Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 683047, 8 pages
http://dx.doi.org/10.1155/2013/683047
Research Article
Jensen’s Inequality for Generalized Peng’s 𝑔-Expectations and
Its Applications
Zhaojun Zong
School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
Correspondence should be addressed to Zhaojun Zong; zongzj001@163.com
Received 12 November 2012; Accepted 7 January 2013
Academic Editor: Xinguang Zhang
Copyright © 2013 Zhaojun Zong. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study Jensen’s inequality for generalized Peng’s 𝑔-expectations and give four equivalent conditions on Jensen’s inequality
for generalized Peng’s 𝑔-expectations without the assumption that the generator 𝑔 is continuous with respect to t. This result
includes and extends some existing results. Furthermore, we give some applications of Jensen’s inequality for generalized Peng’s
𝑔-expectations.
1. Introduction
By Pardoux and Peng [1], we know that there exists a
unique adapted and square integrable solution to a backward
stochastic differential equation (BSDE for short) of the type
𝑇
𝑇
𝑡
𝑡
𝑦𝑡 = 𝜉 + ∫ 𝑔 (𝑠, 𝑦𝑠 , 𝑧𝑠 ) 𝑑𝑠 − ∫ 𝑧𝑠 ⋅ 𝑑𝑊𝑠 ,
𝑡 ∈ [0, 𝑇] , (1)
provided that the function 𝑔 is Lipschitz in both variables
𝑦 and 𝑧, and 𝜉 and (𝑔(𝑡, 0, 0))𝑡∈[0,𝑇] are square integrable.
𝑔 is said to be the generator of BSDE (1). We denote the
unique adapted and square integrable solution of BSDE (1)
(𝑇,𝑔,𝜉) (𝑇,𝑔,𝜉)
, 𝑧𝑡
)𝑡∈[0,𝑇] .
by (𝑦𝑡
Based on such a BSDE, Peng [2] introduced the notion
of 𝑔-expectation. He proved that the 𝑔-expectation preserves
many of properties of the classical mathematical expectation,
but not the linearity property, and thus the 𝑔-expectation
is a type of nonlinear mathematical expectation. Indeed, 𝑔expectation is a kind of nonlinear expectation, which can
be considered as a nonlinear extension of the well-known
Girsanov transformations. The original motivation for studying 𝑔-expectation comes from the theory of expected utility.
Since the notion of 𝑔-expectation was introduced, many
properties of 𝑔-expectation have been investigated by many
researchers. In 1997, Peng [3] introduced the notions of
conditional 𝑔-expectation and 𝑔-martingale. Later, Briand et
al. [4] studied Jensen’s inequality for 𝑔-expectations and gave
a counter example and a proposition to indicate that even
for a linear function, Jensen’s inequality might fail for some
𝑔-expectations. This yields a natural question: under which
conditions on 𝑔 in the 𝑔-expectation does Jensen’s inequality
hold for any convex function? Under the assumptions that 𝑔
does not depend on 𝑦 and is convex, Chen et al. [5, 6] studied
Jensen’s inequality for 𝑔-expectations and gave a necessary
and sufficient condition on 𝑔 under which Jensen’s inequality
holds for convex functions. Provided that 𝑔 only does not
depend on 𝑦, Jiang [7] gave another necessary and sufficient
condition on 𝑔 under which Jensen’s inequality holds for
convex functions. It was an improved result in comparison
with the result that Chen et al. yielded. Later, this result was
improved by Hu [8] and Jiang [9] showing that, in fact, 𝑔 must
be independent of 𝑦. But these results need the assumption
that the generator 𝑔 is continuous with respect to 𝑡.
In this paper, without the assumption that the generator
𝑔 is continuous with respect to 𝑡, we study Jensen’s inequality
for generalized Peng’s 𝑔-expectations and give four equivalent
conditions on Jensen’s inequality for generalized Peng’s 𝑔expectations, which generalize the known results on Jensen’s
inequality for 𝑔-expectations in Chen et al. [5, 6], Jiang [7,
9], and Hu [8]. Furthermore, we give some applications of
Jensen’s inequality for generalized Peng’s 𝑔-expectations.
This paper is organized as follows: in Section 2, we
introduce some notations, assumptions, notions, and lemmas
2
Abstract and Applied Analysis
which will be useful in this paper; in Section 3, we give our
main results including the proofs and applications.
2. Preliminaries
Definition 3 (generalized Peng’s 𝑔-expectation [11]). Suppose
𝑔 satisfies (A.1) and (A.2). For any 𝜉 ∈ L(Ω, F𝑇 , 𝑃), let
(𝑇,𝑔,𝜉) (𝑇,𝑔,𝜉)
, 𝑧𝑡
)𝑡∈[0,𝑇] be the solution of BSDE (1). Consider
(𝑦𝑡
the mapping E𝑔 [⋅] : L(Ω, F𝑇 , 𝑃) 󳨃→ 𝑅, denoted by E𝑔 [𝜉] =
(𝑇,𝑔,𝜉)
Firstly, let us list some notations, assumptions, notions,
lemmas, and propositions that are used in this paper. Let
(Ω, F, 𝑃) be a probability space and let (𝑊𝑡 )𝑡≥0 be a 𝑑dimensional standard Brownian motion with respect to
filtration (F𝑡 )𝑡≥0 generated by Brownian motion and all 𝑃null subsets, that is,
F𝑡 = 𝜎 {𝑊𝑠 ; 𝑠 ≤ 𝑡} ∨ N,
. One calls E𝑔 [𝜉] the generalized Peng’s 𝑔-expectation
𝑦0
of 𝜉.
Definition 4 (generalized Peng’s conditional 𝑔-expectation
[11]). Suppose 𝑔 satisfies (A.1) and (A.2). The generalized
Peng’s conditional 𝑔-expectation of 𝜉 with respect to F𝑡 is
defined by
(2)
where N is the set of all 𝑃-null subsets. Fix a real number
𝑇 > 0. For any positive integer 𝑛 and 𝑧 ∈ 𝑅𝑛 , |𝑧| denotes its
Euclidean norm.
We define the following usual spaces of processes (random variables):
(i) Consider 𝐿𝑝 (Ω, F𝑇 , 𝑃) = {𝜉 : 𝜉 is F𝑇 -measurable
random variable such that 𝐸[|𝜉|𝑝 ] < ∞, 𝑝 ≥ 1};
(ii) Consider L(Ω, F𝑇 , 𝑃) = ⋃𝑝>1 𝐿𝑝 (Ω, F𝑇 , 𝑃);
(𝑇,𝑔,𝜉)
E𝑔 [𝜉 | F𝑡 ] = 𝑦𝑡
𝑝
(iv) Consider SF (0, 𝑇; 𝑃; 𝑅) = ⋃𝑝>1 SF (0, 𝑇; 𝑃; 𝑅);
𝑝
(v) Consider LF (0, 𝑇; 𝑃; 𝑅𝑛 ) = {𝑉 : 𝑉 is a progressively
measurable process with
∞, 𝑝 ≥ 1};
𝑇
𝐸[(∫0 |𝑉𝑠 |2 𝑑𝑠)
𝑝/2
]
<
E𝑔 [1𝐴𝜉] = E𝑔 [1𝐴 𝜂] ,
(A.2) 𝑃-a.s., ∀(𝑡, 𝑦) ∈ [0, 𝑇] × 𝑅, 𝑔(𝑡, 𝑦, 0) ≡ 0.
