogeneous if for each (𝑦,𝑧)∈𝑅×𝑅^𝑑 and any real number 𝜆, then 𝑔(𝑡,𝜆𝑦,𝜆𝑧)≥𝜆𝑔(𝑡,𝑦,𝑧), 𝑑𝑃×𝑑𝑡 a.s. The function 𝑔 is said to be positively homogeneous if for each (𝑦,𝑧)∈𝑅×𝑅^𝑑 and any real number 𝜆≥0, then 𝑔(𝑡,𝜆𝑦,𝜆𝑧)=𝜆𝑔(𝑡,𝑦,𝑧), 𝑑𝑃×𝑑𝑡 a.s.

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 𝐸𝜉^𝑝1=^∞_𝑝1𝑥exp²2−𝑝₁𝑥1√2𝜋e^{−(1/2)𝑥2}=1𝑑𝑥√2𝜋𝑝₁e^−𝑝21<∞,𝐸𝜉^𝑝=∞,∀𝑝>𝑝₁. (15) Hence, 𝜉∈ℒ(Ω,ℱ₁,𝑃), but 𝜉∉𝐿²(Ω,ℱ₁,𝑃).
Let 𝜉^𝑛=𝜉∧𝑛, 𝑛=1,2,…. Clearly, for each 𝑛, 𝜉^𝑛∈𝐿²(Ω,ℱ,𝑃). For simplicity, we will write ℰ^𝜇[⋅]≡ℰ_𝑔[⋅] for 𝑔=𝜇|𝑧|. From Theorem 1 in Chen and Kulperger’s [12], we know that ℰ^𝜇[𝜉^𝑛]=𝐸_𝑄[𝜉^𝑛], where 𝑑𝑄/𝑑𝑃=e^{−(1/2)𝜇2+𝜇𝑊1}.
By Remark 8(i), we have ℰ^𝜇[𝜉^𝑛]→ℰ^𝜇[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 𝐸_𝑄𝜉≤𝐸[𝜉^𝑝1]^1/𝑝1𝐸𝑑𝑄𝑑𝑃^𝑞1/𝑞≤e^{(1/2)(𝑞−1)𝜇2}𝐸𝜉^𝑝11/𝑝1<∞, (16) where (1/𝑝₁)+(1/𝑞)=1. It then follows from the monotonic convergence theorem that 𝐸_𝑄𝜉^𝑛⟶𝐸_𝑄𝜉,as𝑛⟶∞. (17) Thus ℰ^𝜇𝜉=𝐸_𝑄𝜉. (18)
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 ℰ^𝜇𝜑(𝜉)=𝐸_𝑄𝜑(𝜉). (19)
From (18), (19), and the classical Jensen's inequality, we have 𝜑ℰ^𝜇𝜉=𝜑𝐸_𝑄𝜉≤𝐸_𝑄𝜑(𝜉)=ℰ^𝜇𝜑(𝜉). (20)
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.

Theorem 13. Let 𝑔 satisfy (𝐴.1) and (𝐴.2). Then the following four statements are equivalent.(i)Jensen's inequality for generalized Peng's 𝑔-expectation ℰ_𝑔[⋅∣ℱ_𝑡] holds in general, that is, for each convex function 𝜑(𝑥)∶𝑅↦𝑅 and each 𝜉∈ℒ(Ω,ℱ_𝑇,𝑃), if 𝜑(𝜉)∈ℒ(Ω,ℱ_𝑇,𝑃), then one has ℰ_𝑔𝜑(𝜉)ℱ_𝑡≥𝜑ℰ_𝑔𝜉∣ℱ_𝑡𝑎.𝑠.; (21)(ii)consider ∀(𝜉,𝑎,𝑏)∈𝐿²(Ω,ℱ_𝑇,𝑃)×𝑅×𝑅, ℰ_𝑔[𝑎𝜉+𝑏]≥𝑎ℰ_𝑔[𝜉]+𝑏;(iii)consider ∀(𝜉,𝑎,𝑏)∈𝐿²(Ω,ℱ_𝑇,𝑃)×𝑅×𝑅, ℰ_𝑔[𝑎𝜉+𝑏∣ℱ_𝑡]≥𝑎ℰ_𝑔[𝜉∣ℱ_𝑡]+𝑏 a.s.;(iv)consider 𝑔 is independent of 𝑦, superhomogeneous, and positively homogeneous with respect to 𝑧.

