Zong et al. Journal of Inequalities and Applications (2015) 2015:152 DOI 10.1186/s13660-015-0677-5
RESEARCH
Open Access
On Jensen’s inequality, Hölder’s inequality, and Minkowski’s inequality for dynamically consistent nonlinear evaluations Zhaojun Zong1 , Feng Hu1* , Chuancun Yin1 and Helin Wu2 *
Correspondence:
[email protected] 1 School of Statistics, Qufu Normal University, Qufu, 273165, People’s Republic of China Full list of author information is available at the end of the article
Abstract In this paper, the dynamically consistent nonlinear evaluations that were introduced by Peng are considered in probability space L2 (, F, (Ft )t≥0 , P). We investigate the n-dimensional (n ≥ 1) Jensen inequality, Hölder inequality, and Minkowski inequality for dynamically consistent nonlinear evaluations in L1 (, F, (Ft )t≥0 , 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 Lp (, F, (Ft )0≤t≤T , P) (1 < p ≤ 2). 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. Keywords: dynamically consistent nonlinear evaluation; g-evaluation; g-expectation; Jensen’s inequality; Hölder’s inequality; Minkowski’s inequality
1 Introduction It is well known that (see Peng [, ]) a dynamically consistent nonlinear evaluation in probability space L (, F , (Ft )t≥ , P), where {Ft }t≥ is a given filtration, is a system of operators: Es,t [X] : X ∈ L (, Ft , P) → L (, Fs , P),
≤ s ≤ t < ∞,
which satisfies the following properties: (i) Es,t [X ] ≥ Es,t [X ], if X ≥ X ; (ii) Et,t [X] = X; (iii) Er,s [Es,t [X]] = Er,t [X], if ≤ r ≤ s ≤ t < ∞; (iv) A Es,t [X] = A Es,t [A X], ∀A ∈ Fs . Of course, we can define this notion in L (, F , (Ft )t≥ , 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 gevaluation (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 © 2015 Zong et al.; licensee Springer. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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Pardoux and Peng [], who proved the existence and uniqueness of adapted solutions, when the coefficient g is Lipschitz in (y, z) uniformly in (t, ω), with square-integrability assumptions on the coefficient g(t, ω, 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 [], Lepeltier and San Martin [], El Karoui et al. [], Pardoux [, ], Briand et al. [] and the references therein) under some other assumptions on the coefficients but for a fixed terminal time T > . In , Chen and Wang [] obtained the existence and uniqueness theorem for L solutions of infinite time interval BSDEs when T = ∞, by the martingale representation theorem and fixed point theorem. Recently, Zong [] have obtained the result on Lp ( < p < ) 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 [] gave an application of dynamically consistent nonlinear evaluation (expectation) to recursive utility, Peng [, , –] and Rosazza Gianin [] 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 [, , , –]. In [, ], Peng obtained an important result: he proved that if a dynamically consistent nonlinear evaluation Es,t [·] can be dominated by a kind of g-evaluation, then Es,t [·] must be a g-evaluation. Thus, in this case, many problems on dynamically consistent nonlinear evaluations Es,t [·] 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. [], 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 [, , , , ] 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. [, ] 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 [] 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 [] and Jiang [], in fact, Jiang [] showed that g must be independent of y. In addition, Fan [] studied Jensen’s inequality for filtration-consistent nonlinear expectations without domination condition. Jia [] studied the n-dimensional (n > ) Jensen’s inequality for g-expectations and got the result that the n-dimensional (n > ) 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:
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For more general dynamically consistent nonlinear evaluation Es,t [·], what are the sufficient and necessary conditions under which Jensen’s inequality for Es,t [·] holds in general? Roughly speaking, what conditions on Es,t [·] are equivalent with the inequality Es,t ϕ(ξ ) ≥ ϕ Es,t [ξ ] a.s. holding for any convex function ϕ : R → 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 Es,t [·] under which the n-dimensional (n > ) Jensen inequality holds. As applications of these two results, we give four equivalent conditions on the -dimensional Jensen inequality and the n-dimensional (n > ) Jensen inequality for g-evaluations induced by BSDEs with non-uniform Lipschitz coefficients in Lp (, F , (Ft )≤t≤T , P) ( < p ≤ ), respectively. The remainder of this paper is organized as follows: In Section , we study the n-dimensional (n ≥ ) Jensen inequality, Hölder inequality, and Minkowski inequality for dynamically consistent nonlinear evaluations in L (, F , (Ft )t≥ , P). In Section , we give four equivalent conditions on the -dimensional Jensen inequality and the n-dimensional (n > ) Jensen inequality for g-evaluations induced by BSDEs with non-uniform Lipschitz coefficients in Lp (, F , (Ft )≤t≤T , P) ( < p ≤ ), respectively. These results generalize the known results on Jensen’s inequality for g-expectation in [, , , –, ]. In Section , 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 (, F , P) be a probability space carrying a standard d-dimensional Brownian motion (Bt )t≥ , and let (Ft )t≥ be the σ -algebra generated by (Bt )t≥ . We always assume that (Ft )t≥ is complete. Let T > be a given real number. In this paper, we always work in the probability space (, FT , P), and only consider processes indexed by t ∈ [, T]. We denote Lp (, Ft , P) (p ≥ ), the space of Ft -measurable random variables satisfying EP [|X|p ] < ∞, p and by L+ (, Ft , P) the space of non-negative random variables in Lp (, Ft , P). Let A denote the indicator of event A. For notational simplicity, we use Lp (Ft ) := Lp (, Ft , P) and p p L+ (Ft ) := L+ (, Ft , P). For the convenience of the reader, we recall the notion of a dynamically consistent nonlinear evaluation, defined in L (FT ) in Peng [, ], but defined in L (FT ) in this section. Definition . An Ft -consistent nonlinear evaluation in L (FT ) is a system of operators:
Es,t [X] : X ∈ L (Ft ) → L (Fs ),
≤ s ≤ t ≤ T,
which satisfies the following properties: (A.) monotonicity: Es,t [X ] ≥ Es,t [X ], if X ≥ X ; (A.) Et,t [X] = X; (A.) dynamical consistency: Er,s [Es,t [X]] = Er,t [X], if ≤ r ≤ s ≤ t ≤ T; (A.) zero one law: A Es,t [X] = A Es,t [A X], ∀A ∈ Fs .
