On Jensen’s inequality, Hölder’s inequality, and Minkowski’s inequality for dynamically consistent nonlinear evaluations

Abstract

In this paper, the dynamically consistent nonlinear evaluations that were introduced by Peng are considered in probability space $L^{2} (\Omega,{\mathcal{F}}, ({\mathcal {F}}_{t} )_{t\geq0},P )$. We investigate the n-dimensional ($n\geq1$) Jensen inequality, Hölder inequality, and Minkowski inequality for dynamically consistent nonlinear evaluations in $L^{1} (\Omega,{\mathcal{F}}, ({\mathcal {F}}_{t} )_{t\geq0},P )$. Furthermore, we give four equivalent conditions on the n-dimensional Jensen inequality for g-evaluations induced by backward stochastic differential equations with non-uniform Lipschitz coefficients in $L^{p} (\Omega,{\mathcal{F}}, ({\mathcal {F}}_{t} )_{0\leq t\leq T},P )$ ($1< p\leq2$). Finally, we give a sufficient condition on g that satisfies the non-uniform Lipschitz condition under which Hölder’s inequality and Minkowski’s inequality for the corresponding g-evaluation hold true. These results include and extend some existing results.

1 Introduction

It is well known that (see Peng [1, 2]) a dynamically consistent nonlinear evaluation in probability space $L^{2} (\Omega,{\mathcal {F}}, ({\mathcal{F}}_{t} )_{t\geq0},P )$, where $\{{\mathcal {F}}_{t}\}_{t\geq0}$ is a given filtration, is a system of operators:

$${\mathcal{E}}_{s,t}[X]:X\in L^{2}(\Omega,{ \mathcal{F}}_{t},P)\mapsto L^{2}(\Omega,{\mathcal{F}}_{s},P), \quad 0\leq s\leq t< \infty, $$

which satisfies the following properties:

• (i) ${\mathcal{E}}_{s,t}[X_{1}]\geq{\mathcal {E}}_{s,t}[X_{2}]$, if $X_{1}\geq X_{2}$;

• (ii) ${\mathcal{E}}_{t,t}[X]=X$;

• (iii) ${\mathcal{E}}_{r,s}[{\mathcal {E}}_{s,t}[X]]={\mathcal{E}}_{r,t}[X]$, if $0\leq r\leq s\leq t<\infty$;

• (iv) $1_{A}{\mathcal{E}}_{s,t}[X]=1_{A}{\mathcal {E}}_{s,t}[1_{A}X]$, $\forall A\in{\mathcal{F}}_{s}$.

Of course, we can define this notion in $L^{1} (\Omega,{\mathcal {F}}, ({\mathcal{F}}_{t} )_{t\geq0},P )$.

In a financial market, the evaluation of the discounted value of a derivative is often treated as a dynamically consistent nonlinear evaluation (expectation). The well-known g-evaluation (g-expectation) induced by backward stochastic differential equations (BSDEs for short), which was put forward by Peng, is a special case of a dynamically consistent nonlinear evaluation (expectation). While nonlinear BSDEs were firstly introduced by Pardoux and Peng [3], who proved the existence and uniqueness of adapted solutions, when the coefficient g is Lipschitz in $(y,z)$ uniformly in $(t,\omega)$, with square-integrability assumptions on the coefficient $g(t,\omega,y,z)$ and terminal condition ξ. Later many researchers developed the theory of BSDEs and their applications in a series of papers (for example see Hu and Peng [4], Lepeltier and San Martin [5], El Karoui et al. [6], Pardoux [7, 8], Briand et al. [9] and the references therein) under some other assumptions on the coefficients but for a fixed terminal time $T>0$. In 2000, Chen and Wang [10] obtained the existence and uniqueness theorem for $L^{2}$ solutions of infinite time interval BSDEs when $T=\infty$, by the martingale representation theorem and fixed point theorem. Recently, Zong [11] have obtained the result on $L^{p}$ ($1< p<2$) solutions of infinite time interval BSDEs. One of the special cases is the existence and uniqueness theorem of BSDEs with non-uniformly Lipschitz coefficients.

The original motivation for studying nonlinear evaluation (expectation) and g-evaluation (g-expectation) comes from the theory of expected utility, which is the foundation of modern mathematical economics. Chen and Epstein [12] gave an application of dynamically consistent nonlinear evaluation (expectation) to recursive utility, Peng [1, 2, 13–15] and Rosazza Gianin [16] investigated some applications of dynamically consistent nonlinear evaluations (expectations) and g-evaluations (g-expectations) to static and dynamic pricing mechanisms and risk measures.

Since the notions of nonlinear evaluation (expectation) and g-evaluation (g-expectation) were introduced, many properties of the nonlinear evaluation (expectation) and g-evaluation (g-expectation) have been studied in [1, 2, 6, 10–31]. In [1, 2], Peng obtained an important result: he proved that if a dynamically consistent nonlinear evaluation ${\mathcal {E}}_{s,t}[\cdot]$ can be dominated by a kind of g-evaluation, then ${\mathcal{E}}_{s,t}[\cdot]$ must be a g-evaluation. Thus, in this case, many problems on dynamically consistent nonlinear evaluations ${\mathcal{E}}_{s,t}[\cdot]$ can be solved through the theory of BSDEs.

It is well known that Jensen’s inequality for classic mathematical expectations holds in general, which is a very important property and has many important applications. But for nonlinear expectation, even for its special case: g-expectation, by Briand et al. [17], we know that Jensen’s inequality for g-expectations usually does not hold in general. So under the assumption that g is continuous with respect to t, some papers, such as [18, 19, 25, 27, 28] have been devoted to Jensen’s inequality for g-expectations, with the help of the theory of BSDEs, they have obtained the necessary and sufficient conditions under which Jensen’s inequality for g-expectations holds in general. Under the assumptions that g does not depend on y and is convex, Chen et al. [18, 19] 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 g only does not depend on y, Jiang and Chen [28] 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. found. Later, this result was improved by Hu [25] and Jiang [27], in fact, Jiang [27] showed that g must be independent of y. In addition, Fan [22] studied Jensen’s inequality for filtration-consistent nonlinear expectations without domination condition. Jia [26] studied the n-dimensional ($n>1$) Jensen’s inequality for g-expectations and got the result that the n-dimensional ($n>1$) Jensen’s inequality holds for g-expectations if and only if g is independent of y and linear with respect to z, in other words, the corresponding g-expectation must be linear. Then the natural question is asked:

For more general dynamically consistent nonlinear evaluation ${\mathcal{E}}_{s,t}[\cdot]$, what are the sufficient and necessary conditions under which Jensen’s inequality for ${\mathcal{E}}_{s,t}[\cdot]$ holds in general? Roughly speaking, what conditions on ${\mathcal{E}}_{s,t}[\cdot]$ are equivalent with the inequality

$${\mathcal {E}}_{s,t}\bigl[\varphi(\xi)\bigr]\geq\varphi\bigl({ \mathcal{E}}_{s,t}[\xi]\bigr) \quad \mbox{a.s.} $$

holding for any convex function $\varphi: \mathcal{R}\mapsto\mathcal{R}$?

One of the objectives of this paper is to investigate this problem. At the same time, this paper will also investigate the sufficient and necessary conditions on ${\mathcal {E}}_{s,t}[\cdot]$ under which the n-dimensional ($n>1$) Jensen inequality holds. As applications of these two results, we give four equivalent conditions on the 1-dimensional Jensen inequality and the n-dimensional ($n>1$) Jensen inequality for g-evaluations induced by BSDEs with non-uniform Lipschitz coefficients in $L^{p} (\Omega,{\mathcal{F}}, ({\mathcal {F}}_{t} )_{0\leq t\leq T},P )$ ($1< p\leq2$), respectively.

The remainder of this paper is organized as follows: In Section 2, we study the n-dimensional ($n\geq1$) Jensen inequality, Hölder inequality, and Minkowski inequality for dynamically consistent nonlinear evaluations in $L^{1} (\Omega,{\mathcal{F}}, ({\mathcal{F}}_{t} )_{t\geq0},P )$. In Section 3, we give four equivalent conditions on the 1-dimensional Jensen inequality and the n-dimensional ($n>1$) Jensen inequality for g-evaluations induced by BSDEs with non-uniform Lipschitz coefficients in $L^{p} (\Omega,{\mathcal{F}}, ({\mathcal {F}}_{t} )_{0\leq t\leq T},P )$ ($1< p\leq2$), respectively. These results generalize the known results on Jensen’s inequality for g-expectation in [18, 19, 22, 25–28, 31]. In Section 4, we give a sufficient condition on g that satisfies the non-uniform Lipschitz condition under which Hölder’s inequality and Minkowski’s inequality for the corresponding g-evaluation hold true.

2 Jensen’s inequality, Hölder’s inequality, and Minkowski’s inequality for dynamically consistent nonlinear evaluations

Let $(\Omega,{\mathcal{F}},P)$ be a probability space carrying a standard d-dimensional Brownian motion $(B_{t})_{t\geq0}$, and let $({\mathcal{F}}_{t} )_{t\geq0}$ be the σ-algebra generated by $(B_{t} )_{t\geq0}$. We always assume that $({\mathcal{F}}_{t} )_{t\geq0}$ is complete. Let $T > 0$ be a given real number. In this paper, we always work in the probability space $(\Omega,{\mathcal{F}}_{T},P)$, and only consider processes indexed by $t\in[0, T ]$. We denote $L^{p}(\Omega,{\mathcal{F}}_{t} ,P)$ ($p\geq1$), the space of ${\mathcal {F}}_{t}$-measurable random variables satisfying $E_{P}[|X|^{p}]<\infty$, and by $L^{p}_{+}(\Omega,{\mathcal{F}}_{t} ,P)$ the space of non-negative random variables in $L^{p}(\Omega,{\mathcal{F}}_{t} ,P)$. Let $1_{A}$ denote the indicator of event A. For notational simplicity, we use $L^{p}({\mathcal{F}}_{t}):= L^{p}(\Omega,{\mathcal{F}}_{t} ,P)$ and $L^{p}_{+}({\mathcal{F}}_{t}):=L^{p}_{+}(\Omega,{\mathcal{F}}_{t} ,P)$. For the convenience of the reader, we recall the notion of a dynamically consistent nonlinear evaluation, defined in $L^{2}({\mathcal{F}}_{T})$ in Peng [1, 2], but defined in $L^{1}({\mathcal{F}}_{T})$ in this section.

