A comonotonic theorem for backward stochastic differential equations in Lp and its applications
We study backward stochastic differential equations (BSDE) under weak assumptions on the data. We obtain a comonotonic theorem for BSDE in Lp; 1 < p ≤ 2: As applications of this theorem, we study the relation between Choquet expectations and minimax expectations and the relation between Choquet e...
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irk-123456789-1644122020-02-23T20:42:01Z A comonotonic theorem for backward stochastic differential equations in Lp and its applications Zong, Z.J. Статті We study backward stochastic differential equations (BSDE) under weak assumptions on the data. We obtain a comonotonic theorem for BSDE in Lp; 1 < p ≤ 2: As applications of this theorem, we study the relation between Choquet expectations and minimax expectations and the relation between Choquet expectations and generalized Peng’s g-expectations. These results generalize the well-known results of Chen et al. Дослiджено зворотнi стохастичнi диференцiальнi рiвняння при слабких припущеннях щодо вихiдних даних. Отримано теорему про комонотоннiсть для зворотних стохастичних диференцiальних рiвнянь у просторi Lp, 1 < p ≤ 2. Як застосування цiєї теореми, вивчено спiввiдношення мiж сподiваннями Шоке i мiнiмаксними сподiваннями та спiввiдношення мiж сподiваннями Шоке й узагальненими g-сподiваннями Пенга. Цi результати узагальнюють вiдомi результати Чена та iн. 2012 Article A comonotonic theorem for backward stochastic differential equations in Lp and its applications / Z.J. Zong // Український математичний журнал. — 2012. — Т. 64, № 6. — С. 752-765. — Бібліогр.: 18 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164412 519.21 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Zong, Z.J. A comonotonic theorem for backward stochastic differential equations in Lp and its applications Український математичний журнал |
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We study backward stochastic differential equations (BSDE) under weak assumptions on the data. We obtain a comonotonic theorem for BSDE in Lp; 1 < p ≤ 2: As applications of this theorem, we study the relation between Choquet expectations and minimax expectations and the relation between Choquet expectations and generalized Peng’s g-expectations. These results generalize the well-known results of Chen et al. |
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Zong, Z.J. |
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Zong, Z.J. |
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Zong, Z.J. |
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A comonotonic theorem for backward stochastic differential equations in Lp and its applications |
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A comonotonic theorem for backward stochastic differential equations in Lp and its applications |
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A comonotonic theorem for backward stochastic differential equations in Lp and its applications |
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A comonotonic theorem for backward stochastic differential equations in Lp and its applications |
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A comonotonic theorem for backward stochastic differential equations in Lp and its applications |
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comonotonic theorem for backward stochastic differential equations in lp and its applications |
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Інститут математики НАН України |
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2012 |
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A comonotonic theorem for backward stochastic differential equations in Lp and its applications / Z.J. Zong // Український математичний журнал. — 2012. — Т. 64, № 6. — С. 752-765. — Бібліогр.: 18 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT zongzj acomonotonictheoremforbackwardstochasticdifferentialequationsinlpanditsapplications AT zongzj comonotonictheoremforbackwardstochasticdifferentialequationsinlpanditsapplications |
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2025-07-14T16:58:41Z |
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2025-07-14T16:58:41Z |
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1837642360156061696 |
| fulltext |
UDC 519.21
Z.-J. Zong (School Math. Sci., Qufu Normal Univ., Shandong, China)
A COMONOTONIC THEOREM FOR BACKWARD STOCHASTIC
DIFFERENTIAL EQUATIONS IN Lp AND ITS APPLICATIONS *
ТЕОРЕМА ПРО КОМОНОТОННIСТЬ ДЛЯ ЗВОРОТНИХ СТОХАСТИЧНИХ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ У Lp ТА ЇЇ ЗАСТОСУВАННЯ
We study backward stochastic differential equations (BSDEs) under weak assumptions on the data. We obtain a comonotonic
theorem for BSDEs in Lp, 1 < p ≤ 2. As applications of this theorem, we study the relation between Choquet expectations
and minimax expectations and the relation between Choquet expectations and generalized Peng’s g-expectations. These
results generalize the known results of Chen et al.
Дослiджено зворотнi стохастичнi диференцiальнi рiвняння при слабких припущеннях щодо вихiдних даних. Отри-
мано теорему про комонотоннiсть для зворотних стохастичних диференцiальних рiвнянь у просторi Lp, 1 < p ≤ 2.
Як застосування цiєї теореми, вивчено спiввiдношення мiж сподiваннями Шоке i мiнiмаксними сподiваннями та
спiввiдношення мiж сподiваннями Шоке й узагальненими g-сподiваннями Пенга. Цi результати узагальнюють вiдомi
результати Чена та iн.
1. Introduction. By Pardoux and Peng [14], we know that there exists a unique adapted and square
integrable solution to a backward stochastic differential equation (BSDE for short) of type
yt = ξ +
T∫
t
g(s, ys, zs)ds−
T∫
t
zs · dWs, t ∈ [0, T ], (1.1)
providing that the function g is Lipschitz in both variables y and z, and that ξ and the pro-
cess (g(t, 0, 0))t∈[0,T ] are square integrable. We denote the unique solution of BSDE (1.1) by(
y
(T,g,ξ)
t , z
(T,g,ξ)
t
)
t∈[0,T ]
.
