# Nonlinear Decomposition of Doob-Meyer's Type for Continuousg-Supermartingale with Uniformly Continuous Coefficient

Authors: Xuejun Shi; Long Jiang; Ronglin Ji
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We prove that a continuousg-supermartingale with uniformly continuous coeffcient g on finite or infinite horizon, is a g-supersolution of the corresponding backward stochastic differential equation. It is a new nonlinear Doob-Meyer decomposition theorem for the g-supermartingale with continuous trajectory.


---

## Body

## 1. Introduction

In 1990, Pardoux-Peng [1] proposed the following nonlinear backward stochastic differential equation (BSDE) driven by a Brownian motion:
(1)
y
t
=
ξ
+
∫
t
T
‍
g
(
s
,
y
s
,
z
s
)
d
s
-
∫
t
T
‍
z
s
d
B
s
,
t
∈
[
0
,
T
]
,
where the positive real numberT, the random variableξ, and the functiongare called the time horizon, the terminal data, and the generator, respectively, and the pair of adapted processes(
y
t
,
z
t
)
t
∈
[
0
,
T
]to be known is called the solution of the BSDE (1). In this paper, we study a more generalized BSDE with a given increasing process(
V
t
)
t
∈
[
0
,
T
]withV
0
=
0:
(2)
y
t
=
ξ
+
∫
t
T
‍
g
(
s
,
y
s
,
z
s
)
d
s
-
∫
t
T
‍
z
s
d
B
s
+
V
T
-
V
t
,
y
t
=
ξ
+
∫
t
T
‍
g
(
s
,
y
s
,
z
s
)
d
s
-
∫
t
T
‍
z
s
d
B
s
t
∈
[
0
,
T
]
.
If(
V
t
)
t
∈
[
0
,
T
]
≡
0, the first component(
y
t
)
t
∈
[
0
,
T
]of solution of (2) is called theg-solution of (1); otherwise, it is called theg-supersolution. Subsequently, Peng [2] introduced the nonlinear expectation and nonlinear martingale theories via BSDEs. In [3], Peng first obtained the monotonic limit theorem; that is, under some mild conditions, the limit of a monotonically increasing sequence ofg-supersolutions is also ag-supersolution. And applying this result, he proved that a càdlàgg-martingale, which is right continuous with left limits, had a nonlinear decomposition of Doob-Meyer’s type, corresponding to the classical martingale theory. Later, Lin [4, 5] extended Peng’s result and got this decomposition for theg-supermartingale with respect to a general continuous filtration and that with jumps, respectively. It should be pointed out that, in Peng [3] and Lin [4, 5], the monotonic limit theorem for BSDEs plays a key role, and it is also useful in other problems. For example, in [6], Peng-Xu put forward a generalized version of monotonic limit theorem and proved that solving the reflected BSDE with a given lower barrier process was equivalent to finding the smallestg-supermartingale dominating the barrier. And Peng-Xu [7] used this technique to treat the problems of the BSDE with generalized constraints and solve the American option pricing problem in an incomplete market. On the other hand, motivated by the theories of the classical martingale and the nonlinear martingale, Chen-Wang [8] showed that the BSDEs on infinite time horizon were solvable, under the Lipschitz assumption ong, whose Lipschitzian coefficient is a function depending ont, and they obtained the convergence theorem of the nonlinearg-martingale. Afterward, Fan et al. [9] explored the BSDEs on finite or infinite horizon, without the Lipschitz assumption, and got an existence and uniqueness result and a comparison theorem.Based on these results, a natural question is, under the generalized uniformly continuous assumption on the coefficientg, does theg-martingale still have a nonlinear decomposition of Doob-Meyer’s type? Our answer is yes. We prove that if ag-supermartingale has a continuous trajectory on finite or infinite time interval, then it is ag-supersolution of the corresponding BSDE; that is, it has a nonlinear Doob-Meyer decomposition. It should be noted that our results are based on the conditions without the Lipschitz assumption on the coefficientg. And our results do not depend on the infinite time version of the monotonic limit but only on the penalization method.The outline of this paper is as follows. Section2 provides some assumptions, definitions, and the existence and uniqueness theorem and comparison theorem for a generalized BSDE with generalized uniformly continuous generatorg. Then, Section 3 devotes to the main result a new version of nonlinear Doob-Meyer’s decomposition theorem for the continuousg-supermartingale with the generalized uniformly continuous coefficient.


## 2. Preliminaries

LetTbe a finite or infinite nonnegative extended real number, and let(
B
t
)
t
≥
0be a standardd-dimensional Brownian motion defined on a complete probability space(
Ω
,
ℱ
,
P
)endowed with a filtration(
ℱ
t
)
t
≥
0generated by this Brownian motion:
(3)
ℱ
t
≜
σ
{
B
s
:
0
≤
s
≤
t
}
∨
𝒩
,
ℱ
∞
=
⋁
t
≥
0
ℱ
t
,
where 𝒩 is the set of allP-null subsets.For simplicity of presentation, we use|
x
|to denote the Euclidean norm ofxin ℝ or ℝ
d, and letL
2
(
Ω
,
ℱ
t
,
P
)be the space of all theℱ
tmeasurable square integrable real valued random variables, and define the adapted process spaces as follows:ℋ
2
(
0
,
τ
;
ℝ
d
)
∶
=
{
(
ϕ
t
)
t
∈
[
0
,
τ
] is a ℝ
d-valued process such that 𝔼
[
∫
0
τ
‍
|
ϕ
t
|
2
d
t
]
<
+
∞
};
𝒮
2
(
0
,
τ
;
ℝ
)
∶
=
{
(
ϕ
t
)
t
∈
[
0
,
τ
] is a càdlàg ℝ- valued process such that 𝔼
[
sup
⁡
0
≤
t
≤
τ
⁡
|
ϕ
t
|
2
]
<
+
∞
};
𝒜
2
(
0
,
τ
;
ℝ
)
∶
=
{
(
ϕ
t
)
t
∈
[
0
,
τ
] is an increasing process in 𝒮
2
(
0
,
τ
;
ℝ
) with ϕ
(
0
)
=
0
}.Clearly, all the above spaces of stochastic processes are completed Banach spaces.Furthermore, we denote the set of linear increasing functionsϕ
(
·
)
:
ℝ
+
↦
ℝ
+withϕ
(
0
)
=
0by 𝒳. Here the linear increasing means that, for any elementϕ
∈
𝒳, there exists a pair of positive real numbers(
a
,
b
)depending onϕsuch that, for all x
∈
ℝ
+, ϕ
(
x
)
≤
a
x
+
b.The generatorg
(
t
,
ω
,
y
,
z
)
:
[
0
,
T
]
×
Ω
×
ℝ
×
ℝ
d
↦
ℝ is a random function which is a progressively measurable stochastic process for any(
y
,
z
). We assume that it satisfies the following two assumptions, where (H2) is a generalized uniformly continuous condition; that is, its modulus of continuity may depend ont:(H1)
𝔼
[
(
∫
0
T
|
g
(
t
,
0,0
)
|
d
t
)
2
]
<
∞;
(H2)
|
g
(
t
,
y
,
z
)
-
g
(
t
,
y
′
,
z
′
)
|
≤
u
(
t
)
φ
(
|
y
-
y
′
|
)
+
v
(
t
)
ψ
(
|
z
-
z
′
|
), whereu
(
·
)andv
(
·
)are two positive functions mapping fromℝ
+toℝ
+, such that∫
0
T
[
u
(
t
)
+
v
2
(
t
)
]
d
t
<
∞; the functionsφandψbelong to𝒳 and φ
(
·
)is a concave function, with∫
0
+
‍
φ
(
t
)
d
t
=
+
∞. And in addition, we assume that∫
0
T
‍
v
(
t
)
d
t
<
∞, ifψcannot be dominated by a linear function; that is, we cannot find a real numbera, such thatϕ
(
x
)
≤
a
x.Remark 1.