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First, we consider Jensen’s inequality for Ft -consistent nonlinear evaluations. We have the following results. Theorem . Suppose that Es,t [·], ≤ s ≤ t ≤ T is an Ft -consistent nonlinear evaluation in L (FT ), then the following two statements are equivalent: (i) Jensen’s inequality for Ft -consistent evaluation Es,t [·] holds in general, i.e., for each convex function ϕ : R → R and ξ ∈ L (Ft ), if ϕ(ξ ) ∈ L (Ft ), then we have Es,t ϕ(ξ ) ≥ ϕ Es,t [ξ ] a.s.; (ii) ∀(ξ , a, b) ∈ L (Ft ) × R × R, Es,t [aξ + b] ≥ aEs,t [ξ ] + b a.s. Proof First, we prove (i) implies (ii). Suppose (i) holds, for each (ξ , a, b) ∈ L (Ft ) × R × R, let ϕ(x) := ax + b. Obviously, ϕ(x) is a convex function and ϕ(ξ ) ∈ L (Ft ), then we have Es,t [aξ + b] = Es,t ϕ(ξ ) ≥ ϕ Es,t [ξ ] = aEs,t [ξ ] + b a.s. In the following, we prove (ii) implies (i). Suppose (ii) holds, for each (ξ , a, b) ∈ L (Ft ) × R × R, we have
Es,t [aξ + b] ≥ aEs,t [ξ ] + b a.s.
(.)
But, for any convex function ϕ : R → R, there exists a countable set D ⊆ R such that ϕ(x) = sup (ax + b). (a,b)∈D
(.)
In view of (.), for any (a, b) ∈ D , we have Es,t ϕ(ξ ) ≥ Es,t [aξ + b] ≥ aEs,t [ξ ] + b a.s., which implies (i) by taking into consideration of (.).
Theorem . Suppose that Es,t [·], ≤ s ≤ t ≤ T is an Ft -consistent nonlinear evaluation in L (FT ) and n > , then the following two statements are equivalent: (i) the n-dimensional Jensen inequality for a Ft -consistent evaluation Es,t [·] holds in general, i.e., for each convex function ϕ : Rn → R and ξi ∈ L (Ft ) (i = , , . . . , n), if ϕ(ξ , ξ , . . . , ξn ) ∈ L (Ft ), then we have Es,t ϕ(ξ , ξ , . . . , ξn ) ≥ ϕ Es,t [ξ ], Es,t [ξ ], . . . , Es,t [ξn ] a.s.; (ii) Es,t (a) (b) (c)
is linear, i.e., Es,t [λX] = λEs,t [X] a.s., ∀(X, λ) ∈ L (Ft ) × R; Es,t [X + Y ] = Es,t [X] + Es,t [Y ] a.s., ∀(X, Y ) ∈ L (Ft ) × L (Ft ); Es,t [μ] = μ a.s., ∀μ ∈ R.
Proof We prove (i) implies (ii).
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First, we prove (i) implies (ii)(a). For each (X, λ) ∈ L (Ft ) × R, let ϕ(x , x , . . . , xn ) := λx and ξ := X. Obviously, ϕ(x , x , . . . , xn ) is a convex function and ϕ(ξ , ξ , . . . , ξn ) ∈ L (Ft ), then we have Es,t [λX] = Es,t ϕ(ξ , ξ , . . . , ξn ) ≥ ϕ Es,t [ξ ], Es,t [ξ ], . . . , Es,t [ξn ] = λEs,t [X] a.s.
(.)
On the other hand, let ϕ(x , x , . . . , xn ) := x – (λ – )x , ξ := λX, and ξ := X. By (i), we can deduce that Es,t [X] = Es,t ϕ(ξ , ξ , . . . , ξn ) ≥ ϕ Es,t [ξ ], Es,t [ξ ], . . . , Es,t [ξn ] = Es,t [λX] – (λ – )Es,t [X] a.s., i.e.,
Es,t [λX] ≤ λEs,t [X] a.s.
(.)
It follows from (.) and (.) that (ii)(a) holds true. Next we prove (ii)(b) holds. For each (X, Y ) ∈ L (Ft ) × L (Ft ), let ϕ(x , x , . . . , xn ) := x + x , ξ := X, and ξ := Y , then we have Es,t [X + Y ] = Es,t ϕ(ξ , ξ , . . . , ξn ) ≥ ϕ Es,t [ξ ], Es,t [ξ ], . . . , Es,t [ξn ] = Es,t [X] + Es,t [Y ] a.s.
(.)