Since then, many researchers have been working on this subject and related properties of the so-
lutions of BSDEs, due to the connection of this subject with mathematical finance, stochastic control,
partial differential equation, stochastic game and stochastic geometry and mathematical economics;
for example, see References [2 – 5, 7 – 13, 15 – 18]. Among these results, the comparison theorem of
BSDEs with respect to y(T,g,ξ)t plays an important role.
An interesting study is to obtain a comparison result applicable to the second part of the z(T,g,ξ)t
of the solution
(
y
(T,g,ξ)
t , z
(T,g,ξ)
t
)
t∈[0,T ]
of BSDE (1.1). In fact, because z(T,g,ξ)t in BSDE (1.1) is a
speed (volatility in mathematical finance), it is not easy to make comparisons regarding z(T,g,ξ)t in
the same way as to make comparisons regarding y(T,g,ξ)t .
Chen et al. [4] studied the comonotonicity of z(T,g,ξ)t . That is, let
(
y
(T,g,ξ1)
t , z
(T,g,ξ1)
t
)
and(
y
(T,g,ξ2)
t , z
(T,g,ξ2)
t
)
be the solutions of BSDE (1.1) corresponding to terminal values ξ = ξ1 and
ξ = ξ2, respectively. A sufficient condition on ξ1 and ξ2 has been given, under which
z
(T,g,ξ1)
t � z(T,g,ξ2)t ≥ 0, dP × dt-a.s.
*This work was supported partially by the National Natural Science Foundation of China (No. 11171179) and the
Research Foundation for the Doctoral Program of Higher Education (No. 20093705110002).
c© Z.-J. ZONG, 2012
752 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
A COMONOTONIC THEOREM FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS. . . 753
Here for any z, x ∈ Rd, denote z � x = (z1x1, z2x2, . . . , zdxd), where zi and xi are the ith com-
ponents of z and x, i = 1, 2, . . . , d. Furthermore, z � x ≥ 0 means zixi ≥ 0, i = 1, 2, . . . , d. As
applications of this result, Chen et al. provide a sufficient condition of Choquet expectations be-
ing equal to minimax expectations in [5] and give a necessary and sufficient condition of Choquet
expectations being equal to g-expectations in [2].
In this paper, we investigate the comonotonicity of zt under weak assumptions on the data.
Furthermore, we give some applications of the comonotonic theorem. These results generalize the
known results of Chen et al. [2, 4, 5].
This paper is organized as follows. In Section 2, we give some notations, lemmas and notions
that are useful in this paper. In Section 3, we investigate the comonotonic theorem for BSDEs in
Lp. In Section 4, using the comonotonic theorem, we give some results such as the relation between
Choquet expectations and minimax expectations and the relation between Choquet expectations and
generalized Peng’s g-expectations.
2. Preliminaries. In this section, we shall present some notations, lemmas and notions that are
used in this paper.
Let (Ω,F , P ) be a probability space and (Wt)t≥0 be a d-dimensional standard Brownian motion
with respect to filtration (Ft)t≥0 generated by the Brownian motion and all P -null subsets, i. e.,
Ft = σ{Ws; s ≤ t} ∨ N ,
where N is the set of all P -null subsets. Fix a real number T > 0. We assume that FT = F .
Define
Lp(Ω,F , P ) := {ξ : ξ is F-measurable random variable such that E[|ξ|p] <∞, p ≥ 1},
L(Ω,F , P ) :=
⋃
p>1 L
p(Ω,F , P ),
SpT (R) :=
{
V : (Vt)t∈[0,T ] is (Ft)t∈[0,T ]-adapted continuous R-valued process with
E
[
sup0≤t≤T |Vt|p
]
<∞, p ≥ 1
}
,
ST (R) :=
⋃
p>1 S
p
T (R),
Lp(0, T ;P ;Rn) :=
{
V : (Vt)t∈[0,T ] is (Ft)t∈[0,T ]-adapted Rn-valued process with
E
[(∫ T
0
|Vs|2ds
)p/2 ]
<∞, p ≥ 1
}
,
L(0, T ;P ;Rn) :=
⋃
p>1 L
p(0, T ;P ;Rn).
Throughout this paper, we assume that 1 < p ≤ 2.
Suppose function g : Ω× [0, T ]×R×Rd 7−→ R satisfies the following conditions:
(H.1) g(·, 0, 0) ∈ Lp(0, T ;P ;R);
(H.1′) g(·, 0, 0) ∈ L(0, T ;P ;R);
(H.2) g satisfies a uniform Lipschitz condition, that is: there exists a constant µ > 0 such that for
any y1, y2 ∈ R, z1, z2 ∈ Rd, |g(t, y1, z1)− g(t, y2, z2)| ≤ µ(|y1 − y2|+ |z1 − z2|), t ∈ [0, T ];
(H.3) g(·, y, 0) = 0 ∀y ∈ R.
Lemma 2.1 (see Briand et al. [1]). Suppose g satisfies (H.1) and (H.2). Then for any ξ ∈
∈ Lp(Ω,F , P ), the BSDE
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
754 Z.-J. ZONG
yt = ξ +
T∫
t
g(s, ys, zs)ds−
T∫
t
zs · dWs, (2.1)
has a unique pair of adapted processes
(
y
(T,g,ξ)
t , z
(T,g,ξ)
t
)
t∈[0,T ]
∈ SpT (R)× Lp(0, T ;P ;Rd).