In (H2),φ
(
·
)is a concave function which means thatφ
(
λ
t
1
+
(
1
-
λ
)
t
2
)
≥
λ
φ
(
t
1
)
+
(
1
-
λ
)
φ
(
t
2
), forλ
∈
[
0,1
]andt
1,t
2
∈
[
0
,
T
]. And the equality∫
0
+
‍
φ
(
t
)
d
t
=
∞means that the value of the integration∫
0
δ
‍
φ
(
t
)
d
twill be infinite on any interval[
0
,
δ
]withδ
>
0. For simplicity, we also useu
sandv
sto denoteu
(
s
)andv
(
s
), respectively, in the remaining of this paper.Now, we consider the following problem. Suppose that the time horizonT, generatorg, terminal datay
T, and the increasing càdlàg process(
V
t
)
t
∈
[
0
,
T
]
∈
𝒮
2
(
0
,
T
;
ℝ
) are given in advance; let us find a pair of processes(
y
t
,
z
t
)
t
∈
[
0
,
T
]
∈
𝒮
2
(
0
,
T
;
ℝ
)
×
ℋ
2
(
0
,
T
;
ℝ
d
)satisfying
(4)
y
t
=
y
T
+
∫
t
T
‍
g
(
s
,
y
s
,
z
s
)
d
s
-
∫
t
T
‍
z
s
d
B
s
+
V
T
-
V
t
,
y
t
=
y
T
+
∫
t
T
‍
g
(
s
,
y
s
,
z
s
)
d
s
-
∫
t
T
‍
z
s
d
B
s
t
∈
[
0
,
T
]
.IfV
t
≡
0, the above equation (4) will be a classical BSDE on finite or infinite horizon; the existence and uniqueness result is already obtained, which is stated by Theorem 3 in Fan et al. [9]. Otherwise we can sety
¯
t
:
=
y
t
+
V
tand treat the following BSDE as
(5)
y
¯
t
=
y
T
+
V
T
+
∫
t
T
‍
g
(
s
,
y
¯
s
-
V
s
,
z
s
)
d
s
-
∫
t
T
‍
z
s
d
B
s
.
It is a classical BSDE with the terminal dataξ
:
=
y
T
+
V
Tand the generatorg
¯
:
=
g
(
t
,
y
-
V
t
,
z
). Since the assumptions (H1) and (H2) hold for generatorg, it is easy to verify thatg
¯still satisfies the two conditions. So we have the following existence and uniqueness theorem.Lemma 2 (existence and uniqueness).
One assumes that the generatorgof the BSDE (4) satisfies the conditions (H1) and (H2). Then, for any random variabley
T
∈
L
2
(
Ω
,
ℱ
T
,
P
), and a process(
V
t
)
t
∈
[
0
,
T
]
∈
𝒮
2
(
0
,
T
;
ℝ
), there exists a unique pair of processes(
y
t
,
z
t
)
t
∈
[
0
,
T
]
∈
𝒮
2
(
0
,
T
;
ℝ
)
×
ℋ
2
(
0
,
T
;
ℝ
d
), which is a solution of the BSDE (4), such that(
y
t
+
V
t
)
t
∈
[
0
,
T
]is continuous and
(6)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
y
t
|
2
]
<
∞
.We can also have the following comparison theorem, which will be used in the latter part of this section and the next one.Proposition 3 (comparison).
Suppose that the assumptions in Lemma2 hold. Let(
y
¯
t
,
z
¯
t
)
t
∈
[
0
,
T
]be the solution of another BSDE:
(7)
y
¯
t
=
y
¯
T
+
∫
t
T
‍
g
¯
s
d
s
-
∫
t
T
‍
z
¯
s
d
B
s
+
V
¯
T
-
V
¯
t
,
t
∈
[
0
,
T
]
,
where(
V
t
¯
)
t
∈
[
0
,
T
]
∈
𝒮
2
(
0
,
T
;
ℝ
),y
¯
T
∈
L
2
(
Ω
,
ℱ
T
,
P
), and (
g
¯
t
)
t
∈
[
0
,
T
]are given such that(1)
y
^
T
:
=
y
T
-
y
¯
T
≥
0;
(2)
g
^
t
:
=
g
(
t
,
y
¯
t
,
z
¯
t
)
-
g
¯
t
≥
0, d
P
×
d
t
-a
.
e.;
(3)
V
^
t
:
=
V
t
-
V
¯
t is a càdlàg increasing process;
(4)
𝔼
[
(
∫
0
T
‍
|
g
¯
t
|
d
t
)
2
]
<
∞.Then we have, P-a.s.,
(8)
y
t
≥
y
¯
t
,
∀
t
∈
[
0
,
T
]
.Proof.