∀𝐴 ∈ F𝑡 .
(4)
Proposition 6 (see [11]). Suppose 𝑔 satisfies (A.1) and (A.2). If
𝑔 does not depend on 𝑦, that is, 𝑔(𝜔, 𝑡, 𝑧) : Ω×[0, 𝑇]×𝑅𝑑 󳨃→ 𝑅,
then
E𝑔 [𝜉 + 𝜂 | F𝑡 ] = E𝑔 [𝜉 | F𝑡 ] + 𝜂,
∀𝜉 ∈ L (Ω, F𝑇 , 𝑃) ,
∀𝜂 ∈ L (Ω, F𝑡 , 𝑃) .
(5)
𝑝
(A.1) there exists a constant 𝜇 > 0, such that 𝑃-a.s., we have:
∀𝑡 ∈ [0, 𝑇], ∀𝑦1 , 𝑦2 ∈ 𝑅, 𝑧1 , 𝑧2 ∈ 𝑅𝑑 , |𝑔(𝑡, 𝑦1 , 𝑧1 ) −
𝑔(𝑡, 𝑦2 , 𝑧2 )| ≤ 𝜇(|𝑦1 − 𝑦2 | + |𝑧1 − 𝑧2 |);
(3)
Proposition 5 (see [11]). Consider E𝑔 [𝜉 | F𝑡 ] is the unique
random variable 𝜂 in L(Ω, F𝑡 , 𝑃) such that
(vi) Consider LF (0, 𝑇; 𝑃; 𝑅𝑛 ) = ⋃𝑝>1 𝐿 F (0, 𝑇; 𝑃; 𝑅𝑛 ).
Suppose the generator 𝑔(𝜔, 𝑡, 𝑦, 𝑧) : Ω × [0, 𝑇] × 𝑅 × 𝑅𝑑 󳨃→
𝑅 satisfies the following assumptions:
𝑡 ∈ [0, 𝑇] .
Then, let us list some basic properties of generalized
Peng’s 𝑔-expectation.
𝑝
(iii) Consider SF (0, 𝑇; 𝑃; 𝑅) = {𝑉 : 𝑉 is a continuous
process with 𝐸[sup0≤𝑡≤𝑇 |𝑉𝑡 |𝑝 ] < ∞, 𝑝 ≥ 1};
,
Proposition 7 (see [11]). Suppose 𝑔 satisfies (A.1) and (A.2).
For 𝜉, 𝜂𝑛 ∈ 𝐿𝑝 (Ω, F𝑇 , 𝑃), where 𝑛 = 1, 2, . . . and 𝑝 > 1, if
𝐸[|𝜉 − 𝜂𝑛 |𝑝 | F𝑡 ] → 0, a.s., 𝑡 ∈ [0, 𝑇], then
lim E𝑔 [𝜂𝑛 | F𝑡 ] = E𝑔 [𝜉 | F𝑡 ] ,
𝑛→∞
a.s., 𝑡 ∈ [0, 𝑇] .
(6)
The following lemma is a special case of Theorem 4.2 in
Briand et al. [10].
Applying Proposition 7, one can immediately obtain the following.
Lemma 1. Suppose 𝑔 satisfies (A.1) and (A.2). Then for each
given 𝜉 ∈ 𝐿𝑝 (Ω, F𝑇 , 𝑃), where 1 < 𝑝 < 2, the BSDE (1)
(𝑇,𝑔,𝜉) (𝑇,𝑔,𝜉)
has a unique pair of adapted processes (𝑦𝑡
, 𝑧𝑡
)𝑡∈[0,𝑇] ∈
𝑝
𝑝
𝑑
SF (0, 𝑇; 𝑃; 𝑅) × 𝑙F (0, 𝑇; 𝑃; 𝑅 ).
Remark 8. (i) For any 𝜉 ∈ L(Ω, F𝑇 , 𝑃), let 𝜉𝑛 = (𝜉∧𝑛)∨(−𝑛),
𝑛 = 1, 2, . . ., then lim𝑛 → ∞ E𝑔 [𝜉𝑛 | F𝑡 ] = E𝑔 [𝜉 | F𝑡 ], a.s.,
∀𝑡 ∈ [0, 𝑇].
(ii) For any 𝜉𝑛 ∈ L(Ω, F𝑇 , 𝑃), if lim𝑛 → ∞ 𝜉𝑛 = 𝜉 a.s. and
|𝜉𝑛 | ≤ 𝜂 a.s. with 𝜂 ∈ L(Ω, F𝑇 , 𝑃), then lim𝑛 → ∞ E𝑔 [𝜉𝑛 |
F𝑡 ] = E𝑔 [𝜉 | F𝑡 ], a.s., ∀𝑡 ∈ [0, 𝑇].
From Lemma 1, we have the following.
Remark 2. Suppose 𝑔 satisfies (A.1) and (A.2). Then for each
given 𝜉 ∈ L(Ω, F𝑇 , 𝑃), the BSDE (1) has a unique pair of
(𝑇,𝑔,𝜉) (𝑇,𝑔,𝜉)
adapted processes (𝑦𝑡
, 𝑧𝑡
)𝑡∈[0,𝑇] ∈ SF (0, 𝑇; 𝑃; 𝑅) ×
𝑑
LF (0, 𝑇; 𝑃; 𝑅 ).
Now, we introduce the notions of generalized Peng’s
𝑔-expectation and generalized conditional Peng’s 𝑔expectation.
Lemma 9. Suppose 𝑔 satisfies (A.1) and (A.2). Let {𝐴 𝑖 }𝑚
𝑖=1 be
a F𝑡 -measurable partition of Ω (i.e., 𝐴 𝑖 ∈ F𝑡 , 𝐴 𝑖 ⋂ 𝐴 𝑗 = 0
if 𝑖 ≠ 𝑗 and ⋃𝑚
𝑖=1 𝐴 𝑖 = Ω), where 𝑡 ≤ 𝑇. Then for each 𝑋𝑖 ∈
L(Ω, F𝑇 , 𝑃), 𝑖 = 1, . . . , 𝑚, one has
𝑚
𝑚
𝑖=1
𝑖=1
∑1𝐴 𝑖 E𝑔 [𝑋𝑖 | F𝑡 ] = E𝑔 [∑1𝐴 𝑖 𝑋𝑖 | F𝑡 ]
a.s.