Definition 2.1

An ${\mathcal{F}}_{t}$-consistent nonlinear evaluation in $L^{1}({\mathcal{F}}_{T})$ is a system of operators:

$${\mathcal{E}}_{s,t}[X]:X\in L^{1}({\mathcal{F}}_{t}) \mapsto L^{1}({\mathcal {F}}_{s}), \quad 0\leq s\leq t\leq T, $$

which satisfies the following properties:

• (A.1) monotonicity: ${\mathcal {E}}_{s,t}[X_{1}]\geq{\mathcal{E}}_{s,t}[X_{2}]$, if $X_{1}\geq X_{2}$;

• (A.2) ${\mathcal{E}}_{t,t}[X]=X$;

• (A.3) dynamical consistency: ${\mathcal {E}}_{r,s}[{\mathcal{E}}_{s,t}[X]]={\mathcal{E}}_{r,t}[X]$, if $0\leq r\leq s\leq t\leq T$;

• (A.4) zero one law: $1_{A}{\mathcal {E}}_{s,t}[X]=1_{A}{\mathcal{E}}_{s,t}[1_{A}X]$, $\forall A\in{\mathcal{F}}_{s}$.

First, we consider Jensen’s inequality for ${\mathcal{F}}_{t}$-consistent nonlinear evaluations. We have the following results.

Theorem 2.1

Suppose that ${\mathcal {E}}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is an ${\mathcal {F}}_{t}$-consistent nonlinear evaluation in $L^{1}({\mathcal{F}}_{T})$, then the following two statements are equivalent:

• (i) Jensen’s inequality for ${\mathcal{F}}_{t}$-consistent evaluation ${\mathcal{E}}_{s,t}[\cdot]$ holds in general, i.e., for each convex function $\varphi: \mathcal{R}\mapsto\mathcal{R}$ and $\xi\in L^{1}({\mathcal{F}}_{t})$, if $\varphi(\xi)\in L^{1}({\mathcal{F}}_{t})$, then we have

  $${\mathcal{E}}_{s,t}\bigl[\varphi(\xi)\bigr]\geq\varphi\bigl({\mathcal {E}}_{s,t}[\xi]\bigr) \quad \textit{a.s.}; $$

• (ii) $\forall(\xi,a,b)\in L^{1}({\mathcal{F}}_{t})\times \mathcal{R}\times\mathcal{R}$, ${\mathcal{E}}_{s,t}[a\xi+b]\geq a{\mathcal{E}}_{s,t}[\xi]+b$ a.s.

Proof

First, we prove (i) implies (ii). Suppose (i) holds, for each $(\xi,a, b)\in L^{1}({\mathcal {F}}_{t} )\times\mathcal{R} \times\mathcal{R}$, let $\varphi(x):=ax + b$. Obviously, $\varphi(x)$ is a convex function and $\varphi(\xi)\in L^{1}({\mathcal {F}}_{t} )$, then we have

$${\mathcal{E}}_{s,t}[a\xi+b]={\mathcal {E}}_{s,t}\bigl[\varphi( \xi)\bigr]\geq\varphi\bigl({\mathcal{E}}_{s,t}[\xi]\bigr)= a{ \mathcal{E}}_{s,t}[\xi]+b\quad \mbox{a.s.} $$

In the following, we prove (ii) implies (i). Suppose (ii) holds, for each $(\xi,a, b)\in L^{1}({\mathcal{F}}_{t} )\times \mathcal{R} \times\mathcal{R}$, we have

$$ {\mathcal{E}}_{s,t}[a\xi+b]\geq a{\mathcal {E}}_{s,t}[\xi]+b \quad \mbox{a.s.} $$

(2.1)

But, for any convex function $\varphi: \mathcal{R}\mapsto\mathcal{R}$, there exists a countable set $\mathcal{D}\subseteq\mathcal{R}^{2}$ such that

$$ \varphi(x)=\sup_{(a,b)\in \mathcal{D}}(ax+b). $$

(2.2)

In view of (2.1), for any $(a,b)\in\mathcal{D}$, we have

$${\mathcal{E}}_{s,t}\bigl[\varphi(\xi)\bigr]\geq{\mathcal {E}}_{s,t}[a\xi+b]\geq a{\mathcal{E}}_{s,t}[\xi]+b \quad \mbox{a.s.}, $$

which implies (i) by taking into consideration of (2.2). □

Theorem 2.2

Suppose that ${\mathcal {E}}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is an ${\mathcal {F}}_{t}$-consistent nonlinear evaluation in $L^{1}({\mathcal{F}}_{T})$ and $n>1$, then the following two statements are equivalent:

• (i) the n-dimensional Jensen inequality for a ${\mathcal{F}}_{t}$-consistent evaluation ${\mathcal{E}}_{s,t}[\cdot]$ holds in general, i.e., for each convex function $\varphi: \mathcal{R}^{n}\mapsto\mathcal{R}$ and $\xi_{i}\in L^{1}({\mathcal{F}}_{t})$ ($i=1,2,\ldots,n$), if $\varphi(\xi_{1},\xi_{2},\ldots,\xi_{n})\in L^{1}({\mathcal{F}}_{t})$, then we have

  $${\mathcal{E}}_{s,t} \bigl[\varphi(\xi_{1},\xi_{2}, \ldots,\xi_{n}) \bigr]\geq \varphi \bigl({\mathcal {E}}_{s,t}[ \xi_{1}],{\mathcal{E}}_{s,t}[\xi_{2}],\ldots,{ \mathcal {E}}_{s,t}[\xi_{n}] \bigr) \quad \textit{a.s.}; $$

• (ii) ${\mathcal{E}}_{s,t}$ is linear, i.e.,

  • (a) ${\mathcal{E}}_{s,t}[\lambda X]=\lambda{\mathcal {E}}_{s,t}[X]$ a.s., $\forall(X,\lambda)\in L^{1}({\mathcal{F}}_{t})\times \mathcal{R}$;

  • (b) ${\mathcal{E}}_{s,t}[X+Y]={\mathcal {E}}_{s,t}[X]+{\mathcal{E}}_{s,t}[Y]$ a.s., $\forall(X,Y)\in L^{1}({\mathcal{F}}_{t})\times L^{1}({\mathcal{F}}_{t})$;

  • (c) ${\mathcal{E}}_{s,t}[\mu]=\mu$ a.s., $\forall\mu\in \mathcal{R}$.

Proof

We prove (i) implies (ii).

First, we prove (i) implies (ii)(a). For each $(X,\lambda)\in L^{1}({\mathcal{F}}_{t})\times\mathcal{R}$, let $\varphi(x_{1},x_{2},\ldots,x_{n}):=\lambda x_{1}$ and $\xi_{1}:=X$. Obviously, $\varphi(x_{1},x_{2},\ldots,x_{n})$ is a convex function and $\varphi(\xi_{1},\xi_{2},\ldots,\xi_{n})\in L^{1}({\mathcal{F}}_{t} )$, then we have

$$ {\mathcal{E}}_{s,t}[\lambda X]={\mathcal{E}}_{s,t} \bigl[ \varphi(\xi _{1},\xi_{2},\ldots,\xi_{n}) \bigr]\geq \varphi \bigl({\mathcal {E}}_{s,t}[\xi_{1}],{ \mathcal{E}}_{s,t}[\xi_{2}],\ldots,{\mathcal {E}}_{s,t}[\xi_{n}] \bigr)=\lambda{\mathcal{E}}_{s,t}[X] \quad \mbox{a.s.} $$

(2.3)

On the other hand, let $\varphi(x_{1},x_{2},\ldots,x_{n}):=x_{1}-(\lambda-1)x_{2}$, $\xi_{1}:=\lambda X$, and $\xi_{2}:=X$. By (i), we can deduce that

$$\begin{aligned} {\mathcal {E}}_{s,t}[X] =&{\mathcal {E}}_{s,t} \bigl[\varphi( \xi_{1},\xi_{2},\ldots,\xi_{n}) \bigr]\geq\varphi \bigl({\mathcal {E}}_{s,t}[\xi_{1}],{\mathcal{E}}_{s,t}[ \xi_{2}],\ldots,{\mathcal {E}}_{s,t}[\xi_{n}] \bigr) \\ =&{\mathcal{E}}_{s,t}[\lambda X]-(\lambda-1){\mathcal{E}}_{s,t}[X]\quad \mbox{a.s.}, \end{aligned}$$

i.e.,

$$ {\mathcal{E}}_{s,t}[\lambda X]\leq\lambda{\mathcal {E}}_{s,t}[X] \quad \mbox{a.s.} $$

(2.4)

It follows from (2.3) and (2.4) that (ii)(a) holds true.

Next we prove (ii)(b) holds. For each $(X,Y)\in L^{1}({\mathcal {F}}_{t})\times L^{1}({\mathcal{F}}_{t})$, let $\varphi(x_{1},x_{2},\ldots,x_{n}):=x_{1}+x_{2}$, $\xi_{1}:=X$, and $\xi_{2}:=Y$, then we have

$$\begin{aligned} {\mathcal{E}}_{s,t}[X+Y] =&{\mathcal{E}}_{s,t} \bigl[\varphi( \xi_{1},\xi _{2},\ldots,\xi_{n}) \bigr]\geq\varphi \bigl({\mathcal {E}}_{s,t}[\xi_{1}],{\mathcal{E}}_{s,t}[ \xi_{2}],\ldots,{\mathcal {E}}_{s,t}[\xi_{n}] \bigr) \\ =&{\mathcal{E}}_{s,t}[X]+{\mathcal {E}}_{s,t}[Y] \quad \mbox{a.s.} \end{aligned}$$

(2.5)

On the other hand, let $\varphi(x_{1},x_{2},\ldots,x_{n}):=x_{1}-x_{2}$, $\xi_{1}:=X+Y$, and $\xi_{2}:=Y$. By (i), we have

$$\begin{aligned} {\mathcal {E}}_{s,t}[X] =&{\mathcal {E}}_{s,t} \bigl[\varphi( \xi_{1},\xi_{2},\ldots,\xi_{n}) \bigr]\geq\varphi \bigl({\mathcal {E}}_{s,t}[\xi_{1}],{\mathcal{E}}_{s,t}[ \xi_{2}],\ldots,{\mathcal {E}}_{s,t}[\xi_{n}] \bigr) \\ =&{\mathcal{E}}_{s,t}[X+Y]-{\mathcal {E}}_{s,t}[Y] \quad \mbox{a.s.}, \end{aligned}$$

i.e.,

$$ {\mathcal{E}}_{s,t}[X+Y]\leq{\mathcal {E}}_{s,t}[X]+{ \mathcal{E}}_{s,t}[Y] \quad \mbox{a.s.} $$

(2.6)

Thus, from (2.5) and (2.6), we can see that (ii)(b) holds.