We sketch the proof as follows. Sety
^
t
=
y
¯
t
-
y
t and z
^
t
=
z
¯
t
-
z
t; applying Itô-Meyer’s formula to(
y
¯
t
-
y
t
)
+leads to
(9)
y
^
t
+
≤
(
y
¯
T
-
y
T
)
+
+
∫
t
T
‍
1
{
y
^
s
>
0
}
(
g
¯
s
-
g
(
s
,
y
s
,
z
s
)
)
d
s
-
∫
t
T
‍
1
{
y
^
s
>
0
}
z
^
s
d
B
s
+
∫
t
T
‍
1
{
y
^
s
-
>
0
}
(
d
V
¯
s
-
d
V
s
)
.
SinceV
^
tis an increasing process, we see that
(10)
∫
t
T
‍
1
{
y
^
s
-
>
0
}
(
d
V
¯
s
-
d
V
s
)
≤
-
∫
t
T
‍
1
{
y
^
s
-
>
0
}
d
V
^
s
≤
0
.
Recalling thatg
^
t
:
=
g
(
t
,
y
¯
t
,
z
¯
t
)
-
g
¯
t
≥
0
,
d
P
×
d
t
-a
.
e., and the assumption (H2), we can get
(11)
1
{
y
^
s
>
0
}
(
g
¯
s
-
g
(
s
,
y
s
,
z
s
)
)
=
1
{
y
^
s
>
0
}
(
g
¯
s
-
g
(
s
,
y
¯
s
,
z
¯
s
)
+
g
(
s
,
y
¯
s
,
z
¯
s
)
-
g
(
s
,
y
s
,
z
s
)
)
≤
u
s
φ
(
y
^
s
+
)
+
1
{
y
^
s
>
0
}
v
s
ψ
(
|
z
^
s
|
)
.
Thus, it follows that
(12)
y
^
t
+
≤
∫
t
T
‍
u
s
φ
(
y
^
s
+
)
+
1
{
y
^
s
>
0
}
v
s
ψ
(
|
z
^
s
|
)
d
s
-
∫
t
T
‍
1
{
y
^
s
>
0
}
z
^
s
d
B
s
.
Now we are in the same position with Theorem 2 in Fan et al. [9]. Then we can prove that, for all t
∈
[
0
,
T
],y
^
t
+
≤
0, P-a.s. Therefore, for anyt
∈
[
0
,
T
], we have
(13)
y
¯
t
≤
y
t
,
P-a
.
s
.
Observing thaty
tandy
¯
tare càdlàg processes, we can conclude that, P-a.s.,
(14)
y
¯
t
≤
y
t
,
∀
t
∈
[
0
,
T
]
.Remark 4.
If we replace the deterministic terminal timeTby aℱ
t-stopping timeτ
≤
T, then, by Lemma 2, existence and uniqueness theorem and the above comparison theorem still hold true.For a given stopping timeτ
≤
T, we now consider the following BSDE:
(15)
y
t
=
y
τ
+
∫
t
∧
τ
τ
‍
g
(
s
,
y
s
,
z
s
)
d
s
-
∫
t
∧
τ
τ
‍
z
s
d
B
s
+
A
τ
-
A
t
∧
τ
,
wherey
τ
∈
L
2
(
Ω
,
ℱ
τ
,
P
) and (
A
t
)
t
∈
[
0
,
T
]
∈
𝒜
2
(
0
,
τ
;
ℝ
) is a given càdlàg increasing process withA
0
=
0.Next, we introduce the conceptions ofg-solution,g-supersolution,g-martingale, and g-supermartingale closely following Peng’s definitions in [3].Definition 5.
If a process(
y
t
)
t
∈
[
0
,
τ
]can be written in the form of the BSDE (15) with the generatorg, then one call it a g-supersolution on[
0
,
τ
]. Particularly, if(
A
t
)
t
∈
[
0
,
τ
]
≡
0on[
0
,
τ
], then one call(
y
t
)
t
∈
[
0
,
τ
]ag-solution on[
0
,
τ
].Definition 6.
Aℱ
t-progressively measurable real-valued process(
Y
t
)
t
∈
[
0
,
τ
]is called ag-supermartingale (resp.g-martingale), if for each stopping timeτ
≤
T, 𝔼
|
Y
τ
|
2
<
∞, and theg-solution(
y
t
)
t
∈
[
0
,
τ
]with terminal conditiony
τ
=
Y
τsatisfiesy
σ
≤
Y
σ
(resp.y
σ
=
Y
σ
) for all stopping timeσ
≤
τ. Indeed, ag-martingale on[
0
,
T
]is ag-solution on[
0
,
T
].From Proposition3, we know that a g-supersolution is ag-supermartingale. Conversely, a meaningful and interesting question follows immediately. Is ag-supermartingale ag-supersolution? If so, does theg-supermartingale, org-supersolution, has a unique representation of the form (15)?According to Proposition  1.6 in [3], we can assert that, given ag-supersolution(
y
t
)
t
∈
[
0
,
τ
]on[
0
,
τ
], there is a unique pair of processes(
z
t
,
A
t
)
t
∈
[
0
,
τ
]
∈
ℋ
2
(
0
,
τ
;
ℝ
d
)
×
𝒜
2
(
0
,
τ
;
ℝ
) on[
0
,
τ
]such that the triple(
y
t
,
z
t
,
A
t
)
t
∈
[
0
,
τ
]satisfies the BSDE (15). Now, we can propose the next conception as follows.Definition 7.
Provided that the process(
y
t
)
t
∈
[
0
,
T
]is ag-supersolution and the triple of processes(
y
t
,
z
t
,
A
t
)
t
∈
[
0
,
τ
]satisfies the BSDE (15), one call(
z
t
,
A
t
)
t
∈
[
0
,
τ
]the unique decomposition of(
y
t
)
t
∈
[
0
,
τ
].


## 3. Nonlinear Doob-Meyer’s Decomposition forg-Supermartingale with Uniformly Continuous Coefficient

In this section, we provide and prove the main result of this paper that a continuousg-supermartingale is ag-supersolution; that is, it has a unique decomposition in the sense of Definition 7.Theorem 8.