On the other hand, let ϕ(x , x , . . . , xn ) := x – x , ξ := X + Y , and ξ := Y . By (i), we have Es,t [X] = Es,t ϕ(ξ , ξ , . . . , ξn ) ≥ ϕ Es,t [ξ ], Es,t [ξ ], . . . , Es,t [ξn ] = Es,t [X + Y ] – Es,t [Y ] a.s., i.e.,
Es,t [X + Y ] ≤ Es,t [X] + Es,t [Y ] a.s.
(.)
Thus, from (.) and (.), we can see that (ii)(b) holds. Finally, we prove (ii)(c) holds. For each μ ∈ R, let ϕ(x , x , . . . , xn ) := μ, then we have Es,t [μ] = Es,t ϕ(ξ , ξ , . . . , ξn ) ≥ ϕ Es,t [ξ ], Es,t [ξ ], . . . , Es,t [ξn ] = μ a.s.
(.)
On the other hand, let ϕ(x , x , . . . , xn ) := x – μ and ξ := μ. By (i), we can obtain Es,t [μ] = Es,t ϕ(ξ , ξ , . . . , ξn ) ≥ ϕ Es,t [ξ ], Es,t [ξ ], . . . , Es,t [ξn ] = Es,t [μ] – μ a.s., i.e.,
Es,t [μ] ≤ μ a.s. It follows from (.) and (.) that (ii)(c) holds true.
(.)
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In the following, we prove (ii) implies (i). Suppose (ii) holds, for any (a , a , . . . , an , b) ∈ Rn+ and ξi ∈ L (Ft ) (i = , , . . . , n), we have
Es,t
n
ai ξi + b =
i=
n
ai Es,t [ξi ] + b a.s.
(.)
i=
But, for any convex function ϕ : Rn → R, there exists a countable set D ⊆ Rn+ such that
ϕ(x , x , . . . , xn ) =
n
sup
(a ,a ,...,an ,b)∈D
ai xi + b .
(.)
i=
In view of (.), for any (a , a , . . . , an , b) ∈ D , we have n n Es,t ϕ(ξ , ξ , . . . , ξn ) ≥ Es,t ai ξi + b = ai Es,t [ξi ] + b a.s., i=
i=
which implies (i) by taking into consideration of (.).
The basic version of Hölder’s inequality for the classical mathematical expectation EP defined in (, FT , P) reads EP [XY ] ≤ EP X p p EP Y q q ,
(.)
where X, Y are non-negative random variables in (, FT , P) and < p, q < ∞ is a pair of conjugated exponents, i.e., p + q = . One may proceed in the following way (cf., e.g., Krein et al. [], p.). By elementary calculus, one verifies p r p r–q q ab = inf a + b r> p q p
–q
for any constant a, b ≥ . This yields XY ≤ rp X p + r q Y q a.s. for any r > . Taking the p –q expectation yields EP [XY ] ≤ rp EP [X p ] + r q EP [Y q ] for any r > , and taking the infimum with respect to r again we arrive at (.). By the above argument, we have the following Hölder inequality for Ft -consistent nonlinear evaluations. Theorem . Suppose that Es,t [·], ≤ s ≤ t ≤ T is an Ft -consistent nonlinear evaluation in L (FT ). If Es,t [·] satisfies the following conditions: (d) Es,t [ξ + η] ≤ Es,t [ξ ] + Es,t [η] a.s., ∀(ξ , η) ∈ L+ (Ft ) × L+ (Ft ); (e) Es,t [λξ ] ≤ λEs,t [ξ ] a.s., ∀ξ ∈ L+ (Ft ), λ ≥ , then, for any X, Y ∈ L (Ft ) and |X|p , |Y |q ∈ L (Ft ) (p, q > and /p + /q = ), we have Es,t |XY | ≤ Es,t |X|p p Es,t |Y |q q
a.s.
Similarly, we have the following Minkowski inequality for Ft -consistent nonlinear evaluations.
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Theorem . Suppose that Es,t [·], ≤ s ≤ t ≤ T is an Ft -consistent nonlinear evaluation in L (FT ). If Es,t [·] satisfies the following conditions: (d) Es,t [ξ + η] ≤ Es,t [ξ ] + Es,t [η] a.s., ∀(ξ , η) ∈ L+ (Ft ) × L+ (Ft ); (e) Es,t [λξ ] ≤ λEs,t [ξ ] a.s., ∀ξ ∈ L+ (Ft ), λ ≥ , then, for any X, Y ∈ L (Ft ) and |X|p , |Y |p ∈ L (Ft ) (p > ), we have
Es,t |X + Y |p p ≤ Es,t |X|p p + Es,t |Y |p p
a.s.
(.)
Proof Here h : [, ∞) × [, ∞) → [, ∞) is of the form p p p h(x , x ) = x + x =
–p r x + ( – r)–p x ,
inf
(.)
r∈Q∩(,)
where Q is the set of all rational numbers in R. Let x := |X|p and x := |Y |p . From (.), we have
p |X| + |Y | ≤ r–p |X|p + ( – r)–p |Y |p
a.s.
for all r ∈ Q ∩ (, ). It follows from (d) and (e) that p Es,t |X| + |Y | ≤ r–p Es,t |X|p + ( – r)–p Es,t |Y |p a.s. for all r ∈ Q ∩ (, ). Taking the infimum with respect to r in Q ∩ (, ), we have p p Es,t |X| + |Y | ≤ Es,t |X|p p + Es,t |Y |p p
a.s.