Finally, we prove (ii)(c) holds. For each $\mu\in\mathcal{R}$, let $\varphi(x_{1},x_{2},\ldots,x_{n}):=\mu$, then we have

$$ {\mathcal{E}}_{s,t}[\mu]={\mathcal{E}}_{s,t} \bigl[\varphi( \xi_{1},\xi _{2},\ldots,\xi_{n}) \bigr]\geq\varphi \bigl({\mathcal {E}}_{s,t}[\xi_{1}],{\mathcal{E}}_{s,t}[ \xi_{2}],\ldots,{\mathcal {E}}_{s,t}[\xi_{n}] \bigr)= \mu \quad \mbox{a.s.} $$

(2.7)

On the other hand, let $\varphi(x_{1},x_{2},\ldots,x_{n}):=2x_{1}-\mu$ and $\xi_{1}:=\mu$. By (i), we can obtain

$${\mathcal {E}}_{s,t}[\mu]={\mathcal {E}}_{s,t} \bigl[\varphi( \xi_{1},\xi_{2},\ldots,\xi_{n}) \bigr]\geq\varphi \bigl({\mathcal {E}}_{s,t}[\xi_{1}],{\mathcal{E}}_{s,t}[ \xi_{2}],\ldots,{\mathcal {E}}_{s,t}[\xi_{n}] \bigr)=2{\mathcal{E}}_{s,t}[\mu]-\mu\quad \mbox{a.s.}, $$

i.e.,

$$ {\mathcal{E}}_{s,t}[\mu]\leq\mu\quad \mbox{a.s.} $$

(2.8)

It follows from (2.7) and (2.8) that (ii)(c) holds true.

In the following, we prove (ii) implies (i). Suppose (ii) holds, for any $(a_{1},a_{2},\ldots,a_{n},b)\in\mathcal{R}^{n+1}$ and $\xi_{i}\in L^{1}({\mathcal{F}}_{t})$ ($i=1,2,\ldots,n$), we have

$$ {\mathcal{E}}_{s,t} \Biggl[\sum_{i=1}^{n}a_{i} \xi_{i}+b \Biggr]= \sum_{i=1}^{n}a_{i}{ \mathcal{E}}_{s,t}[\xi_{i}]+b \quad \mbox{a.s.} $$

(2.9)

But, for any convex function $\varphi: \mathcal{R}^{n}\mapsto\mathcal{R}$, there exists a countable set $\mathcal{D}\subseteq\mathcal{R}^{n+1}$ such that

$$ \varphi(x_{1},x_{2},\ldots,x_{n})=\sup _{(a_{1},a_{2},\ldots,a_{n},b)\in \mathcal{D}} \Biggl(\sum_{i=1}^{n}a_{i}x_{i}+b \Biggr). $$

(2.10)

In view of (2.9), for any $(a_{1},a_{2},\ldots,a_{n},b)\in\mathcal{D}$, we have

$${\mathcal {E}}_{s,t}\bigl[\varphi(\xi_{1},\xi_{2}, \ldots,\xi_{n})\bigr]\geq{\mathcal {E}}_{s,t} \Biggl[\sum _{i=1}^{n}a_{i} \xi_{i}+b \Biggr]= \sum_{i=1}^{n}a_{i}{ \mathcal{E}}_{s,t}[\xi_{i}]+b \quad \mbox{a.s.}, $$

which implies (i) by taking into consideration of (2.10). □

The basic version of Hölder’s inequality for the classical mathematical expectation $E_{P}$ defined in $(\Omega,{\mathcal{F}}_{T},P)$ reads

$$ E_{P}[XY]\leq\bigl(E_{P}\bigl[X^{p}\bigr] \bigr)^{\frac{1}{p}}\bigl(E_{P}\bigl[Y^{q}\bigr] \bigr)^{\frac{1}{q}}, $$

(2.11)

where X, Y are non-negative random variables in $(\Omega,{\mathcal {F}}_{T},P)$ and $1< p$, $q<\infty$ is a pair of conjugated exponents, i.e., $\frac{1}{p}+\frac{1}{q}=1$. One may proceed in the following way (cf., e.g., Krein et al. [32], p.43). By elementary calculus, one verifies

$$ab=\inf_{r>0} \biggl(\frac{r^{p}}{p}a^{p}+ \frac{r^{-q}}{q}b^{q} \biggr) $$

for any constant $a, b\geq0$. This yields $XY\leq\frac{r^{p}}{p}X^{p}+\frac{r^{-q}}{q}Y^{q}$ a.s. for any $r>0$. Taking the expectation yields $E_{P}[XY]\leq\frac{r^{p}}{p}E_{P}[X^{p}]+\frac{r^{-q}}{q}E_{P}[Y^{q}]$ for any $r>0$, and taking the infimum with respect to r again we arrive at (2.11).

By the above argument, we have the following Hölder inequality for ${\mathcal{F}}_{t}$-consistent nonlinear evaluations.

Theorem 2.3

Suppose that ${\mathcal {E}}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is an ${\mathcal {F}}_{t}$-consistent nonlinear evaluation in $L^{1}({\mathcal{F}}_{T})$. If ${\mathcal{E}}_{s,t}[\cdot]$ satisfies the following conditions:

• (d) ${\mathcal{E}}_{s,t}[\xi+\eta]\leq{\mathcal {E}}_{s,t}[\xi]+{\mathcal{E}}_{s,t}[\eta]$ a.s., $\forall(\xi,\eta)\in L^{1}_{+}({\mathcal{F}}_{t})\times L^{1}_{+}({\mathcal{F}}_{t})$;

• (e) ${\mathcal{E}}_{s,t}[\lambda \xi]\leq\lambda{\mathcal{E}}_{s,t}[\xi]$ a.s., $\forall\xi\in L^{1}_{+}({\mathcal{F}}_{t})$, $\lambda\geq0$,

then, for any $X,Y\in L^{1}({\mathcal{F}}_{t})$ and $|X|^{p}, |Y|^{q}\in L^{1}({\mathcal{F}}_{t})$ ($p, q>1$ and $1/p+1/q=1$), we have

$${\mathcal{E}}_{s,t}\bigl[\vert XY\vert \bigr]\leq \bigl({\mathcal {E}}_{s,t}\bigl[|X|^{p}\bigr] \bigr)^{\frac{1}{p}} \bigl({ \mathcal {E}}_{s,t}\bigl[|Y|^{q}\bigr] \bigr)^{\frac{1}{q}} \quad \textit{a.s.} $$

Similarly, we have the following Minkowski inequality for ${\mathcal {F}}_{t}$-consistent nonlinear evaluations.

Theorem 2.4

Suppose that ${\mathcal {E}}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is an ${\mathcal {F}}_{t}$-consistent nonlinear evaluation in $L^{1}({\mathcal{F}}_{T})$. If ${\mathcal{E}}_{s,t}[\cdot]$ satisfies the following conditions:

• (d) ${\mathcal{E}}_{s,t}[\xi+\eta]\leq{\mathcal {E}}_{s,t}[\xi]+{\mathcal{E}}_{s,t}[\eta]$ a.s., $\forall(\xi,\eta)\in L^{1}_{+}({\mathcal{F}}_{t})\times L^{1}_{+}({\mathcal{F}}_{t})$;

• (e) ${\mathcal{E}}_{s,t}[\lambda \xi]\leq\lambda{\mathcal{E}}_{s,t}[\xi]$ a.s., $\forall\xi\in L^{1}_{+}({\mathcal{F}}_{t})$, $\lambda\geq0$,

then, for any $X,Y\in L^{1}({\mathcal{F}}_{t})$ and $|X|^{p},|Y|^{p}\in L^{1}({\mathcal{F}}_{t})$ ($p>1$), we have

$$ \bigl({\mathcal {E}}_{s,t}\bigl[|X+Y|^{p}\bigr] \bigr)^{\frac{1}{p}}\leq\bigl({\mathcal {E}}_{s,t}\bigl[|X|^{p} \bigr]\bigr)^{\frac{1}{p}}+\bigl({\mathcal {E}}_{s,t} \bigl[|Y|^{p}\bigr]\bigr)^{\frac{1}{p}} \quad \textit{a.s.} $$

(2.12)

Proof

Here $h:[0,\infty)\times[0,\infty)\mapsto[0,\infty)$ is of the form

$$ h(x_{1},x_{2})= \bigl(x_{1}^{\frac{1}{p}}+x_{2}^{\frac{1}{p}} \bigr)^{p}= \inf_{r\in Q\cap(0,1)} \bigl\{ r^{-p}x_{1}+(1-r)^{-p}x_{2} \bigr\} , $$

(2.13)

where $\mathcal{Q}$ is the set of all rational numbers in $\mathcal {R}$. Let $x_{1}:=|X|^{p}$ and $x_{2}:=|Y|^{p}$. From (2.13), we have

$$\bigl(\vert X\vert +|Y|\bigr)^{p}\leq r^{-p}|X|^{p}+(1-r)^{-p}|Y|^{p} \quad \mbox{a.s.} $$

for all $r\in\mathcal{Q}\cap(0,1)$. It follows from (d) and (e) that

$${\mathcal{E}}_{s,t}\bigl[\bigl(\vert X\vert +|Y| \bigr)^{p}\bigr]\leq r^{-p}{\mathcal {E}}_{s,t} \bigl[|X|^{p}\bigr]+(1-r)^{-p}{\mathcal{E}}_{s,t} \bigl[|Y|^{p}\bigr] \quad \mbox{a.s.} $$

for all $r\in\mathcal{Q}\cap(0,1)$. Taking the infimum with respect to r in $\mathcal{Q}\cap(0,1)$, we have

$${\mathcal {E}}_{s,t}\bigl[\bigl(\vert X\vert +|Y| \bigr)^{p}\bigr]\leq \bigl\{ \bigl({\mathcal {E}}_{s,t} \bigl[|X|^{p}\bigr]\bigr)^{\frac{1}{p}}+\bigl({\mathcal {E}}_{s,t}\bigl[|Y|^{p}\bigr]\bigr)^{\frac{1}{p}} \bigr\} ^{p} \quad \mbox{a.s.} $$

Thus, (2.12) holds true. □

3 Jensen’s inequality for g-evaluations

In this section, first, we present some notations, notions, and propositions which are useful in this paper.