One assumes thatgsatisfies the conditions(
H
1
)and(
H
2
). Let(
Y
t
)
t
∈
[
0
,
T
]be a continuousg-supermartingale on[
0
,
T
]in𝒮
2
(
0
,
T
;
ℝ
). Then(
Y
t
)
t
∈
[
0
,
T
]is ag-supersolution on[
0
,
T
] that is, there is a unique pair of processes(
z
t
,
A
t
)
t
∈
[
0
,
T
]inℋ
2
(
0
,
T
;
ℝ
d
)
×
𝒜
2
(
0
,
T
;
ℝ
), such that(
Y
t
)
t
∈
[
0
,
T
]coincides with the first component(
y
t
)
t
∈
[
0
,
T
]of the solution for the following BSDE:
(16)
y
t
=
Y
T
+
∫
t
T
‍
g
(
s
,
y
s
,
z
s
)
d
s
-
∫
t
T
‍
z
s
d
B
s
+
A
T
-
A
t
,
y
t
=
Y
T
+
∫
t
T
‍
g
(
s
,
y
s
,
z
s
)
d
s
-
∫
t
T
‍
z
s
d
B
s
t
∈
[
0
,
T
]
.In order to prove this theorem, we consider the family of penalization BSDEs parameterized byn
=
1,2
,
3
,
…,
(17)
y
t
n
=
Y
T
+
∫
t
T
‍
g
(
s
,
y
s
n
,
z
s
n
)
d
s
-
∫
t
T
‍
z
s
n
d
B
s
+
n
∫
t
T
‍
v
s
2
(
Y
s
-
y
s
n
)
d
s
,
t
∈
[
0
,
T
]
,
and set
(18)
A
t
n
:
=
n
∫
0
t
‍
v
s
2
(
Y
s
-
y
s
n
)
d
s
,
t
∈
[
0
,
T
]
.
We first claim the next proposition.Proposition 9.
For eachn
=
1,2
,
…, one has, P-a.s.,
(19)
Y
t
≥
y
t
n
,
t
∈
[
0
,
T
]
.Proof.
Using an argument similar to that in Lemma  3.4 in [3], one can carry out the proof by contradiction. We sketch it as follows.
Supposing that it is not the case, then there existδ
>
0and a positive integernsuch that the measure of{
(
ω
,
t
)
∣
y
t
n
-
Y
t
-
δ
≥
0
}
⊂
Ω
×
[
0
,
T
]is nonzero; then we can define the following stopping times:
(20)
σ
:
=
min
⁡
{
T
,
inf
⁡
{
t
∣
y
t
n
≥
Y
t
+
δ
}
}
,
τ
:
=
inf
⁡
{
t
≥
σ
∣
y
t
n
≤
Y
t
}
.
It is observed, from the above definition and the continuous of(
Y
t
)
t
∈
[
0
,
T
], thatσ
≤
τ
≤
T and P
(
τ
>
σ
)
>
0. And furthermore, we have, P-a.s.,
(21)
(
i
)
y
σ
n
≥
Y
σ
+
δ
;
(
ii
)
y
τ
n
≤
Y
τ
.
Now let(
y
t
)
t
∈
[
0
,
τ
](resp.(
y
t
′
)
t
∈
[
0
,
τ
]) be theg-solution on[
0
,
τ
]with terminal conditiony
τ
=
y
τ
n(resp.y
τ
′
=
Y
τ). By Proposition 3, (21)-(ii) implies thaty
σ
n
≤
y
σ
≤
y
σ
′. On the other hand since(
Y
t
)
t
∈
[
0
,
T
]is ag-supermartingale, thus we can get
(22)
Y
σ
≥
y
σ
n
,
P-a
.
s
.
This is a contradiction to (21)-(i). Then by Fubini’s theorem, we have, P-a.s.,
(23)
Y
t
≥
y
t
n
,
dt-a
.
e
.
And the conclusion follows from the continuity of(
Y
t
)
t
∈
[
0
,
T
]
(
y
t
n
)
t
∈
[
0
,
T
]. The proof is completed.Now, we can get the following result; the boundedness of the triple of the processes(
y
t
n
,
z
t
n
,
A
t
n
)
t
∈
[
0
,
T
]can be defined by the penalization BSDEs.Proposition 10.
There exists a positive real number C such that for any positive integern
(24)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
y
t
n
|
2
+
∫
0
T
‍
|
z
t
n
|
2
d
t
+
|
A
T
n
|
2
]
≤
C
.Proof.
From BSDE (17), we have
(25)
A
T
n
=
y
0
n
-
Y
T
-
∫
0
T
‍
g
(
s
,
y
s
n
,
z
s
n
)
d
s
+
∫
0
T
‍
z
s
n
d
B
s
≤
|
y
0
n
|
+
|
Y
T
|
+
∫
0
T
‍
[
|
g
(
s
,
0,0
)
|
+
u
s
φ
(
|
y
s
n
|
)
+
v
s
ψ
(
|
z
s
n
|
)
]
d
s
+
|
∫
0
T
‍
z
s
n
d
B
s
|
≤
|
y
0
n
|
+
|
Y
T
|
+
∫
0
T
[
|
g
(
s
,
0,0
)
|
+
u
s
(
a
φ
|
y
s
n
|
+
b
φ
)
+
v
s
(
a
ψ
|
z
s
n
|
+
b
ϕ
)
]
d
s
+
|
∫
0
T
‍
z
s
n
d
B
s
|
,
where the real numbersa
φ,b
φ and a
ψ,b
ψdepend on the functionsφ and ψ, respectively. From Proposition 9, we see thaty
t
nis dominated by|
y
t
1
|
+
|
Y
t
|. Thus there exists a constantC
0independent ofn, such that
(26)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
y
t
n
|
2
]
≤
C
0
.