Remark 2.1. From Lemma 2.1, we have: suppose g satisfies (H.1′) and (H.2), then for each
given ξ ∈ L(Ω,F , P ), BSDE (2.1) has a unique pair of adapted processes
(
y
(T,g,ξ)
t , z
(T,g,ξ)
t
)
t∈[0,T ]
∈
∈ ST (R)× L(0, T ;P ;Rd).
We give the a priori estimate for BSDEs which is of standard type and taken from [1].
Lemma 2.2. Suppose g satisfies (H.1) and (H.2). For any ξ1, ξ2 ∈ Lp(Ω,F , P ), let
(
y
(T,g,ξ1)
t ,
z
(T,g,ξ1)
t
)
and
(
y
(T,g,ξ2)
t , z
(T,g,ξ2)
t
)
be the solutions of BSDE (2.1) corresponding to ξ = ξ1 and
ξ = ξ2, respectively. Then there exists a constant Cp > 0 depending only on p, T and Lipschitz
constant µ such that
E
[
sup
0≤t≤T
∣∣∣y(T,g,ξ1)t − y(T,g,ξ2)t
∣∣∣p]+ E
T∫
0
∣∣∣z(T,g,ξ1)s − z(T,g,ξ2)s
∣∣∣2 ds
p/2
≤ CpE [|ξ1 − ξ2|p] .
The following comparison theorem is very useful.
Lemma 2.3 (Comparison theorem, see Hu and Chen [10]). Suppose g and g satisfy (H.1) and
(H.2). For any ξ1, ξ2 ∈ Lp(Ω,F , P ), let
(
y
(T,g,ξ1)
t , z
(T,g,ξ1)
t
)
and
(
y
(T,g,ξ2)
t , z
(T,g,ξ2)
t
)
be the solu-
tions of the following two BSDEs:
y1t = ξ1 +
T∫
t
g
(
s, y1s , z
1
s
)
ds−
T∫
t
z1s · dWs, t ∈ [0, T ],
y2t = ξ2 +
T∫
t
g
(
s, y2s , z
2
s
)
ds−
T∫
t
z2s · dWs, t ∈ [0, T ].
If
ξ1 ≥ ξ2 a.e., ĝt = g(t, y, z)− g(t, y, z) ≥ 0 a.e.,
then for each t ∈ [0, T ],
y
(T,g,ξ1)
t ≥ y(T,g,ξ2)t a.e.
In the case, we have
y
(T,g,ξ2)
t = y
(T,g,ξ2)
t a.e. if and only if ξ1 = ξ2 a.e., ĝt = 0 a.e.
Definition 2.1 (Generalized Peng’s g-expectation, see [10]). Suppose g satisfies (H.2) and (H.3).
For any ξ ∈ L(Ω,F , P ), let
(
y
(T,g,ξ)
t , z
(T,g,ξ)
t
)
be the solution of BSDE (2.1) with terminal value
ξ. Consider the mapping Eg[·] : L(Ω,F , P ) 7−→ R, denoted by Eg[ξ] = y
(T,g,ξ)
0 . We call Eg[ξ] the
generalized Peng’s g-expectation of ξ.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
A COMONOTONIC THEOREM FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS. . . 755
Definition 2.2 (Generalized Peng’s g-expectation, see [10]). Suppose g satisfies (H.2) and (H.3).
The generalized Peng’s conditional g-expectation of ξ with respect to Ft is defined by
Eg[ξ|Ft] = y
(T,g,ξ)
t .
The generalized Peng’s conditional g-expectation has the following property.
Proposition 2.1 (see [10]). Eg[ξ|Ft] is the unique random variable η in L(Ω,Ft, P ) such that
Eg[1Aξ] = Eg[1Aη] ∀A ∈ Ft.
3. A comonotonic theorem for BSDEs in Lp. From now on, we further assume that the function
g is deterministic, i.e.,
g : [0, T ]×R×Rd 7−→ R.
Then (H.1) can be rewritten as follows:
T∫
0
|g(s, 0, 0)|2ds <∞. (H)
More specifically, we suppose that g1 and g2 satisfy the assumption (H) and (H.2). For any ξ1 and
ξ2 ∈ L(Ω,F , P ), let
(
y
(T,gi,ξi)
t , z
(T,gi,ξi)
t
)
be the solutions of the following BSDEs:
yit = ξi +
T∫
t
gi
(
s, yis, z
i
s
)
ds−
T∫
t
zis · dWs, t ∈ [0, T ], i = 1, 2. (3.1)
Now we consider the case where random variables ξ1 and ξ2 satisfy that there exist two functions
φ1 and φ2 such that ξ1 and ξ2 are of the form ξi = φi
(
Xi
T
)
, where (Xi
t) are the solutions of the
following SDEs, respectively,
dXi
s = bi
(
s,Xi
s
)
ds+ σi
(
s,Xi
s
)
· dWs,
Xi
0 = x, x ∈ R, i = 1, 2,
and bi and σi satisfy the following assumption for each i = 1, 2.
Assumption A. Let bi(t, x) : [0, T ] × R 7−→ R, σi(t, x) : [0, T ] × R 7−→ Rd be continuous in
(t, x) and uniformly Lipshictz continuous in x ∈ R, for each i = 1, 2.