Let

$$\begin{aligned}& {\mathcal{S}}^{p}(0,t;P;\mathcal{R}):=\Bigl\{ V:V_{s} \mbox{ is } \mathcal{R}\mbox{-valued } {\mathcal{F}}_{s}\mbox{-adapted continuous process with} \\& \hphantom{{\mathcal{S}}^{p}(0,t;P;\mathcal{R}):={}}E_{P}\Bigl[\sup_{0\leq s\leq t}|V_{s}|^{p} \Bigr]< \infty\Bigr\} , \\& {\mathcal{S}}(0,t;P;\mathcal{R}):=\bigcup_{p>1}{ \mathcal {S}}^{p}(0,t;P;\mathcal{R}) , \\& L^{p}\bigl(0,t;P;\mathcal{R}^{d}\bigr):=\biggl\{ V:V_{s} \mbox{ is } \mathcal{R}^{d}\mbox{-valued and } { \mathcal{F}}_{s} \mbox{-adapted process with} \\& \hphantom{L^{p}\bigl(0,t;P;\mathcal{R}^{d}\bigr):={}}E_{P}\biggl[ \biggl(\int_{0}^{t}|V_{s}|^{2} \, \mathrm{d}s\biggr)^{\frac{p}{2}}\biggr]<\infty\biggr\} , \\& {\mathcal{L}}\bigl(0,t;P;\mathcal{R}^{d}\bigr):=\bigcup _{p>1}L^{p}\bigl(0,t;P;\mathcal{R}^{d} \bigr) , \\& {\mathcal{M}}^{p}(0,t;P;\mathcal{R}):=\biggl\{ V:V_{s} \mbox{ is } \mathcal{R} \mbox{-valued } {\mathcal{F}}_{s} \mbox{-adapted process with} \\& \hphantom{{\mathcal{M}}^{p}(0,t;P;\mathcal{R}):={}}E_{P}\biggl[\biggl(\int _{0}^{t}|V_{s}|\, \mathrm{d}s \biggr)^{p}\biggr]<\infty\biggr\} , \\& {\mathcal{M}}(0,t;P;\mathcal{R}):=\bigcup_{p>1}{ \mathcal {M}}^{p}(0,t;P;\mathcal{R}) \end{aligned}$$

and

$${\mathcal{L}}({\mathcal{F}}_{t}):=\bigcup _{p>1}L^{p}({\mathcal{F}}_{t}). $$

For each $t\in[0,T ]$, we consider the following BSDE with terminal time t:

$$ y_{s}=X+\int_{s}^{t} g(r,y_{r},z_{r})\, \mathrm{d}r-\int_{s}^{t} z_{r} \cdot\mathrm{d}B_{r}, \quad s\in[0,t]. $$

(3.1)

Here the function g:

$$g(\omega,t,y,z):\Omega\times[0,T]\times\mathcal{R}\times\mathcal {R}^{d}\mapsto\mathcal{R} $$

satisfies the following assumptions:

• (B.1) there exist two non-negative deterministic functions $\alpha(t)$ and $\beta(t)$ such that for all $y_{1},y_{2}\in\mathcal{R}$, $z_{1},z_{2}\in\mathcal{R}^{d}$,

  $$\bigl\vert g(t,y_{1},z_{1})-g(t,y_{2},z_{2}) \bigr\vert \leq\alpha(t)|y_{1}-y_{2}|+\beta(t)|z_{1}-z_{2}|, \quad \forall t\in[0,T], $$

  where $\alpha(t)$ and $\beta(t)$ satisfy $\int_{0}^{T}\alpha^{2}(t)\, \mathrm{d}t<\infty$, $\int_{0}^{T}\beta^{2}(t)\, \mathrm{d}t<\infty$;

• (B.2) $g(t,0,0)\in{\mathcal{M}}(0,t;P;\mathcal{R})$;

• (B.3) $g(t,y,0)=0$, $\mathrm{d}P\times\mathrm{d}t$-a.s., $\forall y\in\mathcal{R}$.

It is well known that (see Zong [11]) if we suppose that the function g satisfies (B.1) and (B.2), then for each given $X\in{\mathcal {L}}({\mathcal{F}}_{t})$, there exists a unique solution $(Y^{X},Z^{X})\in {\mathcal{S}}(0,t;P;\mathcal{R})\times{\mathcal{L}}(0,t;P;\mathcal {R}^{d})$ of BSDE (3.1).

Example 3.1

For each given $\xi\in{\mathcal {L}}({\mathcal{F}}_{T})$, the BSDE

$$y_{t}=\xi+\int_{t}^{T} \biggl( \frac{1}{\sqrt[5]{s}}y_{s}+\frac{1}{\sqrt[8]{T-s}}|z_{s}| \biggr)\, \mathrm{d}s-\int_{t}^{T} z_{s}\cdot \mathrm{d}B_{s}, \quad t\in[0,T], $$

has a unique solution in ${\mathcal{S}}(0,T;P;\mathcal{R})\times{\mathcal {L}}(0,T;P;\mathcal{R}^{d})$.

We denote ${\mathcal{E}}^{g} _{s,t}[X] :=Y_{s}^{X}$. We thus define a system of operators:

$${\mathcal{E}}^{g}_{s,t}[X]:X\in{\mathcal{L}}({ \mathcal{F}}_{t})\mapsto {\mathcal{L}}({\mathcal{F}}_{s}), \quad 0\leq s\leq t\leq T. $$

This system is completely determined by the above given function g. We have the following.

Proposition 3.1

We assume that the function g satisfies (B.1) and (B.2). Then the system of operators ${\mathcal {E}}^{g}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is an ${\mathcal {F}}_{t}$-consistent nonlinear evaluation defined in ${\mathcal {L}}({\mathcal{F}}_{T})$.

The proof of Proposition 3.1 is very similar to that of Corollary 2.9 in [13], so we omit it.

Remark 3.1

From Proposition 3.1, we know that the dynamically consistent nonlinear evaluation ${\mathcal {E}}^{g}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is completely determined by the given function g. Thus, we call ${\mathcal {E}}^{g}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ a g-evaluation.

Definition 3.1

(g-Expectation) (see Zong [11])

Suppose that the function g satisfies (B.1) and (B.3). The g-expectation ${\mathcal{E}}_{g}[\cdot]:{\mathcal{L}}({\mathcal {F}}_{T})\mapsto\mathcal{R}$ is defined by ${\mathcal {E}}_{g}[\xi]=Y_{0}^{\xi}$.

Definition 3.2

(Conditional g-expectation) (see Zong [11])

Suppose that the function g satisfies (B.1) and (B.3). The conditional g-expectation of ξ with respect to ${\mathcal{F}}_{t}$ is defined by ${\mathcal{E}}_{g}[\xi|{\mathcal{F}}_{t}]=Y_{t}^{\xi}$.

Proposition 3.2

(see Zong [11])

${\mathcal {E}}_{g}[\xi|{\mathcal{F}}_{t}]$ is the unique random variable η in ${\mathcal{L}}({\mathcal{F}}_{t})$ such that

$${\mathcal{E}}_{g}[1_{A}\xi]={\mathcal{E}}_{g}[1_{A} \eta], \quad \forall A\in{\mathcal{F}}_{t}. $$

Proposition 3.3

For any $\xi_{n}\in{\mathcal {L}}({\mathcal{F}}_{t})$, if $\lim_{n\rightarrow\infty}\xi_{n}=\xi$ a.s. and $|\xi_{n}|\leq\eta$ a.s. with $\eta\in{\mathcal{L}}({\mathcal{F}}_{t})$, then for $0\leq s\leq t\leq T$,

$$\lim_{n\rightarrow\infty}{\mathcal{E}}^{g}_{s,t}[ \xi_{n}]={\mathcal {E}}^{g}_{s,t}[\xi] \quad \textit{a.s.} $$

The proof of Proposition 3.3 is very similar to that of Theorem 3.1 in Hu and Chen [24], so we omit it.

In the following, we study Jensen’s inequality for g-evaluations. First, we introduce some notions on g.

Definition 3.3

Let $g: \Omega\times[0,T]\times \mathcal{R}\times\mathcal{R}^{d}\mapsto\mathcal{R}$. The function g is said to be super-homogeneous if for each $(y,z)\in\mathcal{R}\times\mathcal {R}^{d}$ and $\lambda\in\mathcal{R}$, then $g(t,\lambda y,\lambda z)\geq\lambda g(t,y,z)$, $\mathrm{d}P\times\mathrm{d}t$-a.s. The function g is said to be positively homogeneous if for each $(y,z)\in\mathcal{R}\times \mathcal{R}^{d}$ and $\lambda\geq0$, then $g(t,\lambda y,\lambda z)=\lambda g(t,y,z)$, $\mathrm{d}P\times\mathrm{d}t$-a.s. The function g is said to be sub-additive if, for any $(y,z), (\overline{y},\overline{z})\in\mathcal{R}\times\mathcal{R}^{d}$, $g(t,y+\overline{y},z+\overline{z})\leq g(t,y,z) +g(t,\overline{y},\overline{z})$, $\mathrm{d}P\times\mathrm{d}t$-a.s. The function g is said to be super-additive if, for any $(y,z), (\overline{y},\overline{z})\in\mathcal{R}\times\mathcal{R}^{d}$, $g(t,y+\overline{y},z+\overline{z})\geq g(t,y,z) +g(t,\overline{y},\overline{z})$, $\mathrm{d}P\times\mathrm{d}t$-a.s.

Theorem 3.1

Suppose that ${\mathcal {E}}^{g}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is a g-evaluation, then the following three statements are equivalent:

• (i) Jensen’s inequality for g-evaluation ${\mathcal{E}}^{g}_{s,t}[\cdot]$ holds in general, i.e., for each convex function $\varphi(x): \mathcal{R}\mapsto\mathcal{R}$ and each $\xi\in{\mathcal{L}}({\mathcal{F}}_{t})$, if $\varphi(\xi)\in{\mathcal {L}}({\mathcal{F}}_{t})$, then we have

  $${\mathcal{E}}^{g}_{s,t}\bigl[\varphi(\xi)\bigr]\geq\varphi \bigl({\mathcal {E}}^{g}_{s,t}[\xi]\bigr) \quad \textit{a.s.}; $$

• (ii) $\forall(\xi,a,b)\in{\mathcal{L}}({\mathcal {F}}_{t})\times\mathcal{R}\times\mathcal{R}$, ${\mathcal {E}}^{g}_{s,t}[a\xi+b]\geq a{\mathcal{E}}^{g}_{s,t}[\xi]+b$ a.s.;

• (iii) g is independent of y and super-homogeneous with respect to z.