(7)
Abstract and Applied Analysis
3
1, 2, . . . , 𝑚). From Lemma 9 and (i), we deduce that for each
𝑋 ∈ L(Ω, F𝑇 , 𝑃),
Proof. We consider the following BSDEs:
E𝑔 [𝑋𝑖 | F𝑡 ]
𝑚
𝑇
𝑖=1
= 𝑋𝑖 + ∫ 𝑔 (𝑠, E𝑔 [𝑋𝑖 | F𝑠 ] , 𝑧𝑠(𝑇,𝑔,𝑋𝑖 ) ) 𝑑𝑠
−∫
𝑡
𝑧𝑠(𝑇,𝑔,𝑋𝑖 )
⋅ 𝑑𝑊𝑠 ,
𝑖=1
(8)
𝑡
𝑇
𝑚
E𝑔 [𝑋 + ∑𝜆 𝑖 1𝐴 𝑖 | F𝑡 ] = E𝑔 [∑1𝐴 𝑖 (𝑋 + 𝜆 𝑖 ) | F𝑡 ]
𝑚
= ∑1𝐴 𝑖 E𝑔 [𝑋 + 𝜆 𝑖 | F𝑡 ]
𝑖=1
𝑖 = 1, . . . , 𝑚,
𝑚
= ∑1𝐴 𝑖 (E𝑔 [𝑋 | F𝑡 ] + 𝜆 𝑖 )
𝑚
𝑖=1
E𝑔 [∑1𝐴 𝑖 𝑋𝑖 | F𝑡 ]
𝑚
𝑖=1
𝑚
𝑇
𝑚
𝑖=1
𝑡
𝑖=1
= E𝑔 [𝑋 | F𝑡 ] + ∑𝜆 𝑖 1𝐴 𝑖
= ∑1𝐴 𝑖 𝑋𝑖 + ∫ 𝑔 (𝑠, E𝑔 [∑1𝐴 𝑖 𝑋𝑖 | F𝑠 ] ,
(11)
(9)
(𝑇,𝑔,∑𝑚
𝑖=1 1𝐴 𝑖 𝑋𝑖 )
𝑧𝑠
) 𝑑𝑠
−∫
𝑇
𝑡
In other words, for any 𝑋 ∈ L(Ω, F𝑇 , 𝑃) and any simple
function 𝜂 ∈ L(Ω, F𝑡 , 𝑃),
E𝑔 [𝑋 + 𝜂 | F𝑡 ] = E𝑔 [𝑋 | F𝑡 ] + 𝜂
(𝑇,𝑔,∑𝑚
𝑖=1 1𝐴 𝑖 𝑋𝑖 )
𝑧𝑠
a.s.
𝑖=1
⋅ 𝑑𝑊𝑠 .
a.s.
(12)
Let
𝑛2𝑛 −1
| F𝑠 ], 𝑧𝑠(𝑇,𝑔,𝑋𝑖 ) ) =
By the fact that ∑𝑚
𝑖=1 1𝐴 𝑖 𝑔(𝑠, E𝑔 [𝑋𝑖
𝑚
𝑚
(𝑇,𝑔,𝑋𝑖 )
𝑔(𝑠, ∑𝑖=1 1𝐴 𝑖 E𝑔 [𝑋𝑖 | F𝑠 ], ∑𝑖=1 1𝐴 𝑖 𝑧𝑠
), 𝑡 ≤ 𝑠 ≤ 𝑇 and
𝜂𝑛 := ∑
𝑖=0
𝑖
𝑛
1 𝑛
+ 𝑛1{𝜂≥𝑛}
2𝑛 {(𝑖/2 )≤𝜂<((𝑖+1)/2 )}
𝑛2𝑛 −1
+ ∑
from (8), we have
𝑖=0
𝑚
+ (−𝑛) 1{𝜂<−𝑛} ,
∑1𝐴 𝑖 E𝑔 [𝑋𝑖 | F𝑡 ]
(13)
−𝑖
𝑛
𝑛
1
2𝑛 {−((𝑖+1)/2 ) ≤𝜂<−(𝑖/2 )}
𝑛 = 1, 2, . . . .
Obviously, for each 𝑛, 𝜂𝑛 is a simple function in L(Ω, F𝑡 , 𝑃).
From (12), we have
𝑖=1
𝑚
= ∑1𝐴 𝑖 𝑋𝑖
E𝑔 [𝑋 + 𝜂𝑛 | F𝑡 ] = E𝑔 [𝑋 | F𝑡 ] + 𝜂𝑛
𝑖=1
𝑇
𝑚
𝑚
𝑡
𝑖=1
𝑖=1
+ ∫ 𝑔 (𝑠, ∑1𝐴 𝑖 E𝑔 [𝑋𝑖 | F𝑠 ] , ∑1𝐴 𝑖 𝑧𝑠(𝑇,𝑔,𝑋𝑖 ) ) 𝑑𝑠
𝑇 𝑚
− ∫ ∑1𝐴 𝑖 𝑧𝑠(𝑇,𝑔,𝑋𝑖 ) ⋅ 𝑑𝑊𝑠 .
a.s.
(14)
On the other hand, lim𝑛 → ∞ (𝑋 + 𝜂𝑛 ) = 𝑋 + 𝜂, |𝑋 + 𝜂𝑛 | ≤
|𝑋| + |𝜂|. Thus, from Remark 8 (ii), it follows that (ii) is true.
The proof of Proposition 10 is complete.
3. Main Results and Applications
𝑡 𝑖=1
(10)
Comparing this with (9), it follows that ∑𝑚
𝑖=1 1𝐴 𝑖 E𝑔 [𝑋𝑖 |
1
𝑋
|
F
]
a.s.
The
proof
of Lemma 9 is
F𝑡 ] = E𝑔 [∑𝑚
𝑡
𝑖=1 𝐴 𝑖 𝑖
complete.
Proposition 10. Suppose 𝑔 satisfies (A.1) and (A.2). Then the
following two statements are equivalent:
(i) consider ∀(𝑋, 𝑘) ∈ L(Ω, F𝑇 , 𝑃)×𝑅, E𝑔 [𝑋+𝑘 | F𝑡 ] =
E𝑔 [𝑋 | F𝑡 ] + 𝑘 a.s.,
(ii) consider ∀(𝑋, 𝜂) ∈ L(Ω, F𝑇 , 𝑃) × L(Ω, F𝑡 , 𝑃),
E𝑔 [𝑋 + 𝜂 | F𝑡 ] = E𝑔 [𝑋 | F𝑡 ] + 𝜂 a.s.
Proof. It is obvious that (ii) implies (i). We only need to
prove that (i) implies (ii). Suppose (i) holds. Let {𝐴 𝑖 }𝑚
𝑖=1
be a F𝑡 -measurable partition of Ω and let 𝜆 𝑖 ∈ 𝑅 (𝑖 =
Definition 11. Let 𝑔: Ω × [0, 𝑇] × 𝑅 × 𝑅𝑑 󳨃→ 𝑅. The function
𝑔 is said to be superhomogeneous if for each (𝑦, 𝑧) ∈ 𝑅 × 𝑅𝑑
and any real number 𝜆, then 𝑔(𝑡, 𝜆𝑦, 𝜆𝑧) ≥ 𝜆𝑔(𝑡, 𝑦, 𝑧), 𝑑𝑃×𝑑𝑡
a.s. The function 𝑔 is said to be positively homogeneous if
for each (𝑦, 𝑧) ∈ 𝑅 × 𝑅𝑑 and any real number 𝜆 ≥ 0, then
𝑔(𝑡, 𝜆𝑦, 𝜆𝑧) = 𝜆𝑔(𝑡, 𝑦, 𝑧), 𝑑𝑃 × 𝑑𝑡 a.s.
Before we give our main results, let us see an example.
Example 12. Fix 𝑇 = 1 and 𝑑 = 1. Let 𝜉 = 𝑓(𝑊1 ), where
𝑓(𝑥) = exp((𝑥2 /2𝑝1 ) − 𝑥)1(𝑥≥𝑝1 ) , 1 < 𝑝1 < 2.