Now, noticing the boundedness of(
y
t
n
)
t
∈
[
0
,
T
]in the above sense, from the basic algebraic inequality, Jensen’s inequality and Hölder’s inequality, we can get that there exists another constantC
1such that
(27)
𝔼
|
A
T
n
|
2
≤
8
𝔼
[
|
y
0
n
|
2
+
|
Y
T
|
2
+
(
∫
0
T
‍
|
g
(
s
,
0,0
)
|
d
s
)
2
+
a
φ
2
(
∫
0
T
‍
u
s
d
s
)
2
sup
⁡
0
≤
t
≤
T
|
y
t
n
|
2
+
b
φ
2
(
∫
0
T
‍
u
s
d
s
)
2
+
a
ψ
2
∫
0
T
‍
v
s
2
d
s
∫
0
T
‍
|
z
s
n
|
2
d
s
+
b
ψ
2
(
∫
0
T
‍
v
s
d
s
)
2
+
∫
0
T
‍
|
z
s
n
|
2
d
s
]
≤
C
1
+
C
1
𝔼
∫
0
T
‍
|
z
s
n
|
2
d
s
.
On the other hand, in the light of (H2), applying Itô’s formula to|
y
t
n
|
2on[
0
,
T
]will lead to
(28)
|
y
0
n
|
2
+
𝔼
∫
0
T
‍
|
z
s
n
|
2
d
s
=
𝔼
|
Y
T
|
2
+
2
𝔼
∫
0
T
‍
y
s
n
g
(
s
,
y
s
n
,
z
s
n
)
d
s
+
2
𝔼
∫
0
T
‍
y
s
n
d
A
s
n
≤
𝔼
|
Y
T
|
2
+
2
𝔼
∫
0
T
‍
[
|
y
s
n
|
(
|
g
(
s
,
0,0
)
|
+
u
s
φ
(
|
y
s
n
|
)
+
v
s
ψ
(
|
z
s
n
|
)
)
|
y
s
n
|
]
d
s
+
2
𝔼
∫
0
T
‍
y
s
n
d
A
s
n
.
Then, the Hölder inequality and the inequalitya
b
≤
ϵ
a
2
+
1
/
ϵ
b
2, for alla,b,ϵ
>
0, imply that
(29)
|
y
0
n
|
2
+
𝔼
∫
0
T
‍
|
z
s
n
|
2
d
s
≤
𝔼
|
Y
T
|
2
+
(
1
+
b
φ
+
b
ψ
+
2
a
φ
∫
0
T
‍
u
s
d
s
+
2
a
ψ
2
∫
0
T
‍
v
s
2
d
s
)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
y
s
n
|
2
]
+
𝔼
[
(
∫
0
T
‍
|
g
(
s
,
0,0
)
|
d
s
)
2
]
+
b
φ
(
∫
0
T
‍
u
s
d
s
)
2
+
b
ψ
(
∫
0
T
‍
v
s
d
s
)
2
+
1
2
𝔼
∫
0
T
|
z
s
n
|
2
d
s
+
1
4
C
1
𝔼
[
|
A
T
n
|
2
]
+
4
C
1
𝔼
[
sup
⁡
0
≤
t
≤
T
|
y
s
n
|
2
]
.
Thus, we can choose a constantC
2satisfying
(30)
𝔼
∫
0
T
‍
|
z
s
n
|
2
d
s
≤
C
2
+
1
2
C
1
𝔼
|
A
T
n
|
2
.
Combining the inequalities (27) and (30), we can conclude that 𝔼
|
A
T
n
|
2
≤
2
C
1
(
1
+
C
2
) and 𝔼
∫
0
T
‍
|
z
s
n
|
2
d
s
≤
1
+
2
C
2. The proof is completed.Then, we give a proposition which plays a key role in the procedure to prove the main theorem.Proposition 11.
(31)
lim
⁡
n
→
∞
𝔼
[
sup
⁡
0
≤
t
≤
T
(
Y
t
-
y
t
n
)
2
]
=
0
.Proof.
Since the family of the processes(
y
t
n
)
t
∈
[
0
,
T
]is increasing inn and dominated by the process(
Y
t
)
t
∈
[
0
,
T
]from the above, we can define a process(
y
t
)
t
∈
[
0
,
T
]pointwise by the limit of the processes sequence. Then we have, P-a.s.,
(32)
y
t
:
=
lim
⁡
n
→
∞
y
t
n
,
∀
t
∈
[
0
,
T
]
.
And according to Lemma2, for any integern, the following BSDE has a unique solution, denoted by(
y
~
t
n
,
z
~
t
n
)
t
∈
[
0
,
T
]:
(33)
y
~
t
n
=
Y
T
+
∫
t
T
‍
g
(
s
,
y
s
n
,
y
s
n
)
d
s
-
∫
t
T
‍
z
~
s
n
d
B
s
+
n
∫
t
T
‍
v
s
2
(
Y
s
-
y
~
s
n
)
d
s
.
Letτbe a stopping time such that0
≤
τ
≤
T; then we have
(34)
y
~
τ
n
=
𝔼
ℱ
τ
[
e
-
n
∫
τ
T
‍
v
r
2
d
r
Y
T
+
n
∫
τ
T
‍
v
s
2
e
-
n
∫
τ
s
‍
v
r
2
d
r
Y
s
d
s
+
∫
τ
T
‍
e
-
n
∫
τ
s
‍
v
r
2
d
r
g
(
s
,
y
s
n
,
z
s
n
)
d
s
]
.
For the first two terms within the bracket on the right-hand side of (34), with the property of the vague convergence for the distribution functions, it is easily seen that
(35)
e
-
n
∫
τ
T
‍
v
r
2
d
r
Y
T
+
n
∫
τ
T
‍
v
s
2
e
-
n
∫
τ
s
‍
v
r
2
d
r
Y
s
d
s
⟶
Y
τ
,
P-a
.
s
.
,
and then, by dominated convergence, it converges in mean square; that is,
(36)
𝔼
[
(
e
-
n
∫
τ
T
‍
v
r
2
d
r
Y
T
+
n
∫
τ
T
‍
v
s
2
e
-
n
∫
τ
s
‍
v
r
2
d
r
Y
s
d
s
-
Y
τ
)
2
]
⟶
0
.
Now, we come to treat the third term. From the assumption (H2), we can deduce that
(37)
∫
τ
T
‍
e
-
n
∫
τ
s
‍
v
r
2
d
r
|
g
(
s
,
y
s
n
,
z
s
n
)
|
d
s
≤
a
ψ
∫
τ
T
‍
e
-
n
∫
τ
s
‍
v
r
2
d
r
v
s
|
z
s
n
|
d
s
+
∫
τ
T
‍
e
-
n
∫
τ
s
‍
v
r
2
d
r
(
|
g
(
s
,
0,0
)
|
+
a
φ
u
(
s
)
|
y
s
n
|
+
b
φ
u
s
+
b
ψ
v
s
)
d
s
.