Thus, (.) 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 S p (, t; P; R) := V : Vs is R-valued Fs -adapted continuous process with EP sup |Vs |p < ∞ ,
S (, t; P; R) :=
≤s≤t
S p (, t; P; R),
p>
Lp , t; P; Rd := V : Vs is Rd -valued and Fs -adapted process with
t
|Vs | ds
EP
p
<∞ ,
p L , t; P; Rd := L , t; P; Rd , p>
Mp (, t; P; R) := V : Vs is R-valued Fs -adapted process with
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t
EP
M(, t; P; R) :=
p |Vs | ds
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<∞ ,
Mp (, t; P; R)
p>
and
L(Ft ) :=
Lp (Ft ).
p>
For each t ∈ [, T], we consider the following BSDE with terminal time t:
t
ys = X +
t
g(r, yr , zr ) dr – s
zr · dBr ,
s ∈ [, t].
(.)
s
Here the function g: g(ω, t, y, z) : × [, T] × R × Rd → R satisfies the following assumptions: (B.) there exist two non-negative deterministic functions α(t) and β(t) such that for all y , y ∈ R, z , z ∈ Rd , g(t, y , z ) – g(t, y , z ) ≤ α(t)|y – y | + β(t)|z – z |,
∀t ∈ [, T],
T T where α(t) and β(t) satisfy α (t) dt < ∞, β (t) dt < ∞; (B.) g(t, , ) ∈ M(, t; P; R); (B.) g(t, y, ) = , dP × dt-a.s., ∀y ∈ R. It is well known that (see Zong []) if we suppose that the function g satisfies (B.) and (B.), then for each given X ∈ L(Ft ), there exists a unique solution (Y X , ZX ) ∈ S (, t; P; R) × L(, t; P; Rd ) of BSDE (.). Example . For each given ξ ∈ L(FT ), the BSDE
T
yt = ξ + t
T + | ds – zs · dBs , y |z √ √ s s s T –s t
t ∈ [, T],
has a unique solution in S (, T; P; R) × L(, T; P; Rd ). g
We denote Es,t [X] := YsX . We thus define a system of operators: g
Es,t [X] : X ∈ L(Ft ) → L(Fs ),
≤ s ≤ t ≤ T.
This system is completely determined by the above given function g. We have the following. Proposition . We assume that the function g satisfies (B.) and (B.). Then the system of g operators Es,t [·], ≤ s ≤ t ≤ T is an Ft -consistent nonlinear evaluation defined in L(FT ).
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The proof of Proposition . is very similar to that of Corollary . in [], so we omit it. Remark . From Proposition ., we know that the dynamically consistent nonlinear g evaluation Es,t [·], ≤ s ≤ t ≤ T is completely determined by the given function g. Thus, g we call Es,t [·], ≤ s ≤ t ≤ T a g-evaluation. Definition . (g-Expectation) (see Zong []) Suppose that the function g satisfies (B.) and (B.). The g-expectation Eg [·] : L(FT ) → R is defined by Eg [ξ ] = Yξ . Definition . (Conditional g-expectation) (see Zong []) Suppose that the function g satisfies (B.) and (B.). The conditional g-expectation of ξ with respect to Ft is defined ξ by Eg [ξ |Ft ] = Yt . Proposition . (see Zong []) Eg [ξ |Ft ] is the unique random variable η in L(Ft ) such that
Eg [A ξ ] = Eg [A η],
∀A ∈ Ft .
Proposition . For any ξn ∈ L(Ft ), if limn→∞ ξn = ξ a.s. and |ξn | ≤ η a.s. with η ∈ L(Ft ), then for ≤ s ≤ t ≤ T, g
g
lim Es,t [ξn ] = Es,t [ξ ] a.s.
n→∞
The proof of Proposition . is very similar to that of Theorem . in Hu and Chen [], so we omit it. In the following, we study Jensen’s inequality for g-evaluations. First, we introduce some notions on g. Definition . Let g : × [, T] × R × Rd → R. The function g is said to be superhomogeneous if for each (y, z) ∈ R × Rd and λ ∈ R, then g(t, λy, λz) ≥ λg(t, y, z), dP × dta.s. The function g is said to be positively homogeneous if for each (y, z) ∈ R × Rd and λ ≥ , then g(t, λy, λz) = λg(t, y, z), dP × dt-a.s. The function g is said to be sub-additive if, for any (y, z), (y, z) ∈ R × Rd , g(t, y + y, z + z) ≤ g(t, y, z) + g(t, y, z), dP × dt-a.s. The function g is said to be super-additive if, for any (y, z), (y, z) ∈ R × Rd , g(t, y+y, z +z) ≥ g(t, y, z)+g(t, y, z), dP × dt-a.s. g
Theorem . Suppose that Es,t [·], ≤ s ≤ t ≤ T is a g-evaluation, then the following three statements are equivalent: g (i) Jensen’s inequality for g-evaluation Es,t [·] holds in general, i.e., for each convex function ϕ(x) : R → R and each ξ ∈ L(Ft ), if ϕ(ξ ) ∈ L(Ft ), then we have g g Es,t ϕ(ξ ) ≥ ϕ Es,t [ξ ] a.s.; g
g
(ii) ∀(ξ , a, b) ∈ L(Ft ) × R × R, Es,t [aξ + b] ≥ aEs,t [ξ ] + b a.s.; (iii) g is independent of y and super-homogeneous with respect to z. g
Theorem . Suppose that Es,t [·], ≤ s ≤ t ≤ T is a g-evaluation, then the following three statements are equivalent:
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g
(i) the n-dimensional (n > ) Jensen inequality for the g-evaluation Es,t [·] holds in general, i.e., for each convex function ϕ : Rn → R and ξi ∈ L(Ft ) (i = , , . . . , n), if ϕ(ξ , ξ , . . . , ξn ) ∈ L(Ft ), then we have g g g g Es,t ϕ(ξ , ξ , . . . , ξn ) ≥ ϕ Es,t [ξ ], Es,t [ξ ], . . . , Es,t [ξn ] a.s.; g