Obviously, 𝑓 is an increasing function. We can easily get
∞
𝑥2
1 −(1/2)𝑥2
󵄨 󵄨𝑝
𝐸 [󵄨󵄨󵄨𝜉󵄨󵄨󵄨 1 ] = ∫ exp ( − 𝑝1 𝑥)
𝑑𝑥
e
√2𝜋
2
𝑝1
=
2
1
e−𝑝1 < ∞,
√2𝜋𝑝1
󵄨 󵄨𝑝
𝐸 [󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ] = ∞,
Hence, 𝜉 ∈ L(Ω, F1 , 𝑃), but 𝜉 ∉ 𝐿2 (Ω, F1 , 𝑃).
∀𝑝 > 𝑝1 .
(15)
4
Abstract and Applied Analysis
Let 𝜉𝑛 = 𝜉 ∧ 𝑛, 𝑛 = 1, 2, . . .. Clearly, for each 𝑛, 𝜉𝑛 ∈
𝐿 (Ω, F, 𝑃). For simplicity, we will write E𝜇 [⋅] ≡ E𝑔 [⋅] for
𝑔 = 𝜇|𝑧|. From Theorem 1 in Chen and Kulperger’s [12], we
2
know that E𝜇 [𝜉𝑛 ] = 𝐸𝑄[𝜉𝑛 ], where 𝑑𝑄/𝑑𝑃 = e−(1/2)𝜇 +𝜇𝑊1 .
By Remark 8(i), we have E𝜇 [𝜉𝑛 ] → E𝜇 [𝜉], as 𝑛 → ∞.
On the other hand, applying Hölder’s inequality and noting
2
2 2
that 𝐸[e−(1/2)𝜇 +𝜇𝑊1 ] = 1 and 𝐸[e−(1/2)𝜇 𝑞 +𝜇𝑞𝑊1 ] = 1, we
obtain
2
1/𝑞
𝑞
𝑑𝑄
󵄨 󵄨𝑝 1/𝑝1
𝐸𝑄 [𝜉] ≤ (𝐸[󵄨󵄨󵄨𝜉󵄨󵄨󵄨 1 ]) (𝐸 [(
) ])
𝑑𝑃
(16)
󵄨 󵄨𝑝 1/𝑝1
≤ e(1/2)(𝑞−1)𝜇 (𝐸 [󵄨󵄨󵄨𝜉󵄨󵄨󵄨 1 ])
< ∞,
2
where (1/𝑝1 ) + (1/𝑞) = 1. It then follows from the monotonic
convergence theorem that
𝐸𝑄 [𝜉𝑛 ] 󳨀→ 𝐸𝑄 [𝜉] ,
as 𝑛 󳨀→ ∞.
Proof. (i)⇒(ii) is obvious.
(ii)⇒(iii): let 𝜂 = 𝜉 + 𝑏. By (ii), we have
𝜑 (E𝜇 [𝜉]) = 𝜑 (𝐸𝑄 [𝜉]) ≤ 𝐸𝑄 [𝜑 (𝜉)] = E𝜇 [𝜑 (𝜉)] .
E𝑔 [1𝐴 (𝑋 + 𝑘)] = E𝑔 [1𝐴 𝑋 + 1𝐴 𝑘 − 𝑘] + 𝑘
= E𝑔 [1𝐴 𝑋 + 1𝐴𝐶 (−𝑘)] + 𝑘
= E𝑔 [E𝑔 [1𝐴 𝑋 + 1𝐴𝐶 (−𝑘) | F𝑡 ]] + 𝑘
= E𝑔 [1𝐴 E𝑔 [𝑋 | F𝑡 ] + 1𝐴𝐶 (−𝑘)] + 𝑘
= E𝑔 [1𝐴 E𝑔 [𝑋 | F𝑡 ] + 1𝐴𝐶 (−𝑘) + 𝑘]
The following theorem will answer this question.
Theorem 13. Let 𝑔 satisfy (A.1) and (A.2). Then the following
four statements are equivalent.
(i) Jensen’s inequality for generalized Peng’s 𝑔-expectation
E𝑔 [⋅ | F𝑡 ] holds in general, that is, for each convex
function 𝜑(𝑥) : 𝑅 󳨃→ 𝑅 and each 𝜉 ∈ L(Ω, F𝑇 , 𝑃), if
𝜑(𝜉) ∈ L(Ω, F𝑇 , 𝑃), then one has
E𝑔 [𝜑 (𝜉) F𝑡 ] ≥ 𝜑 (E𝑔 [𝜉 | F𝑡 ])
a.s.;
= E𝑔 [1𝐴 (E𝑔 [𝑋 | F𝑡 ] + 𝑘)] .
(25)
Thus,
E𝑔 [𝑋 + 𝑘 | F𝑡 ] = E𝑔 [𝑋 | F𝑡 ] + 𝑘
(21)
2
(ii) consider ∀(𝜉, 𝑎, 𝑏) ∈ 𝐿 (Ω, F𝑇 , 𝑃) × 𝑅 × 𝑅, E𝑔 [𝑎𝜉 +
𝑏] ≥ 𝑎E𝑔 [𝜉] + 𝑏;
(iii) consider ∀(𝜉, 𝑎, 𝑏) ∈ 𝐿2 (Ω, F𝑇 , 𝑃) × 𝑅 × 𝑅, E𝑔 [𝑎𝜉 + 𝑏 |
F𝑡 ] ≥ 𝑎E𝑔 [𝜉 | F𝑡 ] + 𝑏 a.s.;
(iv) consider 𝑔 is independent of 𝑦, superhomogeneous, and
positively homogeneous with respect to 𝑧.
a.s., ∀𝑡 ∈ [0, 𝑇] .
(26)
On the other hand, for each 𝜆 ≠ 0, define
(20)
This problem yields a natural question: in general, under
which conditions on 𝑔 do generalized Peng’s 𝑔-expectations
satisfy Jensen’s inequality for convex functions?
(24)
For each (𝑋, 𝑡, 𝑘) ∈ 𝐿2 (Ω, F𝑇 , 𝑃) × [0, 𝑇] × 𝑅, by (24), we
know that for each 𝐴 ∈ F𝑡 ,
Let 𝜑(𝑥) = (𝑥 − 𝑘)+ , where 𝑘 ∈ 𝑅. Obviously, 𝜑(𝑥) is
a convex and increasing function. From this, we know that
𝜑 ∘ 𝑓 is an increasing function. In a similar manner of the
above, we can deduce that
From (18), (19), and the classical Jensen’s inequality, we
have
(23)
E𝑔 [𝜉 + 𝑏] = E𝑔 [𝜉] + 𝑏.
(17)
(19)
E𝑔 [𝜉] + 𝑏 ≥ E𝑔 [𝜉 + 𝑏] .
Thus, for each (𝜉, 𝑏) ∈ 𝐿2 (Ω, F𝑇 , 𝑃) × 𝑅,
(18)
E𝜇 [𝜑 (𝜉)] = 𝐸𝑄 [𝜑 (𝜉)] .
(22)
That is,
Thus
E𝜇 [𝜉] = 𝐸𝑄 [𝜉] .
E𝑔 [𝜂 − 𝑏] ≥ E𝑔 [𝜂] − 𝑏,
E𝜆 [⋅ | F𝑡 ] =
E𝑔 [𝜆⋅ | F𝑡 ]
𝜆
,
∀𝑡 ∈ [0, 𝑇] .