Theorem 3.2

Suppose that ${\mathcal {E}}^{g}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is a g-evaluation, then the following three statements are equivalent:

• (i) the n-dimensional ($n>1$) Jensen inequality for the g-evaluation ${\mathcal{E}}^{g}_{s,t}[\cdot]$ holds in general, i.e., for each convex function $\varphi: R^{n}\mapsto R$ and $\xi_{i}\in{\mathcal{L}}({\mathcal{F}}_{t})$ ($i=1,2,\ldots,n$), if $\varphi(\xi_{1},\xi_{2},\ldots,\xi_{n})\in{\mathcal{L}}({\mathcal{F}}_{t})$, then we have

  $${\mathcal{E}}^{g}_{s,t} \bigl[\varphi(\xi_{1}, \xi_{2},\ldots,\xi_{n}) \bigr]\geq \varphi \bigl({\mathcal {E}}^{g}_{s,t}[\xi_{1}],{\mathcal{E}}^{g}_{s,t}[ \xi_{2}],\ldots,{\mathcal {E}}^{g}_{s,t}[ \xi_{n}] \bigr) \quad \textit{a.s.}; $$

• (ii) ${\mathcal{E}}^{g}_{s,t}$ is linear in ${\mathcal {L}}({\mathcal{F}}_{t})$;

• (iii) g is independent of y and linear with respect to z, i.e., g is of the form $g(t,y,z)=g(t,z)=\alpha_{t}\cdot z$, $\mathrm{d}P\times\mathrm{d}t$-a.s., $\forall(y,z)\in\mathcal{R}\times\mathcal{R}^{d}$, where α is a $R^{d}$-valued progressively measurable process.

In order to prove Theorems 3.1 and 3.2, we need the following lemmas. These lemmas can be found in Zong and Hu [33].

Lemma 3.1

Suppose that the function g satisfies (B.1) and (B.2). Then the following three conditions are equivalent:

• (i) The function g is independent of y.

• (ii) The corresponding dynamically consistent nonlinear evaluation ${\mathcal{E}}^{g}[\cdot]$ satisfies: for each $0\leq s\leq t\leq T$, ${\mathcal{F}}_{t}$ measurable simple function X and $y\in\mathcal{R}$,

  $${\mathcal{E}}^{g}_{s,t}[X+y]={\mathcal{E}}^{g}_{s,t}[X]+y\quad \textit{a.s.} $$

• (iii) The corresponding dynamically consistent nonlinear evaluation ${\mathcal{E}}^{g}[\cdot]$ satisfies: for each $0\leq s\leq t\leq T$, $X\in{\mathcal{L}}({\mathcal{F}}_{t})$, and $\eta \in {\mathcal{L}}({\mathcal{F}}_{s})$,

  $${\mathcal{E}}^{g}_{s,t}[X+\eta]={\mathcal{E}}^{g}_{s,t}[X]+ \eta \quad \textit{a.s.} $$

Lemma 3.2

Suppose that the function g satisfies (B.1) and (B.2). Then the following three conditions are equivalent:

• (i) The function g is positively homogeneous.

• (ii) The corresponding dynamically consistent nonlinear evaluation ${\mathcal{E}}^{g}[\cdot]$ satisfies: for each $0\leq s\leq t\leq T$, $\lambda\geq0$, and ${\mathcal{F}}_{t}$ measurable simple function X,

  $${\mathcal{E}}^{g}_{s,t}[\lambda X]=\lambda{ \mathcal{E}}^{g}_{s,t}[X] \quad \textit{a.s.} $$

• (iii) The corresponding dynamically consistent nonlinear evaluation ${\mathcal{E}}^{g}[\cdot]$ is positively homogeneous: for each $0\leq s\leq t\leq T$, $\lambda\geq0$, and $X\in{\mathcal{L}}({\mathcal{F}}_{t})$,

  $${\mathcal{E}}^{g}_{s,t}[\lambda X]=\lambda{ \mathcal{E}}^{g}_{s,t}[X]\quad \textit{a.s.} $$

Lemma 3.3

Suppose that the function g satisfies (B.1) and (B.2). Then the following three conditions are equivalent:

• (i) The function g is sub-additive (super-additive).

• (ii) The corresponding dynamically consistent nonlinear evaluation ${\mathcal{E}}^{g}[\cdot]$ satisfies: for each $0\leq s\leq t\leq T$ and ${\mathcal{F}}_{t}$ measurable simple functions X and $\overline{X}$,

  $${\mathcal{E}}^{g}_{s,t}[X+\overline{X}]\leq(\geq)\, { \mathcal{E}}^{g}_{s,t}[X] +{\mathcal{E}}^{g}_{s,t}[ \overline{X}] \quad \textit{a.s.} $$

• (iii) The corresponding dynamically consistent nonlinear evaluation ${\mathcal{E}}^{g}[\cdot]$ is sub-additive (super-additive): for each $0\leq s\leq t\leq T$ and X, $\overline{X}\in{\mathcal{L}}({\mathcal{F}}_{t})$,

  $${\mathcal{E}}^{g}_{s,t}[X+\overline{X}]\leq(\geq)\, { \mathcal{E}}^{g}_{s,t}[X] +{\mathcal{E}}^{g}_{s,t}[ \overline{X}] \quad \textit{a.s.} $$

Lemma 3.4

Suppose that the functions g and $\overline{g}$ satisfy (B.1) and (B.2). Then the following three conditions are equivalent:

• (i) $g(t,y,z)\geq\overline{g}(t,y,z)$, $\mathrm{d}P\times\mathrm{d}t$-a.s., $\forall(y,z)\in\mathcal{R}\times\mathcal{R}^{d}$.

• (ii) The corresponding dynamically consistent nonlinear evaluations ${\mathcal{E}}^{g}[\cdot]$ and ${\mathcal {E}}^{\overline{g}}[\cdot]$ satisfy, for each $0\leq s\leq t\leq T$ and ${\mathcal{F}}_{t}$ measurable simple function X,

  $${\mathcal{E}}^{g}_{s,t}[X]\geq{\mathcal{E}}^{\overline{g}}_{s,t}[X] \quad \textit{a.s.} $$

• (iii) The corresponding dynamically consistent nonlinear evaluations ${\mathcal{E}}^{g}[\cdot]$ and ${\mathcal {E}}^{\overline{g}}[\cdot]$ satisfy, for each $0\leq s\leq t\leq T$ and $X\in{\mathcal{L}}({\mathcal{F}}_{t})$,

  $${\mathcal{E}}^{g}_{s,t}[X]\geq{\mathcal{E}}^{\overline{g}}_{s,t}[X] \quad \textit{a.s.} $$

In particular, ${\mathcal {E}}^{g}[\cdot]\equiv{\mathcal{E}}^{\overline{g}}[\cdot]$ if and only if $g\equiv\overline{g}$.

Proof of Theorem 3.1

From Theorem 2.1, we only need to prove (ii) ⇔ (iii). (iii) ⇒ (ii) is obvious.

In the following, we prove (ii) ⇒ (iii). First, we prove that g is independent of y. Suppose (ii) holds, then we have, for any $(\xi,y)\in{\mathcal{L}}({\mathcal {F}}_{t})\times \mathcal{R}$,

$$ {\mathcal{E}}^{g}_{s,t}[\xi+y]={\mathcal{E}}^{g}_{s,t}[ \xi]+y\quad \mbox{a.s.} $$

(3.2)

By Lemma 3.1, we can deduce that g is independent of y.

Next we prove that g is super-homogeneous with respect to z. By (ii), we have, for any $(\xi,\lambda)\in{\mathcal {L}}({\mathcal{F}}_{t})\times R$,

$$ \lambda{\mathcal{E}}^{g}_{s,t}[\xi]\leq{\mathcal {E}}^{g}_{s,t}[\lambda\xi] \quad \mbox{a.s.} $$

(3.3)

For each $(s,z)\in[0,t]\times\mathcal{R}^{d}$, let $Y_{\cdot}^{s,z}$ be the solution of the following stochastic differential equation (SDE for short) defined on $[s,t]$:

$$ Y_{t}^{s,z}=-\int_{s}^{t}g(r,z) \, \mathrm{d}r+z\cdot(B_{t}-B_{s}). $$

(3.4)

From (3.3), we have

$${\mathcal{E}}^{g}_{r,t}\bigl[\lambda Y_{t}^{s,z} \bigr]\geq\lambda{\mathcal {E}}^{g}_{r,t}\bigl[Y_{t}^{s,z} \bigr]=\lambda Y_{r}^{s,z}, \quad 0\leq s\leq r\leq t\leq T. $$

Thus, $(\lambda Y_{r}^{s,z})_{r\in[s,t]}$ is an ${\mathcal{E}}_{g}$-submartingale. From the decomposition theorem of an ${\mathcal {E}}_{g}$-supermartingale (see Zong and Hu [33]), it follows that there exists an increasing process $(A_{r})_{r\in[s,t]}$ such that

$$\lambda Y_{t}^{s,z}=-\int_{s}^{t}g(r,Z_{r}) \, \mathrm{d}r +A_{t}-A_{s}+\int_{s}^{t}Z_{r} \cdot\mathrm{d}B_{r}, \quad t\in[s,T]. $$

This with $\lambda Y_{t}^{s,z}=-\int_{s}^{t}\lambda g(r,z)\, \mathrm{d}r+\int_{s}^{t}\lambda z\cdot\mathrm{d}B_{r}$ yields $Z_{r}\equiv\lambda z$ and

$$ \lambda g(t,z)\leq g(t,\lambda z), \quad \mathrm{d}P\times\mathrm{d}t \mbox{-a.s.} $$

(3.5)

The proof of Theorem 3.1 is complete. □

Remark 3.2

The condition that g is super-homogeneous with respect to z implies that g is positively homogeneous with respect to z. Indeed, for each fixed $\lambda>0$, by (3.5), we have $\frac{1}{\lambda}g(t,\lambda z)\leq g(t,z)$, $\mathrm{d}P\times\mathrm{d}t$-a.s., i.e.,

$$ g(t,\lambda z)\leq\lambda g(t,z), \quad \mathrm{d}P\times\mathrm{d}t \mbox{-a.s.} $$

(3.6)

Thus by (3.5) and (3.6), for any $\lambda>0$,

$$ g(t,\lambda z)=\lambda g(t,z), \quad \mathrm{d}P\times\mathrm{d}t \mbox{-a.s.} $$

(3.7)

In particular, choosing $\lambda=2$, we have $2 g(t,0)=g(t,0)$, $\mathrm{d}P\times\mathrm{d}t$-a.s. Hence $g(t,0)=0$, $\mathrm{d}P\times\mathrm{d}t$-a.s. Thus, for $\lambda=0$ (3.7) still holds.