For the integrand of the second integration term on the right hand of (37), it is dominated by
(38)
P
t
:
=
|
g
(
t
,
0,0
)
|
+
a
φ
u
(
t
)
(
|
y
t
1
|
+
|
Y
t
|
)
+
b
φ
u
t
+
b
ψ
v
t
.
Combining the assumption (H1), and the fact that(
y
t
1
)
t
∈
[
0
,
T
] and (
Y
t
)
t
∈
[
0
,
T
]belong to the space𝒮
2
(
0
,
T
;
R
), we can obtain that this term converges to zero almost surely with respect to probabilityP, by dominated convergence theorem, and then
(39)
𝔼
[
(
|
g
(
s
,
0,0
)
|
+
a
φ
u
(
s
)
|
y
s
n
|
+
b
φ
u
s
+
b
ψ
v
s
)
d
s
∫
τ
T
‍
)
2
(
∫
τ
T
‍
e
-
n
∫
τ
s
‍
v
r
2
d
r
(
|
g
(
s
,
0,0
)
|
+
a
φ
u
(
s
)
|
y
s
n
|
+
b
φ
u
s
+
b
ψ
v
s
)
d
s
∫
τ
T
‍
)
2
]
⟶
0
.
Applying Hölder’s inequality to the first term on the right hand of (37), we can get
(40)
∫
τ
T
‍
e
-
n
∫
τ
s
‍
v
r
2
d
r
v
s
|
z
s
n
|
d
s
≤
1
2
n
(
∫
τ
T
‍
|
z
s
n
|
2
d
s
)
1
/
2
.
Thus, from Proposition 10, it is easy to obtain the following convergence:
(41)
𝔼
[
(
∫
τ
T
‍
e
-
n
∫
τ
s
‍
v
r
2
d
r
v
s
|
z
s
n
|
d
s
)
2
]
≤
1
2
n
C
⟶
0
,
and then
(42)
𝔼
[
(
∫
τ
T
‍
e
-
n
∫
τ
s
‍
v
r
2
d
r
|
g
(
s
,
y
s
n
,
z
s
n
)
|
d
s
)
2
]
⟶
0
.
Consequently, using Jensen’s inequality and the property of conditional expectation, we have𝔼
[
(
y
~
τ
n
-
Y
τ
)
2
]
→
0.
According to the uniqueness of the solutions for BSDE (17) and the definition (32), we can obtainy
t
n
=
y
~
t
n, for all t
∈
[
0
,
T
], P-a.s., and y
τ
=
Y
τ. By section theorem, we have, P-a.s.,
(43)
y
t
=
Y
t
,
∀
t
∈
[
0
,
T
]
.
Therefore, ifT
<
∞, thatY
t
-
y
t
nuniformly converges to zero intalmost surely with respect to probabilityP, is the immediate result of Dini’s theorem. OtherwiseT
=
∞, since the increasing sequence of the continuous process(
Y
t
-
y
t
n
)
t
∈
[
0
,
T
]has the same value0atT; then almost surely, for anynandϵ
>
0, we can choose a real numberM, which may depend only onϵandω, such that ift
>
M, then
(44)
|
Y
t
-
y
t
n
|
≤
|
Y
t
-
y
t
1
|
≤
ϵ
.
On the other hand, by Dini’s theorem,(
Y
t
-
y
t
n
)
t
∈
[
0
,
T
]converges uniformly to zero almost surely on the interval[
0
,
M
]. So we can choose a numberNdepending only onϵ and ωsuch that ifn
>
N, then
(45)
|
Y
t
-
y
t
n
|
≤
ϵ
,
∀
t
∈
[
0
,
M
]
.
Thus,(
Y
t
-
y
t
n
)
t
∈
[
0
,
T
]uniformly converges to zero on the whole interval[
0
,
T
]almost surely with respect to probabilityP. Noticing the fact that|
Y
t
-
y
t
n
|
≤
|
Y
t
|
+
|
y
t
1
|, we can obtain the desired result by dominated convergence theorem. The proof is completed.After that, we can get the following proposition about the two sequences of(
z
t
n
)
t
∈
[
0
,
T
] and (
A
t
n
)
t
∈
[
0
,
T
]parameterized byn.Proposition 12.
The processes(
z
t
n
)
t
∈
[
0
,
T
] and (
A
t
n
)
t
∈
[
0
,
T
], at least their subsequences, are the Cauchy sequences inℋ
2
(
0
,
T
;
ℝ
d
) and 𝒮
2
(
0
,
T
;
ℝ
), respectively.Proof.
Applying Itô’s formula to|
y
t
n
-
y
t
m
|
2on[
0
,
T
], we can obtain
(46)
|
y
0
n
-
y
0
m
|
2
+
∫
0
T
‍
|
z
s
n
-
z
s
m
|
2
d
s
=
2
∫
0
T
‍
(
y
s
n
-
y
s
m
)
(
g
(
s
,
y
s
n
,
z
s
n
)
-
g
(
s
,
y
s
m
,
z
s
m
)
)
d
s
+
2
∫
0
T
‍
(
y
s
n
-
y
s
m
)
d
(
A
s
n
-
A
s
m
)
-
2
∫
0
T
‍
(
y
s
n
-
y
s
m
)
(
z
s
n
-
z
s
m
)
d
B
s
≤
2
∫
0
T
‍
(
y
s
n
-
y
s
m
)
(
g
(
s
,
y
s
n
,
z
s
n
)
-
g
(
s
,
y
s
m
,
z
s
m
)
)
d
s
+
2
∫
0
T
‍
(
Y
s
-
y
s
n
)
d
A
s
m
+
2
∫
0
T
‍
(
Y
s
-
y
s
m
)
d
A
s
n
-
2
∫
0
T
‍
(
y
s
n
-
y
s
m
)
(
z
s
n
-
z
s
m
)
d
B
s
.