(27)
It is easy to check that E𝑔 [⋅ | F𝑡 ] and E𝜆 [⋅ | F𝑡 ] are two Fexpectations on 𝐿2 (Ω, F𝑇 , 𝑃) (the notion of F-expectation
can be seen in [13]). From (ii), we have if 𝜆 > 0, for each
𝜉 ∈ 𝐿2 (Ω, F𝑇 , 𝑃)
E𝜆 [𝜉] ≥ E𝑔 [𝜉] .
(28)
Hence, by Lemma 4.5 in [13], we have
E𝜆 [𝜉 | F𝑡 ] ≥ E𝑔 [𝜉 | F𝑡 ]
a.s., ∀𝑡 ∈ [0, 𝑇] .
(29)
Similarly, if 𝜆 < 0, for each 𝜉 ∈ 𝐿2 (Ω, F𝑇 , 𝑃)
E𝜆 [𝜉] ≤ E𝑔 [𝜉] .
(30)
Hence, by Lemma 4.5 in [13] again, we have
E𝜆 [𝜉 | F𝑡 ] ≤ E𝑔 [𝜉 | F𝑡 ]
a.s., ∀𝑡 ∈ [0, 𝑇] .
(31)
Thus from (29) and (31), we have ∀(𝜉, 𝜆) ∈ 𝐿2 (Ω, F𝑇 , 𝑃) × 𝑅,
E𝑔 [𝜆𝜉 | F𝑡 ] ≥ 𝜆E𝑔 [𝜉 | F𝑡 ]
a.s., ∀𝑡 ∈ [0, 𝑇] .
(32)
Abstract and Applied Analysis
5
Thus, (𝜆𝑌𝑠𝑡,𝑧 )𝑠∈[𝑡,𝑇] is an E𝑔 -submartingale. From the decomposition theorem of E𝑔 -supermartingale (see [15]), it follows
that there exists an increasing process (𝐴 𝑠 )𝑠∈[𝑡,𝑇] such that
From (26) and (32), we have
∀ (𝜉, 𝑎, 𝑏) ∈ 𝐿2 (Ω, F𝑇 , 𝑃) × 𝑅 × 𝑅,
E𝑔 [𝑎𝜉 + 𝑏 | F𝑡 ] ≥ 𝑎E𝑔 [𝜉 | F𝑡 ] + 𝑏
a.s.,
(33)
∀𝑡 ∈ [0, 𝑇] .
(iii)⇒(iv): Firstly, we prove that 𝑔 is independent of 𝑦.
From (iii), we can obtain that for each (𝜉, 𝑦) ∈ 𝐿2 (Ω, F𝑇 , 𝑃)×
𝑅,
E𝑔 [𝜉 − 𝑦 | F𝑡 ] = E𝑔 [𝜉 | F𝑡 ] − 𝑦,
𝑠
𝑌𝑠𝑡,𝑦,𝑧 = 𝑦 − ∫ 𝑔 (𝑟, 𝑌𝑟𝑡,𝑦,𝑧 , 𝑧) 𝑑𝑟 + 𝑧 ⋅ (𝑊𝑠 − 𝑊𝑡 ) .
𝑡
(35)
From (34), we have
𝑌𝑟𝑡,𝑦,𝑧
−𝑦=
𝑠
𝑡
𝑡
(43)
𝑠 ∈ [𝑡, 𝑇] .
𝑠
𝑠
This with 𝜆𝑌𝑠𝑡,𝑧 = − ∫𝑡 𝜆𝑔(𝑟, 𝑧)d𝑟 + ∫𝑡 𝜆𝑧 ⋅ 𝑑𝑊𝑟 yields 𝑍𝑟 ≡ 𝜆𝑧
and
a.s., ∀𝑡 ∈ [0, 𝑇] .
(34)
For each (𝑡, 𝑦, 𝑧) ∈ [0, 𝑇] × 𝑅 × 𝑅𝑑 , let 𝑌⋅𝑡,𝑦,𝑧 be the solution of
the following SDE defined on [𝑡, 𝑇]:
𝑠
𝜆𝑌𝑠𝑡,𝑧 = − ∫ 𝑔 (𝑟, 𝑍𝑟 ) 𝑑𝑟 + 𝐴 𝑠 − 𝐴 𝑡 + ∫ 𝑍𝑟 ⋅ 𝑑𝑊𝑟 ,
𝜆𝑔 (𝑟, 𝑧) ≤ 𝑔 (𝑟, 𝜆𝑧) ,
𝑑𝑃 × 𝑑𝑡 a.s.
(44)
At last, we prove that 𝑔 is positively homogeneous with
respect to 𝑧. From (iii), we can obtain that for each fixed 𝜆 > 0
and 𝜉 ∈ 𝐿2 (Ω, F𝑇 , 𝑃),
1
E [𝜆𝜉 | F𝑡 ] ≤ E𝑔 [𝜉 | F𝑡 ] ,
𝜆 𝑔
a.s., ∀𝑡 ∈ [0, 𝑇] ,
(45)
a.s., ∀𝑡 ∈ [0, 𝑇] .
(46)
a.s., ∀𝑡 ∈ [0, 𝑇] .
(47)
that is,
E𝑔 [𝑌𝑠𝑡,𝑦,𝑧
| F𝑟 ] − 𝑦 =
E𝑔 [𝑌𝑠𝑡,𝑦,𝑧
− 𝑦 | F𝑟 ] ,
𝑡 ≤ 𝑟 ≤ 𝑠 ≤ 𝑇.
(36)
Let 𝑌𝑠 = 𝑌𝑠𝑡,𝑦,𝑧 − 𝑦, 𝑠 ∈ [𝑡, 𝑇] and 𝑍 be the corresponding part
of Itô’s integrand. It then follows that
𝑠
𝑡
𝑠
𝑠
𝑡
𝑡
(37)
= − ∫ 𝑔 (𝑟, 𝑌𝑟 , 𝑍𝑟 ) d𝑟 + ∫ 𝑍𝑟 ⋅ 𝑑𝑊𝑟 .
E𝑔 [𝜆𝜉 | F𝑡 ] = 𝜆E𝑔 [𝜉 | F𝑡 ] ,
Obviously, if 𝜆 = 0, (47) still holds. Thus, for each 𝜆 ≥ 0,
a.s., ∀𝑡 ∈ [0, 𝑇] .
(38)
Then, we can apply Lemma 4.4 in Peng [14] to obtain that for
each (𝑦, 𝑧) ∈ 𝑅 × 𝑅𝑑 ,
(48)
For each (𝑡, 𝑧) ∈ [0, 𝑇] × 𝑅𝑑 , let 𝑌⋅𝑡,𝑧 be the solution of SDE
(34). From (48), for each 𝜆 ≥ 0, we have
E𝑔 [𝜆𝑌𝑠𝑡,𝑧 | F𝑟 ] = 𝜆E𝑔 [𝑌𝑠𝑡,𝑧 | F𝑟 ] = 𝜆𝑌𝑟𝑡,𝑧 ,
Thus, 𝑍𝑟 ≡ 𝑧 and
𝑔 (𝑟, 𝑌𝑟 , 𝑧) = 𝑔 (𝑟, 𝑌𝑟𝑡,𝑦,𝑧 − 𝑦, 𝑧) = 𝑔 (𝑟, 𝑌𝑟𝑡,𝑦,𝑧 , 𝑧) .