Definition 3.1. The functions φ and ψ are said to be comonotonic, if both φ and ψ are of the
same monotonicity, that is, if φ is increasing (or decreasing), so is ψ.
The following theorem is called comonotonic theorem for BSDEs, which plays an important role
in our paper.
Theorem 3.1. Suppose that
(
y
(T,g1,ξ1)
t , z
(T,g1,ξ1)
t
)
and
(
y
(T,g2,ξ2)
t , z
(T,g2,ξ2)
t
)
are the solutions
of BSDE (3.1) corresponding to terminal values ξ1 = φ1(X
1
T ) and ξ2 = φ2(X
2
T ), respectively. If φ1
and φ2 are comonotonic and
σ1
(
t,X1
t
)
� σ2
(
t,X2
t
)
≥ 0, dP × dt-a.s.,
then
z
(T,g1,ξ1)
t � z(T,g2,ξ2)t ≥ 0, dP × dt-a.s.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
756 Z.-J. ZONG
Proof. If φ1(X1
T ) and φ2(X2
T ) ∈ L2(Ω,F , P ), Chen et al. have proved Theorem 3.1 holds in [5].
Otherwise, there exists 1 < p < 2, such that φ1(X1
T ) and φ2(X
2
T ) ∈ Lp(Ω,F , P ). Set ξni =
= φni (Xi
T ) = (φi(X
i
T ) ∧ n) ∨ (−n), i = 1, 2, then φni (Xi
T ) ∈ L2(Ω,F , P ) and both φni (Xi
T ) and
φi(X
i
T ) are of the same monotonicity, for each i = 1, 2. Let z
(T,g1,ξn1 )
t and z
(T,g2,ξn2 )
t be the solutions
of BSDE (3.1) corresponding to φn1 (X1
T ) and φn2 (X2
T ), by Chen et al. [5], we have
z
(T,g1,ξn1 )
t � z(T,g2,ξ
n
2 )
t ≥ 0, dP × dt-a.s. (3.2)
Applying Lemma 2.2, we can obtain z
(T,g1,ξn1 )
t → z
(T,g1,ξ1)
t and z
(T,g2,ξn2 )
t → z
(T,g2,ξ2)
t in Lp(0, T ;P ;
Rd) as n→∞. This with (3.2) implies that
z
(T,g1,ξ1)
t � z(T,g2,ξ2)t ≥ 0, dP × dt-a.s. (3.3)
Theorem 3.1 is proved.
Using Theorem 3.1, immediately, we can obtain the following theorem.
Theorem 3.2. Suppose that b and σ satisfy Assumption A. Let (Xs) be the solution of SDE
dXs = b(s,Xs)ds+ σ(s,Xs) · dWs, X0 = x, s ∈ [0, T ].
Assume φ is a function such that φ(XT ) ∈ L(Ω,F , P ) and g satisfies (H) and (H.2). Let(
y
(T,g,φ(XT ))
t , z
(T,g,φ(XT ))
t
)
be the solution of the BSDE
yt = φ(XT ) +
T∫
t
g(s, ys, zs)ds−
T∫
t
zs · dWs, t ∈ [0, T ].
(i) If φ is an increasing function, then
z
(T,g,φ(XT ))
t � σ(t,Xt) ≥ 0, dP × dt-a.s. (3.4)
(ii) If φ is a decreasing function, then
z
(T,g,φ(XT ))
t � σ(t,Xt) ≤ 0, dP × dt-a.s.. (3.5)
4. Some applications of the comonotonic theorem. 4.1. Additivity of generalized Peng’s
g-expectations. We know that if g(t, y, z) is nonlinear in (y, z), then Eg[·] is usually nonlinear on
L(Ω,F , P ). In this subsection, applying the comonotonic theorem, we give that for some special
random variables, Eg[·] still has the additivity property even when g is nonlinear.
Definition 4.1. (i) A function g(t, y, z) : [0, T ]×R×Rd 7−→ R is called positively additive, if
for any (y1, z1) and (y2, z2) ∈ R×Rd, then
g(t, y1 + y2, z1 + z2) = g(t, y1, z1) + g(t, y2, z2),
whenever y1y2 ≥ 0, z1 � z2 ≥ 0 ∀t ∈ [0, T ].
(ii) A function g(t, y, z) : [0, T ] × R × Rd 7−→ R is called semipositively additive, if for any
(y1, z1) and (y2, z2) ∈ R×Rd, then
g(t, y1 + y2, z1 + z2) = g(t, y1, z1) + g(t, y2, z2),
whenever z1 � z2 ≥ 0 ∀t ∈ [0, T ].
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
A COMONOTONIC THEOREM FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS. . . 757
Remark 4.1. (i) If g is a positively additive (or semipositively additive) function, it is easy to
check that g(t, 0, 0) = 0 ∀t ∈ [0, T ].
(ii) The following two functions are positively additive and semipositively additive, respectively,
g(t, y, z) = at|y|+
d∑
i=1
µit|zi|,
g(t, y, z) = bty +
d∑
i=1
νit |zi|,
where zi is the i th component of z.
Theorem 4.1. Suppose that φ1(X1
T ) and φ2(X2
T ) are the random variables defined in Theo-
rem 3.1 and that g satisfies (H.2) and (H.3).