(ii) Es,t is linear in L(Ft ); (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) = αt · z, dP × dt-a.s., ∀(y, z) ∈ R × Rd , where α is a Rd -valued progressively measurable process. In order to prove Theorems . and ., we need the following lemmas. These lemmas can be found in Zong and Hu []. Lemma . Suppose that the function g satisfies (B.) and (B.). Then the following three conditions are equivalent: (i) The function g is independent of y. (ii) The corresponding dynamically consistent nonlinear evaluation E g [·] satisfies: for each ≤ s ≤ t ≤ T, Ft measurable simple function X and y ∈ R, g
g
Es,t [X + y] = Es,t [X] + y a.s. (iii) The corresponding dynamically consistent nonlinear evaluation E g [·] satisfies: for each ≤ s ≤ t ≤ T, X ∈ L(Ft ), and η ∈ L(Fs ), g
g
Es,t [X + η] = Es,t [X] + η a.s. Lemma . Suppose that the function g satisfies (B.) and (B.). Then the following three conditions are equivalent: (i) The function g is positively homogeneous. (ii) The corresponding dynamically consistent nonlinear evaluation E g [·] satisfies: for each ≤ s ≤ t ≤ T, λ ≥ , and Ft measurable simple function X, g
g
Es,t [λX] = λEs,t [X] a.s. (iii) The corresponding dynamically consistent nonlinear evaluation E g [·] is positively homogeneous: for each ≤ s ≤ t ≤ T, λ ≥ , and X ∈ L(Ft ), g
g
Es,t [λX] = λEs,t [X] a.s. Lemma . Suppose that the function g satisfies (B.) and (B.). Then the following three conditions are equivalent: (i) The function g is sub-additive (super-additive). (ii) The corresponding dynamically consistent nonlinear evaluation E g [·] satisfies: for each ≤ s ≤ t ≤ T and Ft measurable simple functions X and X, g
g
g
Es,t [X + X] ≤ (≥) Es,t [X] + Es,t [X] a.s.
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(iii) The corresponding dynamically consistent nonlinear evaluation E g [·] is sub-additive (super-additive): for each ≤ s ≤ t ≤ T and X, X ∈ L(Ft ), g
g
g
Es,t [X + X] ≤ (≥) Es,t [X] + Es,t [X] a.s. Lemma . Suppose that the functions g and g satisfy (B.) and (B.). Then the following three conditions are equivalent: (i) g(t, y, z) ≥ g(t, y, z), dP × dt-a.s., ∀(y, z) ∈ R × Rd . (ii) The corresponding dynamically consistent nonlinear evaluations E g [·] and E g [·] satisfy, for each ≤ s ≤ t ≤ T and Ft measurable simple function X, g
g
Es,t [X] ≥ Es,t [X] a.s. (iii) The corresponding dynamically consistent nonlinear evaluations E g [·] and E g [·] satisfy, for each ≤ s ≤ t ≤ T and X ∈ L(Ft ), g
g
Es,t [X] ≥ Es,t [X] a.s. In particular, E g [·] ≡ E g [·] if and only if g ≡ g. Proof of Theorem . From Theorem ., 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 (ξ , y) ∈ L(Ft ) × R, g
g
Es,t [ξ + y] = Es,t [ξ ] + y a.s.
(.)
By Lemma ., 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 (ξ , λ) ∈ L(Ft ) × R, g
g
λEs,t [ξ ] ≤ Es,t [λξ ] a.s.
(.)
For each (s, z) ∈ [, t] × Rd , let Y·s,z be the solution of the following stochastic differential equation (SDE for short) defined on [s, t]: Yts,z
t
=–
g(r, z) dr + z · (Bt – Bs ).
(.)
s
From (.), we have g g Er,t λYts,z ≥ λEr,t Yts,z = λYrs,z ,
≤ s ≤ r ≤ t ≤ T.
Thus, (λYrs,z )r∈[s,t] is an Eg -submartingale. From the decomposition theorem of an Eg supermartingale (see Zong and Hu []), it follows that there exists an increasing process (Ar )r∈[s,t] such that λYts,z = –
t
g(r, Zr ) dr + At – As + s
s
t
Zr · dBr ,
t ∈ [s, T].
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This with λYts,z = –
t s
λg(r, z) dr +
λg(t, z) ≤ g(t, λz),
t s
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λz · dBr yields Zr ≡ λz and
dP × dt-a.s.
(.)
The proof of Theorem . is complete.
Remark . 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 λ > , by (.), we have g(t, λz) ≤ g(t, z), dP × dt-a.s., i.e., λ g(t, λz) ≤ λg(t, z),
dP × dt-a.s.
(.)
Thus by (.) and (.), for any λ > , g(t, λz) = λg(t, z),
dP × dt-a.s.
(.)
In particular, choosing λ = , we have g(t, ) = g(t, ), dP × dt-a.s. Hence g(t, ) = , dP × dt-a.s. Thus, for λ = (.) still holds. Proof of Theorem . From Theorem ., we only need to prove (ii) ⇔ (iii). (iii) ⇒ (ii) is obvious. In the following, we prove (ii) ⇒ (iii). From the proof of Theorem ., we can obtain, for any λ ∈ R and (y, z) ∈ R × Rd , g(t, y, λz) = g(t, λz) ≥ λg(t, z), dP × dt-a.s. Using the same method, we have g(t, y, λz) = g(t, λz) ≤ λg(t, z), dP × dt-a.s., ∀λ ∈ R, (y, z) ∈ R × Rd . The above arguments imply that, for any λ ∈ R and (y, z) ∈ R × Rd , g(t, y, λz) = g(t, λz) = λg(t, z),
dP × dt-a.s.