Thus, we have
E𝑔 [𝜆𝜉 | F𝑡 ] = 𝜆E𝑔 [𝜉 | F𝑡 ] ,
𝑠
𝑌𝑠 = − ∫ 𝑔 (𝑟, 𝑌𝑟𝑡,𝑦,𝑧 , 𝑧) 𝑑𝑟 + ∫ 𝑧 ⋅ 𝑑𝑊𝑟
𝑡
E𝑔 [𝜆𝜉 | F𝑡 ] ≤ 𝜆E𝑔 [𝜉 | F𝑡 ] ,
𝑡 ≤ 𝑟 ≤ 𝑠 ≤ 𝑇.
(49)
This implies that there exists a process 𝑍⋅𝑡,𝑧,𝜆 such that
𝑠
𝑠
𝑡
𝑡
𝜆𝑌𝑠𝑡,𝑧 = − ∫ 𝑔 (𝑟, 𝑍𝑟𝑡,𝑧,𝜆 ) 𝑑𝑟 + ∫ 𝑍𝑟𝑡,𝑧,𝜆 ⋅ 𝑑𝑊𝑟 ,
𝑠 ∈ [𝑡, 𝑇] .
(39)
(50)
Namely, 𝑔 is independent of 𝑦.
Now we prove that 𝑔 is superhomogeneous with respect
to 𝑧. From (iii), we can obtain that for each (𝜉, 𝜆) ∈
𝐿2 (Ω, F𝑇 , 𝑃) × 𝑅,
Comparing this with 𝜆𝑌𝑠𝑡,𝑧 = − ∫𝑡 𝜆𝑔(𝑟, 𝑧)𝑑𝑟 + ∫𝑡 𝜆𝑧 ⋅ 𝑑𝑊𝑟 , it
follows that 𝑍𝑟𝑡,𝑧,𝜆 ≡ 𝜆𝑧 and
𝑔 (𝑡, 𝑦, 𝑧) = 𝑔 (𝑡, 0, 𝑧) ,
𝜆E𝑔 [𝜉 | F𝑡 ] ≤ E𝑔 [𝜆𝜉 | F𝑡 ] ,
𝑑𝑃 × 𝑑𝑡 a.s.
a.s., ∀𝑡 ∈ [0, 𝑇] .
(40)
For each (𝑡, 𝑧) ∈ [0, 𝑇] × 𝑅𝑑 , let 𝑌⋅𝑡,𝑧 be the solution of the
following SDE defined on [𝑡, 𝑇]:
𝑠
𝑌𝑠𝑡,𝑧 = − ∫ 𝑔 (𝑟, 𝑧) 𝑑𝑟 + 𝑧 ⋅ (𝑊𝑠 − 𝑊𝑡 ) .
𝑡
(41)
𝜆𝑔 (𝑟, 𝑧) = 𝑔 (𝑟, 𝜆𝑧) ,
𝑠
𝑑𝑃 × 𝑑𝑡 a.s.
𝑡 ≤ 𝑟 ≤ 𝑠 ≤ 𝑇.
(42)
(51)
(iv)⇒(iii): By comparison theorem (for example, we can
see [3]), it is easy to obtain (iii).
(iii)⇒(i): Suppose (iii) holds. From (iii) and by Remark 8
(i), we have
∀ (𝑋, 𝑘) ∈ L (Ω, F𝑇 , 𝑃) × 𝑅,
E𝑔 [𝑋 + 𝑘 | F𝑡 ] = E𝑔 [𝑋 | F𝑡 ] + 𝑘
From (40), we have
E𝑔 [𝜆𝑌𝑠𝑡,𝑧 | F𝑟 ] ≥ 𝜆E𝑔 [𝑌𝑠𝑡,𝑧 | F𝑟 ] = 𝜆𝑌𝑟𝑡,𝑧 ,
𝑠
a.s.,
∀ (𝑋, 𝜆) ∈ L (Ω, F𝑇 , 𝑃) × 𝑅,
E𝑔 [𝜆𝑋 | F𝑡 ] ≥ 𝜆E𝑔 [𝑋 | F𝑡 ]
a.s.
(52)
(53)
6
Abstract and Applied Analysis
From (53), we can deduce that for each bounded variable 𝜁 ∈
F𝑡 ,
∀𝑋 ∈ L (Ω, F𝑇 , 𝑃) ,
E𝑔 [𝜁𝑋 | F𝑡 ] ≥ 𝜁E𝑔 [𝑋 | F𝑡 ]
a.s.
(54)
In fact, let {𝐴 𝑖 }𝑚
𝑖=1 be a F𝑡 -measurable partition of Ω and let
𝜆 𝑖 ∈ 𝑅 (𝑖 = 1, 2, . . . , 𝑚). By (53), we have
𝑚
𝑚
𝑖=1
𝑖=1
E𝑔 [∑𝜆 𝑖 1𝐴 𝑖 𝑋 | F𝑡 ] = ∑1𝐴 𝑖 E𝑔 [𝜆 𝑖 𝑋 | F𝑡 ]
(55)
𝑚
≥ ∑1𝐴 𝑖 𝜆 𝑖 E𝑔 [𝑋 | F𝑡 ]
a.s.
𝑖=1
In other words, for each 𝑋 ∈ L(Ω, F𝑇 , 𝑃) and each simple
function 𝜁 ∈ L(Ω, F𝑡 , 𝑃),
E𝑔 [𝜁𝑋 | F𝑡 ] ≥ 𝜁E𝑔 [𝑋 | F𝑡 ]
a.s.
(56)
Thus, thanks to Remark 8(ii), it follows that (54) is true.
The main idea of the following proof is derived from [7].
Given 𝜉 ∈ L(Ω, F𝑇 , 𝑃) and convex function 𝜑 such that
𝜑(𝜉) ∈ L(Ω, F𝑇 , 𝑃), we set 𝜂𝑡 = 𝜑−󸀠 (E𝑔 [𝜉 | F𝑡 ]). Then 𝜂𝑡
is F𝑡 -measurable. Since 𝜑 is convex, we have
𝜑 (𝑥) − 𝜑 (𝑦) ≥ 𝜑−󸀠 (𝑦) (𝑥 − 𝑦) ,
∀𝑥, 𝑦 ∈ 𝑅.
(57)
Take 𝑥 = 𝜉, 𝑦 = E𝑔 [𝜉 | F𝑡 ]. Then we have
𝜑 (𝜉) − 𝜑 (E𝑔 [𝜉 | F𝑡 ]) ≥ 𝜂𝑡 (𝜉 − E𝑔 [𝜉 | F𝑡 ])
a.s. (58)
For each 𝑛 ∈ 𝑁, we define
󵄨
󵄨 󵄨 󵄨 󵄨
󵄨
Ω𝑡,𝑛 := {󵄨󵄨󵄨󵄨E𝑔 [𝜉 | F𝑡 ]󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝜂𝑡 󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝜑 (E𝑔 [𝜉 | F𝑡 ])󵄨󵄨󵄨󵄨 ≤ 𝑛} , (59)
so we have
Hence, we can deduce that for each 𝑛 ∈ 𝑁,
E𝑔 [1Ω𝑡,𝑛 𝜑 (𝜉) | F𝑡 ] ≥ 1Ω𝑡,𝑛 𝜑 (E𝑔 [𝜉 | F𝑡 ])
a.s.