Proof. (i)⇒(ii) is obvious.
(ii)⇒(iii): let 𝜂=𝜉+𝑏. By (ii), we have ℰ_𝑔𝜂−𝑏≥ℰ_𝑔𝜂−𝑏, (22) That is, ℰ_𝑔𝜉+𝑏≥ℰ_𝑔𝜉+𝑏. (23) Thus, for each (𝜉,𝑏)∈𝐿²(Ω,ℱ_𝑇,𝑃)×𝑅, ℰ_𝑔𝜉+𝑏=ℰ_𝑔𝜉+𝑏. (24) For each (𝑋,𝑡,𝑘)∈𝐿²(Ω,ℱ_𝑇,𝑃)×[0,𝑇]×𝑅, by (24), we know that for each 𝐴∈ℱ_𝑡, ℰ_𝑔1_𝐴(𝑋+𝑘)=ℰ_𝑔1_𝐴𝑋+1_𝐴𝑘−𝑘+𝑘=ℰ_𝑔1_𝐴𝑋+1_𝐴^𝐶(−𝑘)+𝑘=ℰ_𝑔ℰ_𝑔1_𝐴𝑋+1_𝐴^𝐶(−𝑘)∣ℱ_𝑡+𝑘=ℰ_𝑔1_𝐴ℰ_𝑔𝑋∣ℱ_𝑡+1_𝐴^𝐶(−𝑘)+𝑘=ℰ_𝑔1_𝐴ℰ_𝑔𝑋∣ℱ_𝑡+1_𝐴^𝐶(−𝑘)+𝑘=ℰ_𝑔1_𝐴ℰ_𝑔𝑋∣ℱ_𝑡+𝑘. (25) Thus, ℰ_𝑔𝑋+𝑘∣ℱ_𝑡=ℰ_𝑔𝑋∣ℱ_𝑡+𝑘a.s.,∀𝑡∈[0,𝑇]. (26) On the other hand, for each 𝜆≠0, define ℰ^𝜆⋅∣ℱ_𝑡=ℰ_𝑔𝜆⋅∣ℱ_𝑡𝜆,∀𝑡∈[0,𝑇]. (27) It is easy to check that ℰ_𝑔[⋅∣ℱ_𝑡] and ℰ^𝜆[⋅∣ℱ_𝑡] are two ℱ-expectations on 𝐿²(Ω,ℱ_𝑇,𝑃) (the notion of ℱ-expectation can be seen in [13]). From (ii), we have if 𝜆>0, for each 𝜉∈𝐿²(Ω,ℱ_𝑇,𝑃)ℰ^𝜆𝜉≥ℰ_𝑔𝜉. (28) Hence, by Lemma 4.5 in [13], we have ℰ^𝜆𝜉∣ℱ_𝑡≥ℰ_𝑔𝜉∣ℱ_𝑡a.s.,∀𝑡∈[0,𝑇]. (29) Similarly, if 𝜆<0, for each 𝜉∈𝐿²(Ω,ℱ_𝑇,𝑃)ℰ^𝜆𝜉≤ℰ_𝑔𝜉. (30) Hence, by Lemma 4.5 in [13] again, we have ℰ^𝜆𝜉∣ℱ_𝑡≤ℰ_𝑔𝜉∣ℱ_𝑡a.s.,∀𝑡∈[0,𝑇]. (31) Thus from (29) and (31), we have ∀(𝜉,𝜆)∈𝐿²(Ω,ℱ_𝑇,𝑃)×𝑅, ℰ_𝑔𝜆𝜉∣ℱ_𝑡≥𝜆ℰ_𝑔𝜉∣ℱ_𝑡a.s.,∀𝑡∈[0,𝑇]. (32) From (26) and (32), we have ∀(𝜉,𝑎,𝑏)∈𝐿²Ω,ℱ_𝑇ℰ,𝑃×𝑅×𝑅,_𝑔𝑎𝜉+𝑏∣ℱ_𝑡≥𝑎ℰ_𝑔𝜉∣ℱ_𝑡+𝑏a.s.,∀𝑡∈[0,𝑇]. (33)