Due to the fact that the part of Itô integration is uniformly integrable martingale, we have
(47)
𝔼
∫
0
T
‍
|
z
s
n
-
z
s
m
|
2
d
s
≤
2
𝔼
∫
0
T
‍
(
y
s
n
-
y
s
m
)
(
g
(
s
,
y
s
n
,
z
s
n
)
-
g
(
s
,
y
s
m
,
z
s
m
)
)
d
s
+
2
𝔼
∫
0
T
‍
(
Y
s
-
y
s
n
)
d
A
s
m
+
2
𝔼
∫
0
T
‍
(
Y
s
-
y
s
m
)
d
A
s
n
.
As for the last two terms of the above inequality, Propositions 10 and 11 lead to the fact that ifm
,
n
→
∞, then
(48)
𝔼
∫
0
T
‍
(
Y
s
-
y
s
n
)
d
A
s
m
+
𝔼
∫
0
T
‍
(
Y
s
-
y
s
m
)
d
A
s
n
⟶
0
.
Next, we will show that, asm
,
n
→
∞,
(49)
𝔼
∫
0
T
‍
(
y
s
n
-
y
s
m
)
(
g
(
s
,
y
s
n
,
z
s
n
)
-
g
(
s
,
y
s
m
,
z
s
m
)
)
d
s
⟶
0
.
Because the generatorgsatisfies the assumption (H2), by Hölder’s inequality, we have
(50)
𝔼
∫
0
T
‍
|
(
y
s
n
-
y
s
m
)
(
g
(
s
,
y
s
n
,
z
s
n
)
-
g
(
s
,
y
s
m
,
z
s
m
)
)
|
d
s
≤
𝔼
∫
0
T
‍
|
y
s
n
-
y
s
m
|
(
a
φ
u
s
|
y
s
n
-
y
s
m
|
+
b
φ
u
s
+
a
ψ
v
s
|
z
s
n
-
z
s
m
|
+
b
ψ
v
s
)
d
s
≤
a
ψ
(
𝔼
[
∫
0
T
‍
|
z
s
n
-
z
s
m
|
2
d
s
]
)
1
/
2
×
(
𝔼
[
∫
0
T
‍
|
y
s
n
-
y
s
m
|
2
v
s
2
d
s
]
)
1
/
2
+
b
ψ
𝔼
∫
0
T
‍
|
y
s
n
-
y
s
m
|
v
s
d
s
+
a
φ
𝔼
[
sup
⁡
0
≤
s
≤
T
|
y
s
n
-
y
s
m
|
2
]
∫
0
T
‍
u
s
d
s
+
b
φ
𝔼
∫
0
T
‍
|
y
s
n
-
y
s
m
|
u
s
d
s
.
According to Proposition 10 and the algebraic inequality, we can conclude
(51)
𝔼
∫
0
T
|
(
y
s
n
-
y
s
m
)
(
g
(
s
,
y
s
n
,
z
s
n
)
-
g
(
s
,
y
s
m
,
z
s
m
)
)
|
d
s
‍
≤
2
C
2
(
𝔼
[
∫
0
T
‍
|
y
s
n
-
y
s
m
|
2
v
s
2
d
s
]
)
1
/
2
+
b
φ
𝔼
∫
0
T
‍
|
y
s
n
-
y
s
m
|
u
s
d
s
+
a
φ
𝔼
[
sup
⁡
0
≤
s
≤
T
|
y
s
n
-
y
s
m
|
2
]
∫
0
T
‍
u
s
d
s
+
b
ψ
𝔼
∫
0
T
‍
|
y
s
n
-
y
s
m
|
v
s
d
s
.
Now setG
s
=
|
y
s
1
|
+
|
Y
s
|,H
s
=
4
G
s
2
v
s
2, and F
s
=
2
G
s
u
s; then, for anym,n
≥
1, we have
(52)
v
s
2
|
y
s
n
-
y
s
m
|
2
≤
H
s
,
u
s
|
y
s
n
-
y
s
m
|
≤
F
s
,
s
∈
[
0
,
T
]
;
𝔼
[
∫
0
T
‍
H
s
d
s
]
≤
4
𝔼
[
sup
⁡
0
≤
s
≤
T
|
G
s
|
2
]
∫
0
T
‍
v
s
2
d
s
<
+
∞
;
𝔼
[
∫
0
T
‍
F
s
d
s
]
≤
2
(
𝔼
[
sup
⁡
0
≤
s
≤
T
|
G
s
|
2
]
)
1
/
2
∫
0
T
‍
u
s
d
s
<
+
∞
.
The first two terms of the right-hand side of (51) converge to zero by the Lebesgue dominated theorem. And Proposition 11 implies that(
y
t
n
)
t
∈
[
0
,
T
]is a Cauchy sequence in𝒮
2
(
0
,
T
;
ℝ
); then the third term converges to zero. The convergence of the last term can be proved in a similar way to the second one.
Now, coming back to the inequality (47), we can conclude that 𝔼
∫
0
T
‍
|
z
s
n
-
z
s
m
|
2
d
s
→
0. This meansthat
(
z
t
n
)
t
∈
[
0
,
T
]is a Cauchy sequence inℋ
2
(
0
,
T
;
ℝ
d
), and we denoted its limit by(
z
t
)
t
∈
[
0
,
T
].
From (17), we know that
(53)
A
t
n
-
A
t
m
=
y
0
n
-
y
0
m
-
∫
0
t
‍
(
g
(
s
,
y
s
n
,
z
s
n
)
-
g
(
s
,
y
s
m
,
z
s
m
)
)
d
s
+
∫
0
t
‍
(
z
s
n
-
z
s
m
)
d
B
s
,
and then, from the basic algebraic inequality and BDG’s inequality, we can get
(54)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
A
t
n
-
A
t
m
|
2
]
≤
3
𝔼
|
y
0
n
-
y
0
m
|
2
+
3
𝔼
∫
0
T
‍
|
z
s
n
-
z
s
m
|
2
d
s
+
3
𝔼
[
(
∫
0
T
‍
|
g
(
s
,
y
s
n
,
z
s
n
)
-
g
(
s
,
y
s
m
,
z
s
m
)
|
d
s
)
2
]
.
In order to show that, whenm
,
n
→
∞, the limit of the third term of the right-hand side of (54) is zero, we only need to show that ifn
→
∞, then
(55)
𝔼
[
(
∫
0
T
‍
|
g
(
s
,
y
s
n
,
z
s
n
)
-
g
(
s
,
y
s
,
z
s
)
|
d
s
)
2
]
⟶
0
.