Proof of Theorem 3.2

From Theorem 2.2, we only need to prove (ii) ⇔ (iii). (iii) ⇒ (ii) is obvious.

In the following, we prove (ii) ⇒ (iii). From the proof of Theorem 3.1, we can obtain, for any $\lambda\in\mathcal{R}$ and $(y,z)\in\mathcal{R}\times\mathcal {R}^{d}$, $g(t,y,\lambda z)=g(t,\lambda z)\geq\lambda g(t,z)$, $\mathrm{d}P\times\mathrm{d}t$-a.s. Using the same method, we have $g(t,y,\lambda z)=g(t,\lambda z)\leq\lambda g(t,z)$, $\mathrm{d}P\times\mathrm{d}t$-a.s., $\forall\lambda\in\mathcal{R}$, $(y,z)\in\mathcal{R}\times \mathcal{R}^{d}$. The above arguments imply that, for any $\lambda\in\mathcal{R}$ and $(y,z)\in \mathcal{R}\times\mathcal{R}^{d}$,

$$ g(t,y,\lambda z)=g(t,\lambda z)=\lambda g(t,z), \quad \mathrm{d}P\times \mathrm{d}t\mbox{-a.s.} $$

(3.8)

On the other hand, by Lemma 3.3, we have, for any $(y,z),(\overline{y},\overline{z})\in\mathcal{R}\times\mathcal{R}^{d}$,

$$ g(t,y+\overline{y},z+\overline{z})=g(t,y,z) +g(t,\overline{y},\overline{z}), \quad \mathrm{d}P\times\mathrm{d}t\mbox{-a.s.} $$

(3.9)

It follows from (3.8) and (3.9) that (iii) holds true. The proof of Theorem 3.2 is complete. □

From Theorem 3.1(iii), we know that, for any $y\in\mathcal{R}$, $g(t,y,0)=g(t,0)=0$, $\mathrm{d}P\times\mathrm{d}t$-a.s. Hence, ${\mathcal{E}}^{g}_{s,t}[\cdot]={\mathcal{E}}_{g}[\cdot|{\mathcal {F}}_{s}]$. Thus, Theorem 3.1 can be rewritten as follows.

Corollary 3.1

Suppose that ${\mathcal {E}}^{g}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is a g-evaluation, then the following four statements are equivalent:

• (i) Jensen’s inequality for the g-evaluation ${\mathcal{E}}^{g}_{s,t}[\cdot]$ holds in general, i.e., for each convex function $\varphi(x): \mathcal{R}\mapsto\mathcal{R}$ and each $\xi\in{\mathcal{L}}({\mathcal{F}}_{t})$, if $\varphi(\xi)\in{\mathcal {L}}({\mathcal{F}}_{t})$, then we have

  $${\mathcal{E}}^{g}_{s,t}\bigl[\varphi(\xi)\bigr]\geq\varphi \bigl({\mathcal {E}}^{g}_{s,t}[\xi]\bigr) \quad \textit{a.s.}; $$

• (ii) $\forall(\xi,a,b)\in L^{2}({\mathcal{F}}_{T})\times \mathcal{R}\times\mathcal{R}$, ${\mathcal{E}}^{g}_{0,T}[a\xi+b]\geq a{\mathcal {E}}^{g}_{0,T}[\xi]+b$, and, for any $y\in\mathcal{R}$, $g(t,y,0)=0$, $\mathrm{d}P\times\mathrm{d}t$-a.s.;

• (iii) $\forall(\xi,a,b)\in L^{2}({\mathcal{F}}_{t})\times \mathcal{R}\times\mathcal{R}$, ${\mathcal{E}}^{g}_{s,t}[a\xi+b]\geq a{\mathcal {E}}^{g}_{s,t}[\xi]+b$ a.s.;

• (iv) g is independent of y and super-homogeneous with respect to z.

Similarly, Theorem 3.2 can be rewritten as follows.

Corollary 3.2

Suppose that ${\mathcal {E}}^{g}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is a g-evaluation, then the following four statements are equivalent:

• (i) the n-dimensional ($n>1$) Jensen inequality for g-evaluation ${\mathcal{E}}^{g}_{s,t}[\cdot]$ holds in general, i.e., for each convex function $\varphi: \mathcal {R}^{n}\mapsto\mathcal{R}$ and $\xi_{i}\in{\mathcal{L}}({\mathcal{F}}_{t})$ ($i=1,2,\ldots,n$), if $\varphi(\xi_{1},\xi_{2},\ldots,\xi_{n})\in{\mathcal{L}}({\mathcal{F}}_{t})$, then we have

  $${\mathcal{E}}^{g}_{s,t} \bigl[\varphi(\xi_{1}, \xi_{2},\ldots,\xi_{n}) \bigr]\geq \varphi \bigl({\mathcal {E}}^{g}_{s,t}[\xi_{1}],{\mathcal{E}}^{g}_{s,t}[ \xi_{2}],\ldots,{\mathcal {E}}^{g}_{s,t}[ \xi_{n}] \bigr) \quad \textit{a.s.}; $$

• (ii) ${\mathcal{E}}^{g}_{0,T}$ is linear in $L^{2}({\mathcal{F}}_{T})$ and, for any $y\in\mathcal{R}$, $g(t,y,0)=0$, $\mathrm{d}P\times\mathrm{d}t$-a.s.;

• (iii) ${\mathcal{E}}^{g}_{s,t}$ is linear in $L^{2}({\mathcal{F}}_{t})$;

• (iv) for each $(y,z)\in\mathcal{R}\times\mathcal{R}^{d}$, $g(t,y,z)=g(t,z)=\alpha_{t}\cdot z$, $\mathrm{d}P\times\mathrm{d}t$-a.s., where α is a $\mathcal{R}^{d}$-valued progressively measurable process.

Proof of Corollary 3.1

From Proposition 3.3 and Theorem 3.1, we only need to prove (ii) ⇔ (iii). It is obvious that (iii) implies (ii).

In the following, we prove that (ii) implies (iii). Suppose (ii) holds. For each $(X,t,k)\in L^{2}({\mathcal{F}}_{T} )\times[0, T]\times\mathcal{R}$, by (ii), we know that for each $A\in{\mathcal{F}}_{t}$,

$$\begin{aligned} {\mathcal {E}}^{g}_{0,T}\bigl[1_{A}(X+k) \bigr] =&{\mathcal{E}}^{g}_{0,T}[1_{A}X +1_{A}k-k]+k \\ =&{\mathcal{E}}^{g}_{0,T}\bigl[1_{A}X +1_{A^{C}}(-k)\bigr]+k \\ =&{\mathcal {E}}^{g}_{0,t}\bigl[{\mathcal {E}}^{g}_{t,T}\bigl[1_{A}X +1_{A^{C}} (-k) \bigr]\bigr]+k \\ =&{\mathcal{E}}^{g}_{0,t}\bigl[1_{A}{\mathcal {E}}^{g}_{t,T}[X]+1_{A^{C}}(-k)\bigr]+k \\ =&{\mathcal{E}}^{g}_{0,t}\bigl[1_{A}{\mathcal {E}}^{g}_{t,T}[X]+1_{A^{C}}(-k)+k\bigr] \\ =&{\mathcal {E}}^{g}_{0,t}\bigl[1_{A}\bigl({ \mathcal{E}}^{g}_{t,T}[X]+k\bigr)\bigr]. \end{aligned}$$

Thus

$$ {\mathcal{E}}^{g}_{t,T}[X+k]={\mathcal{E}}^{g}_{t,T}[X]+k \quad \mbox{a.s.} $$

(3.10)

For each $\lambda\neq0$, define ${\mathcal {E}}^{\lambda}_{t,T}[\cdot]:=\frac{{\mathcal {E}}^{g}_{t,T}[\lambda\cdot]}{\lambda}$, $\forall t\in[0,T]$. It is easy to check that ${\mathcal{E}}^{g}_{t,T}[\cdot]$ and ${\mathcal {E}}^{\lambda}_{t,T}[\cdot]$ are two ${\mathcal{F}}$-expectations in $L^{2}({\mathcal{F}}_{T})$ (the notion of ${\mathcal{F}}$-expectation can be seen in Coquet et al. [20]). If $\lambda>0$, for each $\xi \in L^{2}({\mathcal{F}}_{T})$, ${\mathcal{E}}^{\lambda}_{0,T}[\xi]\geq{\mathcal {E}}^{g}_{0,T}[\xi]$. In a similar manner to Lemma 4.5 in Coquet et al. [20], we can obtain

$$ {\mathcal{E}}^{\lambda}_{t,T}[\xi]\geq{\mathcal {E}}^{g}_{t,T}[\xi] \quad \mbox{a.s.}, \forall t \in[0,T]. $$

(3.11)

If $\lambda<0$, for each $\xi\in L^{2}({\mathcal{F}}_{T})$, ${\mathcal {E}}^{\lambda}_{0,T}[\xi]\leq{\mathcal {E}}^{g}_{0,T}[\xi]$. In a similar manner to Lemma 4.5 in Coquet et al. [20] again, we have

$$ {\mathcal {E}}^{\lambda}_{t,T}[\xi]\leq{\mathcal{E}}^{g}_{t,T}[ \xi] \quad \mbox{a.s.}, \forall t\in[0,T]. $$

(3.12)

From (3.11) and (3.12), we have, for any $(\xi,\lambda)\in L^{2}({\mathcal{F}}_{T})\times \mathcal{R}$,

$$ {\mathcal{E}}^{g}_{t,T}[\lambda\xi]\geq\lambda{\mathcal {E}}^{g}_{t,T}[\xi] \quad \mbox{a.s.}, \forall t \in[0,T]. $$

(3.13)

From (3.10) and (3.13), we have, for any $(\xi,a,b)\in L^{2}({\mathcal {F}}_{T})\times \mathcal{R}\times\mathcal{R}$,

$${\mathcal {E}}^{g}_{t,T}[a\xi+b]\geq a{\mathcal{E}}^{g}_{t,T}[ \xi]+b \quad \mbox{a.s.}, \forall t\in[0,T]. $$

Since, for any $y\in\mathcal{R}$, $g(t,y,0)=0$, $\mathrm{d}P\times\mathrm{d}t$-a.s., we have

$${\mathcal{E}}^{g}_{s,t}[a\xi+b]={\mathcal {E}}^{g}_{s,T}[a\xi+b]\geq a{\mathcal{E}}^{g}_{s,T}[ \xi]+b=a{\mathcal {E}}^{g}_{s,t}[\xi]+b \quad \mbox{a.s.}, \forall(\xi,a,b)\in L^{2}({\mathcal{F}}_{t})\times \mathcal{R} \times\mathcal{R}. $$

Therefore, (iii) holds true. The proof of Corollary 3.1 is complete. □

Proof of Corollary 3.2

From Proposition 3.3 and Theorem 3.2, we only need to prove (ii) ⇔ (iii). It is obvious that (iii) implies (ii).