(iii)⇒(iv): Firstly, we prove that 𝑔 is independent of 𝑦. From (iii), we can obtain that for each (𝜉,𝑦)∈𝐿²(Ω,ℱ_𝑇,𝑃)×𝑅, ℰ_𝑔𝜉−𝑦∣ℱ_𝑡=ℰ_𝑔𝜉∣ℱ_𝑡−𝑦,a.s.,∀𝑡∈[0,𝑇]. (34) For each (𝑡,𝑦,𝑧)∈[0,𝑇]×𝑅×𝑅^𝑑, let 𝑌_⋅^{𝑡,𝑦,𝑧} be the solution of the following SDE defined on [𝑡,𝑇]: 𝑌_𝑠^{𝑡,𝑦,𝑧}=𝑦−^𝑠_𝑡𝑔𝑟,𝑌_𝑟^{𝑡,𝑦,𝑧},𝑧𝑑𝑟+𝑧⋅𝑊_𝑠−𝑊_𝑡. (35) From (34), we have 𝑌_𝑟^{𝑡,𝑦,𝑧}−𝑦=ℰ_𝑔𝑌_𝑠^{𝑡,𝑦,𝑧}∣ℱ_𝑟−𝑦=ℰ_𝑔𝑌_𝑠^{𝑡,𝑦,𝑧}−𝑦∣ℱ_𝑟,𝑡≤𝑟≤𝑠≤𝑇. (36) Let 𝑌_𝑠=𝑌_𝑠^{𝑡,𝑦,𝑧}−𝑦, 𝑠∈[𝑡,𝑇] and 𝑍 be the corresponding part of Itô's integrand. It then follows that 𝑌_𝑠=−^𝑠_𝑡𝑔𝑟,𝑌_𝑟^{𝑡,𝑦,𝑧},𝑧𝑑𝑟+^𝑠_𝑡𝑧⋅𝑑𝑊_𝑟=−^𝑠_𝑡𝑔𝑟,𝑌_𝑟,𝑍_𝑟d𝑟+^𝑠_𝑡𝑍_𝑟⋅𝑑𝑊_𝑟. (37) Thus, 𝑍_𝑟≡𝑧 and 𝑔𝑟,𝑌_𝑟,𝑧=𝑔𝑟,𝑌_𝑟^{𝑡,𝑦,𝑧}−𝑦,𝑧=𝑔𝑟,𝑌_𝑟^{𝑡,𝑦,𝑧},𝑧. (38) Then, we can apply Lemma 4.4 in Peng [14] to obtain that for each (𝑦,𝑧)∈𝑅×𝑅^𝑑, 𝑔𝑡,𝑦,𝑧=𝑔(𝑡,0,𝑧),𝑑𝑃×𝑑𝑡a.s. (39) 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,ℱ_𝑇,𝑃)×𝑅, 𝜆ℰ_𝑔𝜉∣ℱ_𝑡≤ℰ_𝑔𝜆𝜉∣ℱ_𝑡,a.s.,∀𝑡∈[0,𝑇]. (40) For each (𝑡,𝑧)∈[0,𝑇]×𝑅^𝑑, let 𝑌_⋅^𝑡,𝑧 be the solution of the following SDE defined on [𝑡,𝑇]: 𝑌_𝑠^𝑡,𝑧=−^𝑠_𝑡𝑔(𝑟,𝑧)𝑑𝑟+𝑧⋅𝑊_𝑠−𝑊_𝑡. (41) From (40), we have ℰ_𝑔𝜆𝑌_𝑠^𝑡,𝑧∣ℱ_𝑟≥𝜆ℰ_𝑔𝑌_𝑠^𝑡,𝑧∣ℱ_𝑟=𝜆𝑌_𝑟^𝑡,𝑧,𝑡≤𝑟≤𝑠≤𝑇. (42) Thus, (𝜆𝑌_𝑠^𝑡,𝑧)_{𝑠∈[𝑡,𝑇]} is an ℰ_𝑔-submartingale. From the decomposition theorem of ℰ_𝑔-supermartingale (see [15]), it follows that there exists an increasing process (𝐴_𝑠)_{𝑠∈[𝑡,𝑇]} such that 𝜆𝑌_𝑠^𝑡,𝑧=−^𝑠_𝑡𝑔𝑟,𝑍_𝑟𝑑𝑟+𝐴_𝑠−𝐴_𝑡+^𝑠_𝑡𝑍_𝑟⋅𝑑𝑊_𝑟,s∈[𝑡,𝑇]. (43) This with 𝜆𝑌_𝑠^𝑡,𝑧=−∫^𝑠_𝑡𝜆𝑔(𝑟,𝑧)d𝑟+∫^𝑠_𝑡𝜆𝑧⋅𝑑𝑊_𝑟 yields 𝑍_𝑟≡𝜆𝑧 and 𝜆𝑔(𝑟,𝑧)≤𝑔(𝑟,𝜆𝑧),𝑑𝑃×𝑑𝑡a.s. (44)
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,ℱ_𝑇,𝑃), 1𝜆ℰ_𝑔𝜆𝜉∣ℱ_𝑡≤ℰ_𝑔𝜉∣ℱ_𝑡,a.s.,∀𝑡∈[0,𝑇], (45) that is, ℰ_𝑔𝜆𝜉∣ℱ_𝑡≤𝜆ℰ_𝑔𝜉∣ℱ_𝑡,a.s.,∀𝑡∈[0,𝑇]. (46) Thus, we have ℰ_𝑔𝜆𝜉∣ℱ_𝑡=𝜆ℰ_𝑔𝜉∣ℱ_𝑡,a.s.,∀𝑡∈[0,𝑇]. (47) Obviously, if 𝜆=0, (47) still holds. Thus, for each 𝜆≥0, ℰ_𝑔𝜆𝜉∣ℱ_𝑡=𝜆ℰ_𝑔𝜉∣ℱ_𝑡,a.s.,∀𝑡∈[0,𝑇]. (48) For each (𝑡,𝑧)∈[0,𝑇]×𝑅^𝑑, let 𝑌_⋅^𝑡,𝑧 be the solution of SDE (34). From (48), for each 𝜆≥0, we have ℰ_𝑔𝜆𝑌_𝑠^𝑡,𝑧∣ℱ_𝑟=𝜆ℰ_𝑔𝑌_𝑠^𝑡,𝑧∣ℱ_𝑟=𝜆𝑌_𝑟^𝑡,𝑧,𝑡≤𝑟≤𝑠≤𝑇. (49) This implies that there exists a process 𝑍_⋅^{𝑡,𝑧,𝜆} such that 𝜆𝑌_𝑠^𝑡,𝑧=−^𝑠_𝑡𝑔𝑟,𝑍_𝑟^{𝑡,𝑧,𝜆}𝑑𝑟+^𝑠_𝑡𝑍_𝑟^{𝑡,𝑧,𝜆}⋅𝑑𝑊_𝑟,𝑠∈[𝑡,𝑇]. (50) Comparing this with 𝜆𝑌_𝑠^𝑡,𝑧=−∫^𝑠_𝑡𝜆𝑔(𝑟,𝑧)𝑑𝑟+∫^𝑠_𝑡𝜆𝑧⋅𝑑𝑊_𝑟, it follows that 𝑍_𝑟^{𝑡,𝑧,𝜆}≡𝜆𝑧 and 𝜆𝑔(𝑟,𝑧)=𝑔(𝑟,𝜆𝑧),𝑑𝑃×𝑑𝑡a.s. (51)