Finally, thanks to Remark 8 (ii) again, we can get
E𝑔 [𝜑 (𝜉) | F𝑡 ] ≥ 𝜑 (E𝑔 [𝜉 | F𝑡 ])
a.s.
Example 14. Suppose 𝐻 is a bounded, convex, and closed
subset of 𝑅𝑑 and 𝐷 = the set of 𝑅𝑑 -valued continuous
processes (𝑣𝑡 )𝑡∈[0,𝑇] such that for each 𝑡, 𝑣𝑡 ∈ 𝐻 a.s.. Define
the probability measure 𝑄𝑣 by
𝑇
𝑇
2
𝑑𝑄𝑣
= e−(1/2) ∫0 |𝑣𝑠 | 𝑑𝑠+∫0 𝑣𝑠 ⋅𝑑𝑊𝑠 .
𝑑𝑃
Thus, for any convex function 𝜑,
𝑣∈𝐷
𝑣∈𝐷
whenever 𝜉, 𝜑(𝜉) ∈ L(Ω, F𝑇 , 𝑃).
Proof. Let 𝑔(𝑡, 𝑧) = ess sup𝑣∈𝐷𝑣𝑡 ⋅ 𝑧. Obviously, 𝑔(𝑡, 𝑧)
is superhomogeneous and positively homogeneous with
respect to 𝑧. and satisfies (A.1) and (A.2).
From El Karoui and Quenez [16], we have
ess sup𝑣∈𝐷𝐸𝑄𝑣 [𝜉 | F𝑡 ] = E𝑔 [𝜉 | F𝑡 ], a.s., ∀𝜉 ∈
𝐿2 (Ω, F𝑇 , 𝑃). Now we prove ess sup𝑣∈𝐷𝐸𝑄𝑣 [𝜉 | F𝑡 ] =
E𝑔 [𝜉 | F𝑡 ], a.s., ∀𝜉 ∈ L(Ω, F𝑇 , 𝑃). Indeed, for any
𝜉 ∈ L(Ω, F𝑇 , 𝑃), there exists 1 < 𝑝 < 2 such that
𝜉 ∈ 𝐿𝑝 (Ω, F𝑇 , 𝑃). Let 𝜉𝑛 = (𝜉 ∧ 𝑛) ∨ (−𝑛), 𝑛 = 1, 2, . . .. Clearly,
for each 𝑛, 𝜉𝑛 ∈ 𝐿2 (Ω, F𝑇 , 𝑃), then
a.s.
(68)
Since
ess sup 𝐸𝑄𝑣 [𝜉𝑛 | F𝑡 ]
𝑣∈𝐷
a.s.
= ess sup (𝐸𝑄𝑣 [𝜉𝑛 − 𝜉| F𝑡 ] + 𝐸𝑄𝑣 [𝜉| F𝑡 ])
By the definition of 1Ω𝑡,𝑛 , we know
𝑣∈𝐷
≤ ess sup𝐸𝑄𝑣 [𝜉𝑛 − 𝜉 | F𝑡 ] + ess sup𝐸𝑄𝑣 [𝜉 | F𝑡 ] ,
1Ω𝑡,𝑛 𝜑 (E𝑔 [𝜉 | F𝑡 ]) − 1Ω𝑡,𝑛 𝜂𝑡 E𝑔 [𝜉 | F𝑡 ] ∈ L (Ω, F𝑡 , 𝑃) .
(61)
Thus, in view of (52) and from Proposition 10, we can get
E𝑔 [1Ω𝑡,𝑛 𝜑 (𝜉) | F𝑡 ] ≥ 1Ω𝑡,𝑛 𝜑 (E𝑔 [𝜉 | F𝑡 ])
𝑣∈𝐷
𝑣∈𝐷
(69)
we have
E𝑔 [𝜉𝑛 | F𝑡 ] − ess sup𝐸𝑄𝑣 [𝜉 | F𝑡 ]
𝑣∈𝐷
− 1Ω𝑡,𝑛 𝜂𝑡 E𝑔 [𝜉 | F𝑡 ]
(62)
a.s.
Moreover, from (54), considering that 1Ω𝑡,𝑛 𝜂𝑡 ∈ F𝑡 and is
bounded by 𝑛, we can get
E𝑔 [1Ω𝑡,𝑛 𝜂𝑡 𝜉 | F𝑡 ] ≥ 1Ω𝑡,𝑛 𝜂𝑡 E𝑔 [𝜉 | F𝑡 ]
(67)
a.s., ∀𝑡 ∈ [0, 𝑇] ,
𝑣∈𝐷
+ E𝑔 [1Ω𝑡,𝑛 𝜂𝑡 𝜉 | F𝑡 ]
(66)
𝜑 (ess sup𝐸𝑄𝑣 [𝜉 | F𝑡 ]) ≤ ess sup𝐸𝑄𝑣 [𝜑 (𝜉) | F𝑡 ] ,
ess sup𝐸𝑄𝑣 [𝜉𝑛 | F𝑡 ] = E𝑔 [𝜉𝑛 | F𝑡 ] ,
+1Ω𝑡,𝑛 𝜂𝑡 𝜉 | F𝑡 ]
(65)
Hence, Jensen’s inequality for E𝑔 [⋅ | F𝑡 ] holds in general. The
proof of Theorem 13 is complete.
E𝑔 [1Ω𝑡,𝑛 𝜑 (𝜉) | F𝑡 ]
≥ E𝑔 [1Ω𝑡,𝑛 𝜑 (E𝑔 [𝜉 | F𝑡 ]) − 1Ω𝑡,𝑛 𝜂𝑡 E𝑔 [𝜉 | F𝑡 ] (60)
(64)
a.s.
(63)
≤ ess sup𝐸𝑄𝑣 [𝜉𝑛 − 𝜉 | F𝑡 ] .
(70)
𝑣∈𝐷
With an approach similar to the one above, we can get easily
that
E𝑔 [𝜉𝑛 | F𝑡 ] − ess sup 𝐸𝑄𝑣 [𝜉 | F𝑡 ]
𝑣∈𝐷
≥ ess inf 𝐸𝑄𝑣 [𝜉𝑛 − 𝜉 | F𝑡 ] .
𝑣∈𝐷
(71)
Abstract and Applied Analysis
7
Definition 15. Suppose 𝑔 satisfies (A.1) and (A.2). A process
(𝑋𝑡 )𝑡∈[0,𝑇] satisfying that for each 𝑡, 𝑋𝑡 ∈ L(Ω, F𝑡 , 𝑃)
is called a generalized Peng’s 𝑔-martingale (resp., generalized Peng’s 𝑔-supermartingale, generalized Peng’s 𝑔submartingale), if for any 𝑡, 𝑠 satisfying 𝑡 ≤ 𝑠 ≤ 𝑇,
Combining (42) with (43), we have
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨
󵄨󵄨E𝑔 [𝜉𝑛 | F𝑡 ] − ess sup𝐸𝑄𝑣 [𝜉 | F𝑡 ]󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑣∈𝐷
󵄨
󵄨
󵄨󵄨
󵄨󵄨
≤ ( 󵄨󵄨󵄨󵄨ess inf 𝐸𝑄𝑣 [𝜉𝑛 − 𝜉| F𝑡 ]󵄨󵄨󵄨󵄨
󵄨 𝑣∈𝐷
󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨
∨ 󵄨󵄨󵄨󵄨ess sup𝐸𝑄𝑣 [𝜉𝑛 − 𝜉 | F𝑡 ]󵄨󵄨󵄨󵄨)
󵄨󵄨
󵄨󵄨
𝑣∈𝐷
󵄨
󵄨
≤ ess sup 𝐸𝑄𝑣 [󵄨󵄨󵄨𝜉𝑛 − 𝜉󵄨󵄨󵄨 | F𝑡 ] .