In the following, we prove that (ii) implies (iii). Suppose (ii) holds. By Proposition 3.3, we know that for each sequence $\{X_{n}\}_{n=1}^{\infty}\subset L^{2}({\mathcal{F}}_{T})$ such that $X_{n}(\omega)\downarrow0$ for all ω, ${\mathcal {E}}^{g}_{0,T}[X_{n}]\downarrow0$. By the well-known Daniell-Stone theorem (cf., e.g., Yan [34], Theorem 3.6.8, p.83), there exists a unique probability measure $P_{\alpha}$ defined on $(\Omega,{\mathcal {F}}_{T})$ such that

$$ {\mathcal{E}}^{g}_{0,T}[\xi]=E_{P_{\alpha}}[\xi], \quad \forall\xi\in L^{2}({\mathcal{F}}_{T}) $$

(3.14)

holds. Indeed, from (iv), we know that $\frac{\mathrm{d}P_{\alpha}}{\mathrm{d}P}={ \exp} (\int_{0}^{T}\alpha_{t}\cdot\mathrm{d}B_{t}-\frac{1}{2}\int_{0}^{T}|\alpha_{t}|^{2}\, \mathrm{d}t )$.

On the other hand, since, for any $y\in\mathcal{R}$, $g(t,y,0)=0$, $\mathrm{d}P\times\mathrm{d}t$-a.s., we can obtain

$$ {\mathcal {E}}^{g}_{s,t}[\xi]={\mathcal{E}}^{g}_{s,T}[ \xi] \quad \mbox{a.s.}, \forall\xi\in L^{2}({\mathcal{F}}_{t}). $$

(3.15)

It follows from (3.14) and (3.15) that

$${\mathcal{E}}^{g}_{s,t}[\xi]=E_{P_{\alpha}}[\xi|{ \mathcal{F}}_{s}] \quad \mbox {a.s.}, \forall\xi\in L^{2}({ \mathcal{F}}_{t}). $$

Therefore, ${\mathcal {E}}^{g}_{s,t}$ is linear in $L^{2}({\mathcal{F}}_{t})$. The proof of Corollary 3.2 is complete. □

From Corollary 3.2, we can immediately obtain the following.

Theorem 3.3

Suppose that ${\mathcal {E}}^{g}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is a g-evaluation, then the following two statements are equivalent:

• (i) ${\mathcal{E}}^{g}_{s,t}$ is linear in ${\mathcal {L}}({\mathcal{F}}_{t})$;

• (ii) there exists a unique probability measure $P_{\alpha}$ defined on $(\Omega,{\mathcal{F}}_{T})$ such that, for any $\xi\in{\mathcal{L}}({\mathcal{F}}_{t})$,

  $${\mathcal{E}}^{g}_{s,t}[\xi]=E_{P_{\alpha}}[\xi|{ \mathcal{F}}_{s}] \quad \textit{a.s.} $$

The following result can be seen as an extension of Theorem 3.3.

Theorem 3.4

Suppose that ${\mathcal {E}}^{g}_{s,t}[\cdot]$, $0\leq s\leq t\leq T$ is a g-evaluation, then the following two statements are equivalent:

• (i) ${\mathcal{E}}^{g}_{s,t}$ is sublinear in ${\mathcal{L}}({\mathcal{F}}_{t})$, i.e.,

  • (f) ${\mathcal{E}}^{g}_{s,t}[\lambda X]=\lambda{\mathcal {E}}^{g}_{s,t}[X]$ a.s., for any $X\in{\mathcal{L}}({\mathcal{F}}_{t})$ and $\lambda\geq0$;

  • (g) ${\mathcal{E}}_{s,t}[X+Y]\leq{\mathcal {E}}^{g}_{s,t}[X]+{\mathcal{E}}^{g}_{s,t}[Y]$ a.s., for any $(X,Y)\in {\mathcal{L}}({\mathcal{F}}_{t})\times{\mathcal{L}}({\mathcal{F}}_{t})$;

  • (h) ${\mathcal{E}}^{g}_{s,t}[\mu]=\mu$ a.s., for any $\mu\in\mathcal{R}$;

• (ii) for any $\xi\in{\mathcal{L}}({\mathcal{F}}_{t})$,

  $${\mathcal{E}}^{g}_{s,t}[\xi]=\sup_{Q_{\theta}\in\Lambda}E_{Q_{\theta}}[ \xi|{\mathcal{F}}_{s}] \quad \textit{a.s.}, $$

  where Λ is a set of probability measures on $(\Omega,{\mathcal{F}}_{T})$ and defined by

  $$\Lambda:=\bigl\{ Q_{\theta}:E_{Q_{\theta}}[\xi]\leq{\mathcal {E}}^{g}_{0,T}[\xi],\forall\xi\in{\mathcal{L}}({ \mathcal{F}}_{T})\bigr\} . $$

Proof

It is obvious that (ii) implies (i).

In the following, we prove that (i) implies (ii). Suppose (i) holds. Since ${\mathcal{E}}_{0,T}[\cdot]$ is a sublinear expectation in ${\mathcal{L}}({\mathcal{F}}_{T})$, by Lemma 2.4 in Peng [35], we know that there exists a family of linear expectations $\{E_{\theta}:\theta\in\Theta\}$ on $(\Omega,{\mathcal {F}}_{T})$ such that, for any $\xi\in{\mathcal{L}}({\mathcal{F}}_{T})$,

$$ {\mathcal {E}}^{g}_{0,T}[\xi]=\sup_{\theta\in\Theta}E_{\theta}[ \xi]. $$

(3.16)

On the other hand, by Proposition 3.3, we know that for each sequence $\{X_{n}\}_{n=1}^{\infty}\subset{\mathcal{L}}({\mathcal{F}}_{T})$ such that $X_{n}(\omega)\downarrow0$ for all ω, ${\mathcal {E}}^{g}_{0,T}[X_{n}]\downarrow0$. By the well-known Daniell-Stone theorem, we can deduce that for each $\theta\in\Theta$ and $\xi\in{\mathcal{L}}({\mathcal{F}}_{T})$, there exists a unique probability measure $Q_{\theta}$ defined on $(\Omega,{\mathcal{F}}_{T})$ such that

$$ E_{\theta}[\xi]=E_{Q_{\theta}}[\xi]. $$

(3.17)

It follows from (3.16) and (3.17) that, for any $\xi\in{\mathcal{L}}({\mathcal{F}}_{T})$,

$$ {\mathcal {E}}^{g}_{0,T}[\xi]=\sup_{Q_{\theta}\in\Lambda}E_{Q_{\theta}}[ \xi ]. $$

(3.18)

Let Π be a set of probability measures on $(\Omega,{\mathcal {F}}_{T})$ defined by

$$\Pi:= \biggl\{ P_{\alpha}:\alpha\in\Theta^{g}, \frac{\mathrm{d}P_{\alpha}}{\mathrm{d}P}={ \exp} \biggl(\int_{0}^{T} \alpha_{t}\cdot\mathrm{d}B_{t}-\frac{1}{2}\int _{0}^{T}|\alpha_{t}|^{2}\, \mathrm{d}t \biggr) \biggr\} , $$

where $\Theta^{g}$ := {$(\alpha_{t})_{t\in[0,T]}:\alpha$ is $\mathcal{R}^{d}$-valued, progressively measurable and, for any $(y,z)\in\mathcal{R}\times \mathcal{R}^{d}$, $\alpha_{t}\cdot z\leq g(t,y,z)$, $\mathrm{d}P\times\mathrm{d}t$-a.s.}. In order to prove (ii), now we prove that $\Pi=\Lambda$.

For any $\alpha\in\Theta^{g}$, we define $g^{\alpha}(t,y,z):=\alpha_{t}\cdot z$, $\forall t\in[0,T]$, $(y,z)\in \mathcal{R}\times\mathcal{R}^{d}$. Then, for any $\xi\in{\mathcal {L}}({\mathcal{F}}_{T})$, by the well-known Girsanov theorem, we can deduce that

$${\mathcal {E}}^{g^{\alpha}}_{0,T}[\xi]=E_{P_{\alpha}}[\xi]. $$

Since, for any $(y,z)\in\mathcal{R}\times\mathcal{R}^{d}$, $\alpha_{t}\cdot z=g^{\alpha}(t,y,z)\leq g(t,y,z)$, $\mathrm{d}P\times\mathrm{d}t$-a.s., it follows from the well-known comparison theorem for BSDEs that $E_{P_{\alpha}}[\xi]={\mathcal {E}}^{g^{\alpha}}_{0,T}[\xi]\leq{\mathcal{E}}^{g}_{0,T}[\xi]$. Hence $\Pi\subseteq\Lambda$.

Next let us prove that $\Lambda\subseteq\Pi$. For each $Q_{\theta}\in\Lambda$, since $E_{Q_{\theta}}[\cdot]\leq{\mathcal {E}}^{g}_{0,T}[\cdot]$, $\forall\xi, \eta\in L^{2}({\mathcal{F}}_{T})$, we have

$$ E_{Q_{\theta}}[\xi+\eta]-E_{Q_{\theta}}[\eta]=E_{Q_{\theta}}[\xi]\leq{ \mathcal {E}}^{g}_{0,T}[\xi]. $$

(3.19)

Denote $g^{\beta}(t,y,z):=\beta(t)|z|$, $\forall t\in[0,T]$, $(y,z)\in \mathcal{R}\times\mathcal{R}^{d}$. From Lemmas 3.1 and 3.2 and applying the well-known comparison theorem for BSDEs again, we have

$$ {\mathcal {E}}^{g}_{0,T}[\xi]={\mathcal{E}}_{g}[ \xi]\leq{\mathcal {E}}_{g^{\beta}}[\xi]. $$

(3.20)

From (3.19) and (3.20), we can deduce that $E_{Q_{\theta}}[\xi+\eta]-E_{Q_{\theta}}[\eta]\leq{\mathcal {E}}_{g^{\beta}}[\xi]$. Then, in a similar manner to Theorem 7.1 in Coquet et al. [20], we know that there exists a unique function $g^{\theta}$ defined on $\Omega\times[0,T]\times\mathcal{R}\times \mathcal{R}^{d}$ satisfying the following three conditions:

• (H.1) $g^{\theta}(t,y,0)=0$, $\mathrm{d}P\times\mathrm{d}t$-a.s., $\forall y\in\mathcal{R}$;

• (H.2) $|g^{\theta}(t,y_{1},z_{1})-g^{\theta}(t,y_{2},z_{2})|\leq\beta(t)|z_{1}-z_{2}|$, $\forall(y_{1},z_{1}), (y_{2},z_{2})\in\mathcal{R}\times\mathcal{R}^{d}$, where $\beta(t)$ is a non-negative deterministic function satisfying that $\int_{0}^{T}\beta^{2}(t)\, \mathrm{d}t<\infty$;

• (H.3) ${\mathcal{E}}_{g^{\theta}}[\xi|{\mathcal {F}}_{t}]=E_{Q_{\theta}}[\xi|{\mathcal{F}}_{t}]$ a.s., $\forall\xi\in L^{2}({\mathcal{F}}_{T})$.