(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 ∀(𝑋,𝑘)∈ℒΩ,ℱ_𝑇ℰ,𝑃×𝑅,_𝑔𝑋+𝑘∣ℱ_𝑡=ℰ_𝑔𝑋∣ℱ_𝑡+𝑘a.s.,∀(𝑋,𝜆)∈ℒΩ,ℱ_𝑇ℰ,𝑃×𝑅,_𝑔𝜆𝑋∣ℱ_𝑡≥𝜆ℰ_𝑔𝑋∣ℱ_𝑡a.s. (52)(53) From (53), we can deduce that for each bounded variable 𝜁∈ℱ_𝑡, ∀𝑋∈ℒΩ,ℱ_𝑇,𝑃,ℰ_𝑔𝜁𝑋∣ℱ_𝑡≥𝜁ℰ_𝑔𝑋∣ℱ_𝑡a.s. (54) In fact, let {𝐴_𝑖}^𝑚_𝑖=1 be a ℱ_𝑡-measurable partition of Ω and let 𝜆_𝑖∈𝑅 (𝑖=1,2,…,𝑚). By (53), we have ℰ_𝑔^𝑚_𝑖=1𝜆_𝑖1_𝐴𝑖𝑋∣ℱ_𝑡=^𝑚_𝑖=11_𝐴𝑖ℰ_𝑔𝜆_𝑖𝑋∣ℱ_𝑡≥^𝑚_𝑖=11_𝐴𝑖𝜆_𝑖ℰ_𝑔𝑋∣ℱ_𝑡a.s. (55) In other words, for each 𝑋∈ℒ(Ω,ℱ_𝑇,𝑃) and each simple function 𝜁∈ℒ(Ω,ℱ_𝑡,𝑃), ℰ_𝑔𝜁𝑋∣ℱ_𝑡≥𝜁ℰ_𝑔𝑋∣ℱ_𝑡a.s. (56) 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,ℱ_𝑇,𝑃) and convex function 𝜑 such that 𝜑(𝜉)∈ℒ(Ω,ℱ_𝑇,𝑃), we set 𝜂_𝑡=𝜑^′₋(ℰ_𝑔[𝜉∣ℱ_𝑡]). Then 𝜂_𝑡 is ℱ_𝑡-measurable. Since 𝜑 is convex, we have 𝜑(𝑥)−𝜑𝑦≥𝜑^′₋𝑦𝑥−𝑦,∀𝑥,𝑦∈𝑅. (57) Take 𝑥=𝜉, 𝑦=ℰ_𝑔[𝜉∣ℱ_𝑡]. Then we have 𝜑(𝜉)−𝜑ℰ_𝑔𝜉∣ℱ_𝑡≥𝜂_𝑡𝜉−ℰ_𝑔𝜉∣ℱ_𝑡a.s. (58) For each 𝑛∈𝑁, we define Ω_𝑡,𝑛∶=ℰ_𝑔𝜉∣ℱ_𝑡+𝜂_𝑡+𝜑ℰ_𝑔𝜉∣ℱ_𝑡≤𝑛, (59) so we have ℰ_𝑔1_Ω𝑡,𝑛𝜑(𝜉)∣ℱ_𝑡≥ℰ_𝑔1_Ω𝑡,𝑛𝜑ℰ_𝑔𝜉∣ℱ_𝑡−1_Ω𝑡,𝑛𝜂_𝑡ℰ_𝑔𝜉∣ℱ_𝑡+1_Ω𝑡,𝑛𝜂_𝑡𝜉∣ℱ_𝑡a.s. (60) By the