(.)
On the other hand, by Lemma ., we have, for any (y, z), (y, z) ∈ R × Rd , g(t, y + y, z + z) = g(t, y, z) + g(t, y, z),
dP × dt-a.s.
(.)
It follows from (.) and (.) that (iii) holds true. The proof of Theorem . is complete. From Theorem .(iii), we know that, for any y ∈ R, g(t, y, ) = g(t, ) = , dP × dt-a.s. g Hence, Es,t [·] = Eg [·|Fs ]. Thus, Theorem . can be rewritten as follows. g
Corollary . Suppose that Es,t [·], ≤ s ≤ t ≤ T is a g-evaluation, then the following four statements are equivalent: g (i) Jensen’s inequality for the g-evaluation Es,t [·] holds in general, i.e., for each convex function ϕ(x) : R → R and each ξ ∈ L(Ft ), if ϕ(ξ ) ∈ L(Ft ), then we have g g Es,t ϕ(ξ ) ≥ ϕ Es,t [ξ ] a.s.; g
g
(ii) ∀(ξ , a, b) ∈ L (FT ) × R × R, E,T [aξ + b] ≥ aE,T [ξ ] + b, and, for any y ∈ R, g(t, y, ) = , dP × dt-a.s.;
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g
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g
(iii) ∀(ξ , a, b) ∈ L (Ft ) × R × R, Es,t [aξ + b] ≥ aEs,t [ξ ] + b a.s.; (iv) g is independent of y and super-homogeneous with respect to z. Similarly, Theorem . can be rewritten as follows. g
Corollary . Suppose that Es,t [·], ≤ s ≤ t ≤ T is a g-evaluation, then the following four statements are equivalent: g (i) the n-dimensional (n > ) Jensen inequality for g-evaluation Es,t [·] holds in general, i.e., for each convex function ϕ : Rn → R and ξi ∈ L(Ft ) (i = , , . . . , n), if ϕ(ξ , ξ , . . . , ξn ) ∈ L(Ft ), then we have g g g g Es,t ϕ(ξ , ξ , . . . , ξn ) ≥ ϕ Es,t [ξ ], Es,t [ξ ], . . . , Es,t [ξn ] a.s.; g
(ii) E,T is linear in L (FT ) and, for any y ∈ R, g(t, y, ) = , dP × dt-a.s.; g (iii) Es,t is linear in L (Ft ); (iv) for each (y, z) ∈ R × Rd , g(t, y, z) = g(t, z) = αt · z, dP × dt-a.s., where α is a Rd -valued progressively measurable process. Proof of Corollary . From Proposition . and Theorem ., 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) ∈ L (FT ) × [, T] × R, by (ii), we know that for each A ∈ Ft , g g E,T A (X + k) = E,T [A X + A k – k] + k g = E,T A X + AC (–k) + k g g = E,t Et,T A X + AC (–k) + k g g = E,t A Et,T [X] + AC (–k) + k g g = E,t A Et,T [X] + AC (–k) + k g g = E,t A Et,T [X] + k . Thus g
g
Et,T [X + k] = Et,T [X] + k
a.s. E
g
(.) [λ·]
g
λ λ For each λ = , define Et,T [·] := t,Tλ , ∀t ∈ [, T]. It is easy to check that Et,T [·] and Et,T [·] are two F -expectations in L (FT ) (the notion of F -expectation can be seen in Coquet et g λ al. []). If λ > , for each ξ ∈ L (FT ), E,T [ξ ] ≥ E,T [ξ ]. In a similar manner to Lemma . in Coquet et al. [], we can obtain g
λ Et,T [ξ ] ≥ Et,T [ξ ] a.s., ∀t ∈ [, T].
(.) g
λ [ξ ] ≤ E,T [ξ ]. In a similar manner to Lemma . in Coquet If λ < , for each ξ ∈ L (FT ), E,T et al. [] again, we have g
λ Et,T [ξ ] ≤ Et,T [ξ ] a.s., ∀t ∈ [, T].
(.)
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From (.) and (.), we have, for any (ξ , λ) ∈ L (FT ) × R, g
g
Et,T [λξ ] ≥ λEt,T [ξ ] a.s., ∀t ∈ [, T].
(.)
From (.) and (.), we have, for any (ξ , a, b) ∈ L (FT ) × R × R, g
g
Et,T [aξ + b] ≥ aEt,T [ξ ] + b a.s., ∀t ∈ [, T]. Since, for any y ∈ R, g(t, y, ) = , dP × dt-a.s., we have g
g
g
g
Es,t [aξ + b] = Es,T [aξ + b] ≥ aEs,T [ξ ] + b = aEs,t [ξ ] + b a.s., ∀(ξ , a, b) ∈ L (Ft ) × R × R. Therefore, (iii) holds true. The proof of Corollary . is complete.
Proof of Corollary . From Proposition . and Theorem ., 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 ., we g know that for each sequence {Xn }∞ n= ⊂ L (FT ) such that Xn (ω) ↓ for all ω, E,T [Xn ] ↓ . By the well-known Daniell-Stone theorem (cf., e.g., Yan [], Theorem .., p.), there exists a unique probability measure Pα defined on (, FT ) such that g
E,T [ξ ] = EPα [ξ ],
∀ξ ∈ L (FT )
(.)