(i) Suppose φ1 and φ2 are comonotonic with φ1(X
1
T ) ≥ 0, φ2(X
2
T ) ≥ 0 (or φ1(X1
T ) ≤ 0,
φ2(X
2
T ) ≤ 0). If
σ1
(
t,X1
t
)
� σ2
(
t,X2
t
)
≥ 0, dP × dt-a.s.,
and g is a positively additive function, then
Eg[φ1(X1
T ) + φ2(X
2
T )|Ft] = Eg[φ1(X1
T )|Ft] + Eg[φ2(X2
T )|Ft] a.e., t ∈ [0, T ].
In particular,
Eg[φ1(X1
T ) + φ2(X
2
T )] = Eg[φ1(X1
T )] + Eg[φ2(X2
T )].
(ii) If g is a semipositively additive function, then the assumptions φ1(X1
T ) ≥ 0, φ2(X
2
T ) ≥ 0
(or φ1(X1
T ) ≤ 0, φ2(X
2
T ) ≤ 0) in (i) can be dropped.
Proof. (i) For each i = 1, 2, let
(
y
(T,g,φi(X
i
T ))
t , z
(T,g,φi(X
i
T ))
t
)
be the solution of BSDEs
yit = φi(X
i
T ) +
T∫
t
g
(
s, yis, z
i
s
)
ds−
T∫
t
zis · dWs, t ∈ [0, T ].
Since φ1 and φ2 are comonotinic and
σ1
(
t,X1
t
)
� σ2
(
t,X2
t
)
≥ 0, dP × dt-a.s.,
by Theorem 3.1, we have
z
(T,g,φ1(X1
T ))
t � z(T,g,φ2(X
2
T ))
t ≥ 0, dP × dt-a.s. (4.1)
We next show
y
(T,g,φ1(X1
T ))
t y
(T,g,φ2(X2
T ))
t ≥ 0 a.e., t ∈ [0, T ].
Indeed, if φ1(X1
T ) ≥ 0, φ2(X
2
T ) ≥ 0, then applying Lemma 2.3,
y
(T,g,φ1(X1
T ))
t ≥ 0 a.e., y
(T,g,φ2(X2
T ))
t ≥ 0 a.e.
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758 Z.-J. ZONG
If φ1(X1
T ) ≤ 0, φ2(X
2
T ) ≤ 0, applying Lemma 2.3 again,
y
(T,g,φ1(X1
T ))
t ≤ 0 a.e., y
(T,g,φ2(X2
T ))
t ≤ 0 a.e.
Hence
y
(T,g,φ1(X1
T ))
t y
(T,g,φ2(X2
T ))
t ≥ 0 a.e., t ∈ [0, T ]. (4.2)
(4.1), (4.2) and the assumption that g(t, y, z) is a positively additive function imply
y
(T,g,φ1(X1
T ))
t + y
(T,g,φ2(X2
T ))
t = φ1(X
1
T ) + φ2(X
2
T )+
+
T∫
t
g
(
s, y
(T,g,φ1(X1
T ))
s + y
(T,g,φ2(X2
T ))
s , z
(T,g,φ1(X1
T ))
s + z
(T,g,φ2(X2
T ))
s
)
ds−
−
T∫
t
(
z
(T,g,φ1(X1
T ))
s + z
(T,g,φ2(X2
T ))
s
)
· dWs.
It follows that
Eg[φ1(X1
T ) + φ2(X
2
T )|Ft] = Eg[φ1(X1
T )|Ft] + Eg[φ2(X2
T )|Ft] a.e., t ∈ [0, T ].
Choose t = 0, then
Eg[φ1(X1
T ) + φ2(X
2
T )] = Eg[φ1(X1
T )] + Eg[φ2(X2
T )].
The proof of (i) is complete.
(ii) is obvious.
Theorem 4.1. is proved.
4.2. Choquet expectations, minimax expectations and generalized Peng’s g-expectations. 4.2.1.
Minimax expectations versus generalized Peng’s g-expectations. Let
P :=
{
Qθ :
dQθ
dP
= e−
1
2
∫ T
0 |θs|
2ds+
∫ T
0 θs·dWs , |θit| ≤ µ, dP × dt-a.s.
}
, (4.3)
where θit is the ith component of θt.
Referring to [3, 5, 8], for any ξ ∈ L(Ω,F , P ), we define E [ξ] = supQ∈P EQ[ξ], E [ξ] =
= infQ∈P EQ[ξ]. We further define conditional minimax expectations by
E [ξ|Ft] = ess sup
Q∈P
EQ[ξ|Ft], E [ξ|Ft] = ess inf
Q∈P
EQ[ξ|Ft].
Obviously, E [ξ|F0] = E [ξ], E [ξ|F0] = E [ξ], where ess is essential.
The following lemma shows that E [ξ], E [ξ], E [ξ|Ft] and E [ξ|Ft] are well-defined for all ξ ∈
∈ L(Ω,F , P ).
Lemma 4.1. For any ξ ∈ L(Ω,F , P ), then E [ξ|Ft] ∈ L(Ω,Ft, P ) and E [ξ|Ft] ∈ L(Ω,Ft, P ).
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A COMONOTONIC THEOREM FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS. . . 759
Proof. For any Q ∈ P, then there exists an adapted process {at} bounded by µ such that
dQa
dP
= e−
1
2
∫ T
0 |as|
2ds+
∫ T
0 as·dWs .