definition of 1_Ω𝑡,𝑛, we know 1_Ω𝑡,𝑛𝜑ℰ_𝑔𝜉∣ℱ_𝑡−1_Ω𝑡,𝑛𝜂_𝑡ℰ_𝑔𝜉∣ℱ_𝑡∈ℒΩ,ℱ_𝑡,𝑃. (61) Thus, in view of (52) and from Proposition 10, we can get ℰ_𝑔1_Ω𝑡,𝑛𝜑(𝜉)∣ℱ_𝑡≥1_Ω𝑡,𝑛𝜑ℰ_𝑔𝜉∣ℱ_𝑡−1_Ω𝑡,𝑛𝜂_𝑡ℰ_𝑔𝜉∣ℱ_𝑡+ℰ_𝑔1_Ω𝑡,𝑛𝜂_𝑡𝜉∣ℱ_𝑡a.s. (62) Moreover, from (54), considering that 1_Ω𝑡,𝑛𝜂_𝑡∈ℱ_𝑡 and is bounded by 𝑛, we can getℰ_𝑔1_Ω𝑡,𝑛𝜂_𝑡𝜉∣ℱ_𝑡≥1_Ω𝑡,𝑛𝜂_𝑡ℰ_𝑔𝜉∣ℱ_𝑡a.s. (63) Hence, we can deduce that for each 𝑛∈𝑁, ℰ_𝑔1_Ω𝑡,𝑛𝜑(𝜉)∣ℱ_𝑡≥1_Ω𝑡,𝑛𝜑ℰ_𝑔𝜉∣ℱ_𝑡a.s. (64) Finally, thanks to Remark 8 (ii) again, we can get ℰ_𝑔𝜑(𝜉)∣ℱ_𝑡≥𝜑ℰ_𝑔𝜉∣ℱ_𝑡a.s. (65) Hence, Jensen's inequality for ℰ_𝑔[⋅∣ℱ_𝑡] holds in general. The proof of Theorem 13 is complete.

Example 14. Suppose 𝐻 is a bounded, convex, and closed subset of 𝑅^𝑑 and 𝐷= the set of 𝑅^𝑑-valued continuous processes (𝑣_𝑡)_{𝑡∈[0,𝑇]} such that for each 𝑡, 𝑣_𝑡∈𝐻 a.s.. Define the probability measure 𝑄^𝑣 by 𝑑𝑄^𝑣=𝑑𝑃e^{−(1/2)∫𝑇0|𝑣𝑠|2𝑑𝑠+∫𝑇0𝑣𝑠⋅𝑑𝑊𝑠}. (66) Thus, for any convex function 𝜑, 𝜑esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉∣ℱ_𝑡≤esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜑(𝜉)∣ℱ_𝑡,a.s.,∀𝑡∈[0,𝑇], (67) whenever 𝜉, 𝜑(𝜉)∈ℒ(Ω,ℱ_𝑇,𝑃).