It follows from the linearity of $({\mathcal {E}}_{g^{\theta}}[\cdot|{\mathcal{F}}_{t}] )_{t\in[0,T]}$ and Theorem 3.2 that $g^{\theta}$ is linear with respect to z. Therefore, there exists a $\mathcal{R}^{d}$-valued progressively measurable process $(\theta_{t})_{t\in[0,T]}$ such that $g^{\theta}(t,y,z)=\theta_{t}\cdot z$, $\mathrm{d}P\times\mathrm{d}t$-a.s., $\forall(y,z)\in\mathcal{R}\times \mathcal{R}^{d}$. In view of $Q_{\theta}\in\Lambda$ and (H.3), we have for each $\xi\in L^{2}({\mathcal{F}}_{T})$, ${\mathcal {E}}_{g^{\theta}}[\xi]=E_{Q_{\theta}}[\xi]\leq{\mathcal {E}}^{g}_{0,T}[\xi]$. Then in a similar manner to Lemma 4.5 in Coquet et al. [20] and by Lemma 3.4, we can obtain $g^{\theta}(t,y,z)=\theta_{t}\cdot z\leq g(t,y,z)$, $\mathrm{d}P\times\mathrm{d}t$-a.s., $\forall(y,z)\in\mathcal{R}\times\mathcal {R}^{d}$. For θ, we define the probability measure $P_{\theta}$ satisfying $\frac{\mathrm{d}P_{\theta}}{\mathrm{d}P}={ \exp} (\int_{0}^{T}\theta_{t}\cdot\mathrm{d}B_{t}-\frac{1}{2}\int_{0}^{T}|\theta_{t}|^{2}\, \mathrm{d}t )$, then $P_{\theta}\in\Pi$ and $E_{P_{\theta}}[\xi]={\mathcal {E}}_{g^{\theta}}[\xi]=E_{Q_{\theta}}[\xi]$, $\forall\xi\in L^{2}({\mathcal {F}}_{T})$. Hence, $Q_{\theta}=P_{\theta}\in\Pi$. Thus, $\Lambda\subseteq\Pi$. Therefore, we have $\Pi=\Lambda$.

Finally, we prove that, for any $s, t\in[0,T]$ satisfying $s\leq t$ and $\xi\in{\mathcal{L}}({\mathcal{F}}_{t})$, ${\mathcal {E}}^{g}_{s,t}[\xi]=\sup_{Q_{\theta}\in\Lambda}E_{Q_{\theta}}[\xi |{\mathcal{F}}_{s}]$ a.s. It follows from (H.3), the well-known comparison theorem for BSDEs, and Proposition 3.3 that

$${\mathcal {E}}^{g}_{s,t}[\xi]\geq{\mathcal{E}}_{g^{\theta}}[ \xi|{\mathcal {F}}_{s}]=E_{Q_{\theta}}[\xi|{\mathcal{F}}_{s}] \quad \mbox{a.s.}, \forall\xi\in{\mathcal{L}}({\mathcal{F}}_{t}). $$

Hence, for any $s, t\in[0,T]$ satisfying $s\leq t$ and $\xi\in{\mathcal{L}}({\mathcal {F}}_{t})$,

$$ {\mathcal {E}}^{g}_{s,t}[\xi]\geq\sup_{Q_{\theta}\in\Lambda}E_{Q_{\theta}}[ \xi |{\mathcal{F}}_{s}] \quad \mbox{a.s.} $$

(3.21)

On the other hand, by Lemmas 3.1, 3.2, and 3.3, we can deduce that g is independent of y and positively homogeneous, sub-additive with respect to z. For any $\xi\in{\mathcal{L}}({\mathcal{F}}_{T})$, let $(Y^{\xi}_{t},Z^{\xi}_{t} )_{t\in[0,T]}$ denote the solution of the following BSDE:

$$y_{t}=\xi+\int_{t}^{T}g(s,z_{s}) \, \mathrm{d}s-\int_{t}^{T}z_{s}\cdot \mathrm{d}B_{s}, \quad \forall t\in[0,T]. $$

By a measurable selection theorem (cf., e.g., El Karoui and Quenez [21], p.215), we can deduce that there exists a progressively measurable process $\alpha^{\xi}\in\Theta^{g}$ such that

$$ g\bigl(t,Z^{\xi}_{t}\bigr)=\alpha^{\xi}_{t} \cdot Z^{\xi}_{t}, \quad \mathrm{d}P\times\mathrm{d}t \mbox{-a.s.} $$

(3.22)

From (3.22) and applying the well-known Girsanov theorem, we have ${\mathcal{E}}^{g}_{s,t}[\xi]={\mathcal {E}}^{g}_{s,T}[\xi]=E_{P_{\alpha^{\xi}}}[\xi|{\mathcal{F}}_{s}]$ a.s. Hence, for any $\xi\in{\mathcal{L}}({\mathcal{F}}_{t})$,

$$ {\mathcal {E}}^{g}_{s,t}[\xi]\leq\sup_{P_{\alpha}\in\Pi}E_{P_{\alpha}}[ \xi |{\mathcal{F}}_{s}]=\sup_{Q_{\theta}\in\Lambda}E_{Q_{\theta}}[ \xi |{\mathcal{F}}_{s}] \quad \mbox{a.s.} $$

(3.23)

It follows from (3.21) and (3.23) that

$${\mathcal {E}}^{g}_{s,t}[\xi]=\sup_{Q_{\theta}\in\Lambda}E_{Q_{\theta}}[ \xi |{\mathcal{F}}_{s}] \quad \mbox{a.s.}, \forall\xi\in{\mathcal {L}}({\mathcal{F}}_{t}). $$

The proof of Theorem 3.4 is complete. □

4 Hölder’s inequality and Minkowski’s inequality for g-evaluations

In this section, we give a sufficient condition on g under which Hölder’s inequality and Minkowski’s inequality for g-evaluations hold true.

First, we give the following lemma.

Lemma 4.1

Suppose that the function g satisfies (B.1) and (B.2). Let g satisfy the following conditions:

• (i) for any $y_{1}\geq0$, $y_{2}\geq0$, and $(z_{1},z_{2})\in\mathcal{R}^{d}\times\mathcal{R}^{d}$,

  $$g(t,y_{1}+y_{2},z_{1}+z_{2})\leq g(t,y_{1},z_{1})+g(t,y_{2},z_{2}), \quad \mathrm{d}P\times \mathrm{d}t\textit{-a.s.}; $$

• (ii) for any $\lambda\geq0$, $y\geq0$, and $z\in \mathcal{R}^{d}$,

  $$g(t,\lambda y,\lambda z)\leq\lambda g(t,y,z),\quad \mathrm{d}P\times\mathrm{d}t \textit{-a.s.}, $$

then ${\mathcal{E}}^{g}_{s,t}[\cdot]$ satisfies the following conditions:

• (j) ${\mathcal{E}}^{g}_{s,t}[\xi+\eta]\leq{\mathcal {E}}^{g}_{s,t}[\xi]+{\mathcal{E}}^{g}_{s,t}[\eta]$ a.s., for any $(\xi,\eta)\in{\mathcal{L}}_{+}({\mathcal{F}}_{t})\times{\mathcal {L}}_{+}({\mathcal{F}}_{t})$;

• (k) ${\mathcal{E}}^{g}_{s,t}[\lambda\xi]=\lambda{\mathcal {E}}^{g}_{s,t}[\xi]$ a.s., for any $\xi\in{\mathcal{L}}_{+}({\mathcal{F}}_{t})$ and $\lambda\geq0$.

The key idea of the proof of Lemma 4.1 is the well-known comparison theorem for BSDEs. The proof is very similar to that of Proposition 4.2 in Jia [26]. So we omit it.

Applying Lemma 4.1 and Theorems 2.3 and 2.4, we immediately have the following Hölder inequality and Minkowski inequality for g-evaluations.

Theorem 4.1

Let g satisfy the conditions of Lemma  4.1, then, for any $X,Y\in{\mathcal{L}}({\mathcal{F}}_{t})$ and $|X|^{p}, |Y|^{q}\in{\mathcal{L}}({\mathcal{F}}_{t})$ ($p, q>1$ and $1/p+1/q=1$), we have

$${\mathcal{E}}^{g}_{s,t}\bigl[\vert XY\vert \bigr]\leq \bigl({\mathcal {E}}^{g}_{s,t}\bigl[|X|^{p}\bigr] \bigr)^{\frac{1}{p}} \bigl({\mathcal {E}}^{g}_{s,t} \bigl[|Y|^{q}\bigr] \bigr)^{\frac{1}{q}}\quad \textit{a.s.} $$

Theorem 4.2

Let g satisfy the conditions of Lemma  4.1, then, for any $X, Y\in{\mathcal{L}}({\mathcal{F}}_{t})$, and $|X|^{p},|Y|^{p}\in{\mathcal{L}}({\mathcal{F}}_{t})$ ($p>1$), we have

$$\bigl({\mathcal{E}}^{g}_{s,t}\bigl[|X+Y|^{p}\bigr] \bigr)^{\frac{1}{p}}\leq\bigl({\mathcal {E}}^{g}_{s,t} \bigl[|X|^{p}\bigr]\bigr)^{\frac{1}{p}}+\bigl({\mathcal {E}}^{g}_{s,t}\bigl[|Y|^{p}\bigr] \bigr)^{\frac{1}{p}} \quad \textit{a.s.} $$