T T α = exp( αt · dBt – |αt | dt). holds. Indeed, from (iv), we know that dP dP On the other hand, since, for any y ∈ R, g(t, y, ) = , dP × dt-a.s., we can obtain g
g
Es,t [ξ ] = Es,T [ξ ] a.s., ∀ξ ∈ L (Ft ).
(.)
It follows from (.) and (.) that g
Es,t [ξ ] = EPα [ξ |Fs ] a.s., ∀ξ ∈ L (Ft ). g
Therefore, Es,t is linear in L (Ft ). The proof of Corollary . is complete.
From Corollary ., we can immediately obtain the following. g
Theorem . Suppose that Es,t [·], ≤ s ≤ t ≤ T is a g-evaluation, then the following two statements are equivalent: g (i) Es,t is linear in L(Ft ); (ii) there exists a unique probability measure Pα defined on (, FT ) such that, for any ξ ∈ L(Ft ), g
Es,t [ξ ] = EPα [ξ |Fs ] a.s. The following result can be seen as an extension of Theorem .. g
Theorem . Suppose that Es,t [·], ≤ s ≤ t ≤ T is a g-evaluation, then the following two statements are equivalent:
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g
(i) Es,t is sublinear in L(Ft ), i.e., g g (f ) Es,t [λX] = λEs,t [X] a.s., for any X ∈ L(Ft ) and λ ≥ ; g g (g) Es,t [X + Y ] ≤ Es,t [X] + Es,t [Y ] a.s., for any (X, Y ) ∈ L(Ft ) × L(Ft ); g (h) Es,t [μ] = μ a.s., for any μ ∈ R; (ii) for any ξ ∈ L(Ft ), g
Es,t [ξ ] = sup EQθ [ξ |Fs ] a.s., Qθ ∈
where is a set of probability measures on (, FT ) and defined by
g := Qθ : EQθ [ξ ] ≤ E,T [ξ ], ∀ξ ∈ L(FT ) . Proof It is obvious that (ii) implies (i). In the following, we prove that (i) implies (ii). Suppose (i) holds. Since E,T [·] is a sublinear expectation in L(FT ), by Lemma . in Peng [], we know that there exists a family of linear expectations {Eθ : θ ∈ } on (, FT ) such that, for any ξ ∈ L(FT ), g
E,T [ξ ] = sup Eθ [ξ ].
(.)
θ∈
On the other hand, by Proposition ., we know that for each sequence {Xn }∞ n= ⊂ L(FT ) g such that Xn (ω) ↓ for all ω, E,T [Xn ] ↓ . By the well-known Daniell-Stone theorem, we can deduce that for each θ ∈ and ξ ∈ L(FT ), there exists a unique probability measure Qθ defined on (, FT ) such that Eθ [ξ ] = EQθ [ξ ].
(.)
It follows from (.) and (.) that, for any ξ ∈ L(FT ), g
E,T [ξ ] = sup EQθ [ξ ].
(.)
Qθ ∈
Let be a set of probability measures on (, FT ) defined by T dPα T = exp αt · dBt – |αt | dt , := Pα : α ∈ , dP
g
where g := {(αt )t∈[,T] : α is Rd -valued, progressively measurable and, for any (y, z) ∈ R × Rd , αt · z ≤ g(t, y, z), dP × dt-a.s.}. In order to prove (ii), now we prove that = . For any α ∈ g , we define g α (t, y, z) := αt · z, ∀t ∈ [, T], (y, z) ∈ R × Rd . Then, for any ξ ∈ L(FT ), by the well-known Girsanov theorem, we can deduce that gα
E,T [ξ ] = EPα [ξ ]. Since, for any (y, z) ∈ R × Rd , αt · z = g α (t, y, z) ≤ g(t, y, z), dP × dt-a.s., it follows from the gα g well-known comparison theorem for BSDEs that EPα [ξ ] = E,T [ξ ] ≤ E,T [ξ ]. Hence ⊆ . g Next let us prove that ⊆ . For each Qθ ∈ , since EQθ [·] ≤ E,T [·], ∀ξ , η ∈ L (FT ), we have g
EQθ [ξ + η] – EQθ [η] = EQθ [ξ ] ≤ E,T [ξ ].
(.)
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Denote g β (t, y, z) := β(t)|z|, ∀t ∈ [, T], (y, z) ∈ R × Rd . From Lemmas . and . and applying the well-known comparison theorem for BSDEs again, we have g
E,T [ξ ] = Eg [ξ ] ≤ Eg β [ξ ].
(.)