Proof. Let 𝑔(𝑡,𝑧)= ess sup_𝑣∈𝐷𝑣_𝑡⋅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 esssup_𝑣∈𝐷𝐸_𝑄^𝑣[𝜉∣ℱ_𝑡]=ℰ_𝑔[𝜉∣ℱ_𝑡],a.s.,∀𝜉∈𝐿²(Ω,ℱ_𝑇,𝑃). Now we prove esssup_𝑣∈𝐷𝐸_𝑄^𝑣[𝜉∣ℱ_𝑡]=ℰ_𝑔[𝜉∣ℱ_𝑡], a.s., ∀𝜉∈ℒ(Ω,ℱ_𝑇,𝑃). Indeed, for any 𝜉∈ℒ(Ω,ℱ_𝑇,𝑃), there exists 1<𝑝<2 such that 𝜉∈𝐿^𝑝(Ω,ℱ_𝑇,𝑃). Let 𝜉^𝑛=(𝜉∧𝑛)∨(−𝑛), 𝑛=1,2,…. Clearly, for each 𝑛, 𝜉^𝑛∈𝐿²(Ω,ℱ_𝑇,𝑃), then esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉^𝑛∣ℱ_𝑡=ℰ_𝑔𝜉^𝑛∣ℱ_𝑡,a.s. (68)
Since esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉^𝑛∣ℱ_𝑡=esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉^𝑛−𝜉∣ℱ_𝑡+𝐸_𝑄^𝑣𝜉∣ℱ_𝑡≤esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉^𝑛−𝜉∣ℱ_𝑡+esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉∣ℱ_𝑡, (69) we have ℰ_𝑔𝜉^𝑛∣ℱ_𝑡−esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉∣ℱ_𝑡≤esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉^𝑛−𝜉∣ℱ_𝑡. (70) With an approach similar to the one above, we can get easily that ℰ_𝑔𝜉^𝑛∣ℱ_𝑡−esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉∣ℱ_𝑡≥essinf_𝑣∈𝐷𝐸_𝑄^𝑣𝜉^𝑛−𝜉∣ℱ_𝑡. (71) Combining (42) with (43), we have ℰ_𝑔𝜉^𝑛∣ℱ_𝑡−esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉∣ℱ_𝑡≤essinf_𝑣∈𝐷𝐸_𝑄^𝑣𝜉^𝑛−𝜉∣ℱ_𝑡∨esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉^𝑛−𝜉∣ℱ_𝑡≤esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉^𝑛−𝜉∣ℱ_𝑡. (72) By Hölder's inequality and noting that (e^{−(1/2)∫𝑡0|𝑣𝑠|2𝑑𝑠+∫𝑡0𝑣𝑠⋅𝑑𝑊𝑠})_{𝑡∈[0,𝑇]} and (e^{−(1/2)∫𝑡0|𝑞𝑣𝑠|2𝑑𝑠+∫𝑡0𝑞𝑣𝑠⋅𝑑𝑊𝑠})_{𝑡∈[0,𝑇]} are both martingales with respect to (ℱ_𝑡)_{𝑡∈[0,𝑇]}, we can obtain 𝐸_𝑄^𝑣𝜉^𝑛−𝜉∣ℱ_𝑡=𝐸𝜉^𝑛−𝜉𝑑𝑄^𝑣/𝑑𝑃∣ℱ_𝑡𝐸(𝑑𝑄^𝑣/𝑑𝑃)∣ℱ_𝑡≤𝐸𝜉^𝑛−𝜉^𝑝∣ℱ_𝑡^1/𝑝𝐸𝑑𝑄^𝑣/𝑑𝑃^𝑞∣ℱ_𝑡^1/𝑞𝐸(𝑑𝑄^𝑣/𝑑𝑃)∣ℱ_𝑡≤e^{(1/2)(𝑞−1)𝜇2𝑇}𝐸𝜉^𝑛−𝜉^𝑝∣ℱ_𝑡^1/𝑝, (73) where (1/𝑝)+(1/𝑞)=1. It then follows from Lebesgue's dominated convergence theorem that esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉^𝑛−𝜉∣ℱ_𝑡⟶0,as𝑛⟶∞. (74) Hence, ℰ_𝑔𝜉^𝑛∣ℱ_𝑡−esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉∣ℱ_𝑡⟶0,as𝑛⟶∞. (75)
On the other hand, from Remark 8 (i), we have ℰ_𝑔𝜉^𝑛∣ℱ_𝑡⟶ℰ_𝑔𝜉∣ℱ_𝑡,as𝑛⟶∞. (76) Thus, ℰ_𝑔𝜉∣ℱ_𝑡=esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉∣ℱ_𝑡,a.s.,∀𝜉∈ℒΩ,ℱ_𝑇,𝑃. (77)
Applying Theorem 13, we have 𝜑esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜉∣ℱ_𝑡≤esssup_𝑣∈𝐷𝐸_𝑄^𝑣𝜑(𝜉)∣ℱ_𝑡,a.s. (78)

Definition 15. Suppose 𝑔 satisfies (A.1) and (A.2). A process (𝑋_𝑡)_{𝑡∈[0,𝑇]} satisfying that for each 𝑡, 𝑋_𝑡∈ℒ(Ω,ℱ_𝑡,𝑃) is called a generalized Peng's 𝑔-martingale (resp., generalized Peng's 𝑔-supermartingale, generalized Peng's 𝑔-submartingale), if for any 𝑡, 𝑠 satisfying 𝑡≤𝑠≤𝑇, ℰ_𝑔𝑋_𝑠∣ℱ_𝑡=𝑋_𝑡resp.≤𝑋_𝑡,≥𝑋_𝑡,a.s. (79)

Applying Theorem 13, immediately we have the following.