(72)
Hölder’s
𝑡
inequality
𝑡
−(1/2) ∫0 |𝑣𝑠 |2 𝑑𝑠+∫0 𝑣𝑠 ⋅𝑑𝑊𝑠
(e
that
𝑡
𝑡
−(1/2) ∫0 |𝑞𝑣𝑠 |2 𝑑𝑠+∫0 𝑞𝑣𝑠 ⋅𝑑𝑊𝑠
and
noting
)𝑡∈[0,𝑇]
and
)𝑡∈[0,𝑇] are both martingales with
(e
respect to (F𝑡 )𝑡∈[0,𝑇] , we can obtain
󵄨
󵄨
𝐸𝑄𝑣 [󵄨󵄨󵄨𝜉𝑛 − 𝜉󵄨󵄨󵄨 | F𝑡 ]
=
≤
󵄨
󵄨
𝐸 [󵄨󵄨󵄨𝜉𝑛 − 𝜉󵄨󵄨󵄨 (𝑑𝑄𝑣 /𝑑𝑃) | F𝑡 ]
𝐸 [(𝑑𝑄𝑣 /𝑑𝑃) | F𝑡 ]
1/𝑝
1/𝑞
𝑞
󵄨
󵄨𝑝
(𝐸 [󵄨󵄨󵄨𝜉𝑛 − 𝜉󵄨󵄨󵄨 | F𝑡 ]) (𝐸 [(𝑑𝑄𝑣 /𝑑𝑃) | F𝑡 ])
𝐸 [(𝑑𝑄𝑣 /𝑑𝑃) | F𝑡 ]
2
1/𝑝
󵄨
󵄨𝑝
≤ e(1/2)(𝑞−1)𝜇 𝑇 (𝐸 [󵄨󵄨󵄨𝜉𝑛 − 𝜉󵄨󵄨󵄨 | F𝑡 ]) ,
(73)
where (1/𝑝) + (1/𝑞) = 1. It then follows from Lebesgue’s
dominated convergence theorem that
󵄨
󵄨
ess sup 𝐸𝑄𝑣 [󵄨󵄨󵄨𝜉𝑛 − 𝜉󵄨󵄨󵄨 | F𝑡 ] 󳨀→ 0,
𝑣∈𝐷
as 𝑛 󳨀→ ∞.
(74)
Hence,
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨E𝑔 [𝜉𝑛 | F𝑡 ] − ess sup 𝐸𝑄𝑣 [𝜉 | F𝑡 ]󵄨󵄨󵄨 󳨀→ 0,
󵄨󵄨
󵄨󵄨
𝑣∈𝐷
󵄨
󵄨
as 𝑛 󳨀→ ∞.
(75)
On the other hand, from Remark 8(i), we have
E𝑔 [𝜉𝑛 | F𝑡 ] 󳨀→ E𝑔 [𝜉 | F𝑡 ] ,
as 𝑛 󳨀→ ∞.
(76)
Thus,
E𝑔 [𝜉 | F𝑡 ] = ess sup 𝐸𝑄𝑣 [𝜉 | F𝑡 ] ,
𝑣∈𝐷
(77)
a.s., ∀𝜉 ∈ L (Ω, F𝑇 , 𝑃) .
𝑣∈𝐷
𝑣∈𝐷
(79)
Theorem 16. Suppose 𝑔 is independent of 𝑦, superhomogeneous and positively homogeneous with respect to 𝑧 and
satisfies (A.1) and (A.2). If (𝑋𝑡 )𝑡∈[0,𝑇] is a generalized Peng’s
𝑔-martingale and 𝜑 is a convex function such that 𝜑(𝑋𝑡 ) ∈
L(Ω, F𝑡 , 𝑃), then (𝜑(𝑋𝑡 ))𝑡∈[0,𝑇] is a generalized Peng’s 𝑔submartingale.
Remark 17. Suppose 𝑔 is independent of 𝑦, superhomogeneous and positively homogeneous with respect to 𝑧 and
satisfies (A.1) and (A.2). Similarly, we can get the following.
(i) If (𝑋𝑡 )𝑡∈[0,𝑇] is a generalized Peng’s 𝑔-submartingale
and 𝜑 is a convex and increasing function such
that 𝜑(𝑋𝑡 ) ∈ L(Ω, F𝑡 , 𝑃), then (𝜑(𝑋𝑡 ))𝑡∈[0,𝑇] is a
generalized Peng’s 𝑔-submartingale.
(ii) If (𝑋𝑡 )𝑡∈[0,𝑇] is a generalized Peng’s 𝑔-supermartingale
and 𝜑 is a convex and decreasing function such
that 𝜑(𝑋𝑡 ) ∈ L(Ω, F𝑡 , 𝑃), then (𝜑(𝑋𝑡 ))𝑡∈[0,𝑇] is a
generalized Peng’s 𝑔-submartingale.
Example 18. (i) Let 𝑔 = 𝜇|𝑧| and 𝜑(𝑥) = (𝑥−𝑎)+ where 𝑎 ∈ 𝑅.
Obviously, 𝑔 satisfies the assumptions of Remark 17 and 𝜑 is
a convex and increasing function. By Remark 17 (i), we have
the following: suppose (𝑋𝑡 )𝑡∈[0,𝑇] is a E𝜇 -submartingale, then
((𝑋𝑡 − 𝑎)+ )𝑡∈[0,𝑇] is a E𝜇 -submartingale.
(ii) Let 𝑔 = 𝜇|𝑧| and 𝜑(𝑥) = (𝑥 − 𝑏)− where 𝑏 ∈ 𝑅.
With the similar argument, we have the following: suppose
(𝑌𝑡 )𝑡∈[0,𝑇] is a E𝜇 -supermartingale, then ((𝑌𝑡 − 𝑏)− )𝑡∈[0,𝑇] is a
E𝜇 -submartingale.
Acknowledgments
The author would like to thank the anonymous referees for
their careful reading of this paper and valuable suggestions.
The author thanks the partial support from the National
Natural Science Foundation of China (Grant no. 11171179),
the Doctoral Program Foundation of Ministry of Education
of China (Grant no. 20093705110002 and 20123705120005),
and the Natural Science Foundation of Shandong Province
of China (Grant no. ZR2012AQ009).
References
Applying Theorem 13, we have
𝜑 (ess sup𝐸𝑄𝑣 [𝜉 | F𝑡 ]) ≤ ess sup𝐸𝑄𝑣 [𝜑 (𝜉) | F𝑡 ] ,
(resp. ≤ 𝑋𝑡 , ≥ 𝑋𝑡 ) , a.s.
Applying Theorem 13, immediately we have the following.
𝑣∈𝐷
By
E𝑔 [𝑋𝑠 | F𝑡 ] = 𝑋𝑡
a.s.
(78)
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