From (.) and (.), we can deduce that EQθ [ξ + η] – EQθ [η] ≤ Eg β [ξ ]. Then, in a similar manner to Theorem . in Coquet et al. [], we know that there exists a unique function g θ defined on × [, T] × R × Rd satisfying the following three conditions: (H.) g θ (t, y, ) = , dP × dt-a.s., ∀y ∈ R; (H.) |g θ (t, y , z ) – g θ (t, y , z )| ≤ β(t)|z – z |, ∀(y , z ), (y , z ) ∈ R × Rd , where β(t) is T a non-negative deterministic function satisfying that β (t) dt < ∞; (H.) Eg θ [ξ |Ft ] = EQθ [ξ |Ft ] a.s., ∀ξ ∈ L (FT ). It follows from the linearity of (Eg θ [·|Ft ])t∈[,T] and Theorem . that g θ is linear with respect to z. Therefore, there exists a Rd -valued progressively measurable process (θt )t∈[,T] such that g θ (t, y, z) = θt · z, dP × dt-a.s., ∀(y, z) ∈ R × Rd . In view of Qθ ∈ and (H.), g we have for each ξ ∈ L (FT ), Eg θ [ξ ] = EQθ [ξ ] ≤ E,T [ξ ]. Then in a similar manner to Lemma . in Coquet et al. [] and by Lemma ., we can obtain g θ (t, y, z) = θt · z ≤ g(t, y, z), dP × dt-a.s., ∀(y, z) ∈ R × Rd . For θ , we define the probability measure Pθ satT T θ isfying dP = exp( θt · dBt – |θt | dt), then Pθ ∈ and EPθ [ξ ] = Eg θ [ξ ] = EQθ [ξ ], dP ∀ξ ∈ L (FT ). Hence, Qθ = Pθ ∈ . Thus, ⊆ . Therefore, we have = . g Finally, we prove that, for any s, t ∈ [, T] satisfying s ≤ t and ξ ∈ L(Ft ), Es,t [ξ ] = supQθ ∈ EQθ [ξ |Fs ] a.s. It follows from (H.), the well-known comparison theorem for BSDEs, and Proposition . that g
Es,t [ξ ] ≥ Eg θ [ξ |Fs ] = EQθ [ξ |Fs ] a.s., ∀ξ ∈ L(Ft ). Hence, for any s, t ∈ [, T] satisfying s ≤ t and ξ ∈ L(Ft ), g
Es,t [ξ ] ≥ sup EQθ [ξ |Fs ] a.s.
(.)
Qθ ∈
On the other hand, by Lemmas ., ., and ., we can deduce that g is independent of y and positively homogeneous, sub-additive with respect to z. For any ξ ∈ L(FT ), let (Ytξ , Ztξ )t∈[,T] denote the solution of the following BSDE: yt = ξ +
T
g(s, zs ) ds – t
T
zs · dBs ,
∀t ∈ [, T].
t
By a measurable selection theorem (cf., e.g., El Karoui and Quenez [], p.), we can deduce that there exists a progressively measurable process α ξ ∈ g such that g t, Ztξ = αtξ · Ztξ ,
dP × dt-a.s.
(.) g
g
From (.) and applying the well-known Girsanov theorem, we have Es,t [ξ ] = Es,T [ξ ] = EPαξ [ξ |Fs ] a.s. Hence, for any ξ ∈ L(Ft ), g
Es,t [ξ ] ≤ sup EPα [ξ |Fs ] = sup EQθ [ξ |Fs ] a.s. Pα ∈
Qθ ∈
(.)
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It follows from (.) and (.) that g
Es,t [ξ ] = sup EQθ [ξ |Fs ] a.s., ∀ξ ∈ L(Ft ). Qθ ∈
The proof of Theorem . 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 . Suppose that the function g satisfies (B.) and (B.). Let g satisfy the following conditions: (i) for any y ≥ , y ≥ , and (z , z ) ∈ Rd × Rd , g(t, y + y , z + z ) ≤ g(t, y , z ) + g(t, y , z ),
dP × dt-a.s.;
(ii) for any λ ≥ , y ≥ , and z ∈ Rd , g(t, λy, λz) ≤ λg(t, y, z),
dP × dt-a.s.,
g
then Es,t [·] satisfies the following conditions: g g g (j) Es,t [ξ + η] ≤ Es,t [ξ ] + Es,t [η] a.s., for any (ξ , η) ∈ L+ (Ft ) × L+ (Ft ); g g (k) Es,t [λξ ] = λEs,t [ξ ] a.s., for any ξ ∈ L+ (Ft ) and λ ≥ . The key idea of the proof of Lemma . is the well-known comparison theorem for BSDEs. The proof is very similar to that of Proposition . in Jia []. So we omit it. Applying Lemma . and Theorems . and ., we immediately have the following Hölder inequality and Minkowski inequality for g-evaluations. Theorem . Let g satisfy the conditions of Lemma ., then, for any X, Y ∈ L(Ft ) and |X|p , |Y |q ∈ L(Ft ) (p, q > and /p + /q = ), we have g g g Es,t |XY | ≤ Es,t |X|p p Es,t |Y |q q
a.s.
Theorem . Let g satisfy the conditions of Lemma ., then, for any X, Y ∈ L(Ft ), and |X|p , |Y |p ∈ L(Ft ) (p > ), we have
g g g Es,t |X + Y |p p ≤ Es,t |X|p p + Es,t |Y |p p
a.s.
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors read and approved the final manuscript. Author details 1 School of Statistics, Qufu Normal University, Qufu, 273165, People’s Republic of China. 2 School of Mathematics, Chongqing University of Technology, Chongqing, 400054, People’s Republic of China.
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Acknowledgements The authors would like to thank the anonymous referees for their careful reading of this paper, correction of errors, and valuable suggestions. The work of Zhaojun Zong, Feng Hu and Chuancun Yin is supported by the National Natural Science Foundation of China (Nos. 11301295 and 11171179), the Doctoral Program Foundation of Ministry of Education of China (Nos. 20123705120005 and 20133705110002), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province of China and the Program for Scientific Research Innovation Team in Applied Probability and Statistics of Qufu Normal University (No. 0230518). The work of Helin Wu is Supported by the Scientific and Technological Research Program of Chongqing Municipal Education Commission (No. KJ1400922). Received: 4 November 2014 Accepted: 24 April 2015 References 1. Peng, SG: Dynamical evaluations. C. R. Acad. Sci. Paris, Ser. I 339, 585-589 (2004) 2. 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