Theorem 16. Suppose 𝑔 is independent of 𝑦, superhomogeneous and positively homogeneous with respect to 𝑧 and satisfies (A.1) and (A.2). If (𝑋_𝑡)_{𝑡∈[0,𝑇]} is a generalized Peng's 𝑔-martingale and 𝜑 is a convex function such that 𝜑(𝑋_𝑡)∈ℒ(Ω,ℱ_𝑡,𝑃), then (𝜑(𝑋_𝑡))_{𝑡∈[0,𝑇]} is a generalized Peng's 𝑔-submartingale.

Remark 17. Suppose 𝑔 is independent of 𝑦, superhomogeneous and positively homogeneous with respect to 𝑧 and satisfies (A.1) and (A.2). Similarly, we can get the following.(i)If (𝑋_𝑡)_{𝑡∈[0,𝑇]} is a generalized Peng's 𝑔-submartingale and 𝜑 is a convex and increasing function such that 𝜑(𝑋_𝑡)∈ℒ(Ω,ℱ_𝑡,𝑃), then (𝜑(𝑋_𝑡))_{𝑡∈[0,𝑇]} is a generalized Peng's 𝑔-submartingale.(ii)If (𝑋_𝑡)_{𝑡∈[0,𝑇]} is a generalized Peng's 𝑔-supermartingale and 𝜑 is a convex and decreasing function such that 𝜑(𝑋_𝑡)∈ℒ(Ω,ℱ_𝑡,𝑃), then (𝜑(𝑋_𝑡))_{𝑡∈[0,𝑇]} is a generalized Peng's 𝑔-submartingale.

Example 18. (i) Let 𝑔=𝜇|𝑧| and 𝜑(𝑥)=(𝑥−𝑎)⁺ where 𝑎∈𝑅. Obviously, 𝑔 satisfies the assumptions of Remark 17 and 𝜑 is a convex and increasing function. By Remark 17 (i), we have the following: suppose (𝑋_𝑡)_{𝑡∈[0,𝑇]} is a ℰ^𝜇-submartingale, then ((𝑋_𝑡−𝑎)⁺)_{𝑡∈[0,𝑇]} is a ℰ^𝜇-submartingale.
(ii) Let 𝑔=𝜇|𝑧| and 𝜑(𝑥)=(𝑥−𝑏)⁻ where 𝑏∈𝑅. With the similar argument, we have the following: suppose (𝑌_𝑡)_{𝑡∈[0,𝑇]} is a ℰ^𝜇-supermartingale, then ((𝑌_𝑡−𝑏)⁻)_{𝑡∈[0,𝑇]} is a ℰ^𝜇-submartingale.

Acknowledgments

The author would like to thank the anonymous referees for their careful reading of this paper and valuable suggestions. The author thanks the partial support from the National Natural Science Foundation of China (Grant no. 11171179), the Doctoral Program Foundation of Ministry of Education of China (Grant no. 20093705110002 and 20123705120005), and the Natural Science Foundation of Shandong Province of China (Grant no. ZR2012AQ009).