For any ξ ∈ L(Ω,F , P ), then there exists p > 1 such that ξ ∈ Lp(Ω,F , P ). By Hölder’s
inequality, we obtain
EQa [|ξ||Ft] =
E
[
|ξ|dQ
a
dP |Ft
]
E
[
dQa
dP |Ft
] ≤ (E[|ξ|p|Ft])1/p
(
E
[(
dQa
dP
)q
|Ft
])1/q
E
[
dQa
dP |Ft
] ,
where
1
p
+
1
q
= 1. Since
(
e−
1
2
∫ t
0 |as|
2ds+
∫ t
0 as·dWs
)
t∈[0,T ]
and
(
e−
1
2
∫ t
0 |qas|
2ds+
∫ t
0 qas·dWs
)
t∈[0,T ]
are
both martingales with respect to (Ft)t∈[0,T ], hence
(
E
[(
dQa
dP
)q
|Ft
])1/q
E
[
dQa
dP |Ft
] ≤ e
1
2
(q−1)dµ2T
(
e−
1
2
∫ t
0 |qas|
2ds+
∫ t
0 qas·dWs
)1/q
e−
1
2
∫ t
0 |as|2ds+
∫ t
0 as·dWs
≤ e
1
2
(q−1)dµ2T .
Thus
EQa [|ξ||Ft] ≤ e
1
2
(q−1)dµ2T (E[|ξ|p|Ft])1/p,
which implies E [ξ|Ft] ∈ L(Ω,Ft, P ) and E [ξ|Ft] ∈ L(Ω,Ft, P ).
In the following, for simplicity, we write in the sequel Eµ[·|Ft] ≡ Eg[·|Ft] for g = µ
∑d
i=1
|zi|
and E−µ[·|Ft] ≡ Eg[·|Ft] for g = −µ
∑d
i=1
|zi|, where zi is the ith component of z.
The following theorem shows a relation between minimax expectations and generalized Peng’s
g-expectations.
Theorem 4.2 (Martingale representation theorem for minimum and maximum expectations). If
ξ ∈ L(Ω,F , P ), then Eµ[ξ|Ft] = E [ξ|Ft], E−µ[ξ|Ft] = E [ξ|Ft]. In particular, Eµ[ξ] = E [ξ],
E−µ[ξ] = E [ξ].
By Lemma 2.3 and Girsanov’s theorem, it is easy to prove Theorem 4.2. The proof is very similar
to that of Theorem 2.2 in Chen and Epstein [3]. We omit it.
4.2.2. Choquet expectations versus minimax expectations.
Definition 4.2. A capacity is a real valued set function V : F 7→ [0, 1] satisfying:
(i) V (∅) = 0, V (Ω) = 1;
(ii) V (A) ≤ V (B) for any A ⊆ B.
The related Choquet expectation is denoted by
C[ξ] :=
0∫
−∞
(V (ξ ≥ t)− 1)dt+
∞∫
0
V (ξ ≥ t)dt.
From Definition 4.2, we may verify that C[·] satisfies (see [6]):
(1) monotonicity: If ξ ≥ η, then C[ξ] ≥ C[η],
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760 Z.-J. ZONG
(2) positive homogeneity: If λ ≥ 0, then C[λξ] = λC[ξ],
(3) translation invariance: If c ∈ R, then C[ξ + c] = C[ξ] + c.
Define
Vg(A) := Eg(1A) ∀A ∈ F .
It is easy to check that Vg is a capacity. The related Choquet expectation is denoted by
Cg[ξ] :=
0∫
−∞
(Vg(ξ ≥ t)− 1)dt+
∞∫
0
Vg(ξ ≥ t)dt.
We next show that for any ξ ∈ L(Ω,F , P ), Cg[ξ] <∞.
Lemma 4.2. Suppose that g satisfies (H.2) and (H.3), then Cg[ξ] <∞ for each ξ ∈ L(Ω,F , P ).
Proof. We set g(t, y, z) := −g(t, 1 − y,−z) for any (t, y, z) ∈ [0, T ] × R × Rd. Obviously g
satisfies (H.2) and (H.3) with the same Lipschitz constant as g. It is easy to check that Vg(A) =
= 1− Vg(AC) for each A ∈ F and
Cg[ξ] = Cg[ξ+] + Cg[−ξ−] = Cg[ξ+]− Cg[ξ−] ∀ξ ∈ L(Ω,F , P ).
For each ξ ∈ L(Ω,F , P ), there exists 1 < p < 2 such that ξ ∈ Lp(Ω,F , P ). From Lemma 2.2, for
fixed p′ ∈ (1, p), we have Eg[ξ] ≤ L(E[|ξ|p′ ])1/p′ , where L > 0 is a constant depending only on p′,
T and Lipschitz constant µ. Thus,
∞∫
1
V (ξ ≥ t)dt ≤ L
∞∫
1
(P (ξ ≥ t))1/p′dt ≤ L
∞∫
1
tp−1P (ξ ≥ t)dt
1/p′ ∞∫
1
t
− (p−1)q′
p′ dt
1/q′ ,
where
1
p′
+
1
q′
= 1. Since
∫ ∞
1
tp−1P (ξ ≥ t)dt ≤ E[|ξ|p] < ∞ and
∫ ∞
1
t
− (p−1)q′
p′ dt < ∞, we get
Cg[ξ+] <∞. Similarly, Cg[ξ−] <∞. This concludes the proof of the lemma.