Because(
z
t
n
)
t
∈
[
0
,
T
]is a Cauchy sequence inℋ
2
(
0
,
T
;
ℝ
d
), there is at least a subsequence(
z
t
n
k
)
t
∈
[
0
,
T
]such thatd
P
×
d
t-a.e., z
t
n
k
→
z
t, and z
˘
t
:
=
sup
⁡
k
≥
1
|
z
t
n
k
|
∈
ℋ
2
(
0
,
T
;
ℝ
d
). For convenience, we denote the subsequence by(
z
t
n
)
t
∈
[
0
,
T
]itself. According to the assumption (H2), we can deduce that
(56)
|
g
(
s
,
y
s
n
,
z
s
n
)
-
g
(
s
,
y
s
,
z
s
)
|
≤
a
φ
u
s
|
y
s
n
-
y
s
|
+
b
φ
u
s
+
a
ψ
v
s
|
z
s
n
-
z
s
|
+
b
ψ
v
s
.
The right-hand side of the above inequality is dominated by
(57)
R
s
:
=
a
φ
u
s
G
s
+
b
φ
u
s
+
a
ψ
v
s
(
z
˘
s
+
z
s
)
+
b
ψ
v
s
.
It is easy to check that𝔼
[
(
∫
0
T
‍
R
s
d
s
)
2
]
<
∞. Then the convergence of (55) is a direct consequence of the Lebesgue dominated convergence theorem.
From the above argument and Proposition11, we can assert that(
A
t
n
)
t
∈
[
0
,
T
]is also a Cauchy sequence in𝒮
2
(
0
,
T
;
ℝ
)with a unique limit(
A
t
)
t
∈
[
0
,
T
]. The proof is completed.Proof of Theorem8.
From the procedure of the proof of Proposition12, we know that
(58)
∫
t
T
‍
g
(
s
,
y
s
n
,
z
s
n
)
d
s
⟶
∫
t
T
‍
g
(
s
,
y
s
,
z
s
)
d
s
,
uniformly on  [0,T],  in mean square,  that is,
(59)
𝔼
[
(
sup
⁡
0
≤
t
≤
T
|
∫
t
T
g
(
s
,
y
s
n
,
z
s
n
)
d
s
-
∫
t
T
g
(
s
,
y
s
,
z
s
)
d
s
|
)
2
]
⟶
0
.
And, by the property of Itô’s integration, BDG’s inequality, and Proposition 12, we also have
(60)
∫
t
T
z
s
n
d
B
s
⟶
∫
t
T
‍
z
s
d
B
s
,
‍
uniformly on [0,T],  in mean square,  that is,
(61)
𝔼
[
sup
⁡
0
≤
t
≤
T
|
∫
t
T
‍
z
s
n
d
B
s
-
∫
t
T
‍
z
s
d
B
s
|
2
]
≤
C
𝔼
[
∫
0
T
‍
|
z
s
n
-
z
s
|
2
d
s
]
⟶
0
.
Then combining the above convergence and the fact that the sequences(
y
t
n
)
t
∈
[
0
,
T
] and (
A
t
n
)
t
∈
[
0
,
T
]themselves or their subsequences converge to(
y
t
)
t
∈
[
0
,
T
]and(
A
t
)
t
∈
[
0
,
T
]uniformly on[
0
,
T
], in mean square, respectively, we can obtain the following equation:
(62)
y
t
=
Y
T
+
∫
t
T
g
(
s
,
y
s
,
z
s
)
d
s
-
∫
t
T
‍
z
s
d
B
s
+
A
T
-
A
t
,
y
t
=
Y
T
+
∫
t
T
g
(
s
,
y
s
,
z
s
)
d
s
-
∫
t
T
‍
z
s
d
B
s
t
∈
[
0
,
T
]
.
Notice that(
A
t
n
)
t
∈
[
0
,
T
]is an increasing process withA
0
n
=
0; then its limit(
A
t
)
t
∈
[
0
,
T
]will preserve this property. In fact, we have proved the first part of Theorem 8, because, according to (43), theg-supermartingale(
Y
t
)
t
∈
[
0
,
T
]coincides with the first component(
y
t
)
t
∈
[
0
,
T
]of the solution for the BSDE (62). And finally, the uniqueness of the decomposition ofg-supermartingale follows from the uniqueness of the decomposition ofg-supersolution. The proof is completed.Now, in addition, if we assume thatgis independent ofy, then we can write the decomposition of Doob-Meyer’s type forg-supermartingale in a more clear sense like the classical martingale theory.Corollary 13.
Letgbe independent ofyand satisfy the conditions (H1) and (H2). If(
X
t
)
t
∈
[
0
,
T
]is a continuousg-supermartingale in𝒮
2
(
0
,
T
;
ℝ
), then it has the following decomposition:
(63)
X
t
=
M
t
-
A
t
,
where(
M
t
)
t
∈
[
0
,
T
]is ag-martingale and(
A
t
)
t
∈
[
0
,
T
]is an increasing process which belongs to𝒜
2
(
0
,
T
;
ℝ
).Proof.
By Theorem8, ag-supermartingale(
X
t
)
t
∈
[
0
,
T
]on[
0
,
T
]has the following form. There exists a pair of processes(
z
t
,
A
t
)
t
∈
[
0
,
T
]such that
(64)
X
t
=
X
T
+
∫
t
T
g
(
s
,
z
s
)
d
s
-
∫
t
T
z
s
d
B
s
+
A
T
-
A
t
,
X
t
=
X
T
+
∫
t
T
g
(
s
,
z
s
)
d
s
-
∫
t
T
z
s
d
B
s
t
∈
[
0
,
T
]
.
We setM
t
=
X
t
+
A
t; then
(65)
M
t
=
X
T
+
A
T
+
∫
t
T
‍
g
(
s
,
z
s
)
d
s
-
∫
t
T
‍
z
s
d
B
s
,
t
∈
[
0
,
T
]
.
Obviously, the pair of the processes(
M
t
,
z
t
)
t
∈
[
0
,
T
]is a solution of the BSDE with the terminal dataX
T
+
A
Tand the generatorg. Definition 6 implies thatM
tis ag-martingale. The proof is completed.



---

*Source: 743508-2014-03-02.xml*