For any ξ ∈ L(Ω,F , P ), we define
C[ξ] :=
0∫
−∞
(V (ξ ≥ t)− 1)dt+
∞∫
0
V (ξ ≥ t)dt,
C[ξ] :=
0∫
−∞
(V (ξ ≥ t)− 1)dt+
∞∫
0
V (ξ ≥ t)dt,
where V and V are upper and lower probabilities defined by
V (A) := sup
Q∈P
Q(A), V (A) := inf
Q∈P
Q(A),
where P is the same as in (4.3).
Obviously, V (A) = E [1A] = Eµ[1A], V (A) = E [1A] = E−µ[1A].
We have the following theorem.
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A COMONOTONIC THEOREM FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS. . . 761
Theorem 4.3. Suppose (Xs) be the solution of SDE in Theorem 3.2. Let φ be a monotonic
function such that φ(XT ) ∈ L(Ω,F , P ). Assuming that for all t ≥ 0 and x ∈ R, σi(t, x) > 0
(σi(t, x) is the ith component of σ(t, x)), i = 1, 2, . . . , d, then there exist probability measures Q1
and Q2 such that
(a) for any φ that is increasing, then
C[φ(XT )] = E [φ(XT )] = EQ1 [φ(XT )], C[φ(XT )] = E [φ(XT )] = EQ2 [φ(XT )];
(b) for any φ that is decreasing, then
C[φ(XT )] = E [φ(XT )] = EQ2 [φ(XT )], C[φ(XT )] = E [φ(XT )] = EQ1 [φ(XT )].
The probability measures Q1 and Q2 are defined by
dQ1
dP
= e−
1
2dµ
2T+µ
∑d
i=1W
i
T ,
dQ2
dP
= e−
1
2dµ
2T−µ
∑d
i=1W
i
T .
Proof. We only prove (a). The rest of this theorem can be proved in a similar manner.
Proof of part (a). For any φ(XT ) ∈ L(Ω,F , P ), there exists 1 < p < 2, such that φ(XT ) ∈
∈ Lp(Ω,F , P ). Consider the following BSDE:
yt = φ(XT ) +
T∫
t
µ
d∑
i=1
|zis|ds−
T∫
t
zs · dWs, t ∈ [0, T ]. (4.4)
Let (yµt , z
µ
t ) be the unique solution of BSDE (4.4). By Theorem 3.2, noting that φ is an increasing
function, we have
zµt � σ(t,Xt) ≥ 0, dP × dt-a.s.
Since σi(t, x) > 0, we can deduce
zµ,it ≥ 0, dP × dt-a.s., i = 1, 2, . . . , d.
Therefore, (yµt , z
µ
t ) is also the unique solution of the following linear BSDE:
yt = φ(XT ) +
T∫
t
µ
d∑
i=1
zisds−
T∫
t
zs · dWs, t ∈ [0, T ]. (4.5)
Let W t := Wt−µ(1, . . . , 1)T t. By Girsanov’s theorem, (W t)t∈[0,T ] is a Q1-Brownian motion, where
dQ1
dP
= e−
1
2dµ
2T+µ
∑d
i=1W
i
T . Moreover, BSDE (4.5) can be rewritten as
yt = φ(XT )−
T∫
t
zs · dW s, t ∈ [0, T ].
It is easy to check that
(∫ t
0
zsdW s
)
t∈[0,T ]
is a martingale with respect to Q1. Indeed, we have, by
the BDG inequality and Hölder’s inequality,
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762 Z.-J. ZONG
EQ1
∣∣∣∣∣∣
t∫
0
zsdW s
∣∣∣∣∣∣
≤ EQ1
T∫
0
|zs|2ds
1/2
≤
≤
E
T∫
0
|zs|2ds
p/2
1/p(
E
[(
dQ1
dP
)q])1/q
<∞,
where
1
p
+
1
q
= 1. Thus,
E [φ(XT )] = EQ1 [φ(XT )]. (4.6)
On the other hand, let φn(XT ) := (φ(XT ) ∧ n) ∨ (−n). From [5], we have
E [φn(XT )] = C[φn(XT )].
Note that
V (φn(XT ) ≥ t) = E
[
1(φn(XT )≥t)
]
= Eµ
[
1(φn(XT )≥t)
]
,
V (φ(XT ) ≥ t) = E
[
1(φ(XT )≥t)
]
= Eµ
[
1(φ(XT )≥t)
]
and hence
V (φn(XT ) ≥ t)→ V (φ(XT ) ≥ t), as n→∞.
Applying the monotonic convergence theorem, we obtain
C[φn(XT )]→ C[φ(XT )], as n→∞.
From Lemma 2.2, we obtain
E [φn(XT )]→ E [φ(XT )], as n→∞.
Therefore
E [φ(XT )] = C[φ(XT )]. (4.7)
From (4.6) and (4.7), we have
C[φ(XT )] = EQ1 [φ(XT )].
In a similar manner, we can obtain
C[φ(XT )] = EQ2 [φ(XT )].
Now we give an example to illustrate how our result allows one to calculate Choquet expectations.
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A COMONOTONIC THEOREM FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS. . . 763
Example 4.1. For simplicity, let T = 1, d = 1. Suppose b = 0, σ = 1 and x = 0, then
X1 = W1, W1 ∼ N(0, 1).
(i) Let φ(x) = exp
(
x2
2p1
− x
)
1(x≥p1), where 1 < p1 < 2.
Obviously, φ is an increasing function and φ(X1) = φ(W1) = exp
(
W 2
1
2p1
−W1
)
1(W1≥p1).
It is easy to check out
E[|φ(W1)|p1 ] =
∞∫
p1
exp
(
x2
2
− p1x
)
1√
2π
e−
1
2x
2
dx =
1√
2πp1
e−p
2
1 <∞,
and
E[|φ(W1)|p] =∞ ∀p > p1.
Hence, φ(W1) ∈ L(Ω,F , P ), φ(W1) /∈ L2(Ω,F , P ).
Using Theorem 4.3, we obtain
C[φ(W1)] = EQ1 [φ(W1)] =
∞∫
p1
exp
(
x2
2p1
− x
)
e−
1
2µ
2+µx 1√
2π
e−
1
2x
2
dx =
=
1√
2π
e−
1
2µ
2
∞∫
p1
e
−1
2
(
1− 1
p1
)
x2
e−(1−µ)xdx <∞,
C[φ(W1)] = EQ2 [φ(W1)] =
∞∫
p1
exp
(
x2
2p1
− x
)
e−
1
2µ
2−µx 1√
2π
e−
1
2x
2
dx =
=
1√
2π
e−
1
2µ
2
∞∫
p1
e
−1
2
(
1− 1
p1
)
x2
e−(1+µ)xdx <∞.
(ii) Let φ(x) = exp
(
x2
2p1
+ x
)
1(x≤−p1), where 1 < p1 < 2.
Obviously, φ is a decreasing function and φ(X1) = φ(W1) = exp
(
W 2
1
2p1
+W1
)
1(W1≤−p1).
It is easy to check out
E[|φ(W1)|p1 ] =
−p1∫
−∞
exp
(
x2
2
+ p1x
)
1√
2π
e−
1
2x
2
dx =
1√
2πp1
e−p
2
1 <∞,
and
E[|φ(W1)|p] =∞ ∀p > p1.
Hence, φ(W1) ∈ L(Ω,F , P ), φ(W1) /∈ L2(Ω,F , P ).
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764 Z.-J. ZONG
Using Theorem 4.3, we obtain
C[φ(W1)] = EQ2 [φ(W1)] =
−p1∫
−∞
exp
(
x2
2p1
+ x
)
e−
1
2µ
2−µx 1√
2π
e−
1
2x
2
dx =
=
1√
2π
e−
1
2µ
2
−p1∫
−∞
e
−1
2
(
1− 1
p1
)
x2
e−(µ−1)xdx <∞,
C[φ(W1)] = EQ1 [φ(W1)] =
−p1∫
−∞
exp
(
x2
2p1
+ x
)
e−
1
2µ
2+µx 1√
2π
e−
1
2x
2
dx =
=
1√
2π
e−
1
2µ
2
−p1∫
−∞
e
−1
2
(
1− 1
p1
)
x2
e(µ+1)xdx <∞.
Remark 4.2. The Choquet expectations C[φ(W1)], C[φ(W1)] in Example 4.1 can not be calcu-
lated by Chen and Kulperger [5], because φ(W1) /∈ L2(Ω,F , P ). But thanks to Theorem 4.3, since
φ(W1) ∈ L(Ω,F , P ), one can easily calculate them.
4.2.3. Choquet expectations versus generalized Peng’s g-expectations. In this subsection, we
provide a necessary and sufficient condition of Choquet expectations being equal to generalized
Peng’s g-expectations.
We have the following theorem.
Theorem 4.4. Suppose that g satisfies (H.2) and (H.3). Then there exists a Choquet expectation
whose restriction to L(Ω,F , P ) is equal to a generalized Peng’s g-expectation if and only if g does
not depend on y and is linear in z, i.e.,
g(t, y, z) = vt · z =
d∑
i=1
vitz
i.
Proof. Since L2(Ω,F , P ) ⊂ L(Ω,F , P ), the proof of necessity can be seen in Chen et al. [2]
and Hu [11]. We only prove the sufficiency.
For any ξ ∈ L(Ω,F , P ), there exists 1 < p < 2, such that ξ ∈ Lp(Ω,F , P ). If g(t, y, z) = vt · z,
let us consider the BSDE
yt = ξ +
T∫
t
vs · zsds−
T∫
t
zs · dWs.
Set W t = Wt −
∫ t
0
vsds, then
yt = ξ −
T∫
t
zs · dW s.
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A COMONOTONIC THEOREM FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS. . . 765
By Girsanov’s theorem, (W t)t∈[0,T ] is a Q-Brownian motion under Q defined by
dQ
dP
= exp
−1
2
T∫
0
|vs|2ds+
T∫
0
vs · dWs
.
Thus
Eg[ξ] = EQ[ξ].
This implies the generalized Peng’s g-expectation is a classical mathematical expectation. Obviously,
the classical mathematical expectation can be represented by the Choquet expectation. So the proof
of sufficiency is complete.
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Received 10.01.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
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