Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 671643, 6 pages
doi:10.1155/2009/671643
Research Article
Monotonic Limit Properties for Solutions of BSDEs
with Continuous Coefficients
ShengJun Fan, Xing Song, and Ming Ma
College of Sciences, China University of Mining & Technology, Xuzhou, Jiangsu 221008, China
Correspondence should be addressed to ShengJun Fan, f s j@126.com
Received 7 August 2009; Accepted 30 December 2009
Recommended by Andrew Rosalsky
This paper investigates the monotonic limit properties for the minimal and maximal solutions of
certain one-dimensional backward stochastic differential equations with continuous coefficients.
Copyright q 2009 ShengJun Fan et al. 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.
1. Introduction
Let Ω, F, P be a probability space carrying a standard d-dimensional Brownian motion
Bt t≥0 . Fix a terminal time T > 0, let Ft t≥0 be the natural σ-algebra generated by Bt t≥0 ,
and assume FT F. For every positive integer n, we use | · | to denote norm of Euclidean
space Rn . For t ∈ 0, T , let L2 Ω, Ft , P denote the set of all Ft -measurable random variables
ξ such that E|ξ|2 < ∞. Let L2F 0, T ; Rn denote the set of Ft -progressively measurable Rn valued processes {Xt , t ∈ 0, T } such that
T
1/2
2
E |Xt | dt
< ∞.
X2 1.1
0
This paper is concerned with the following one-dimensional BSDE:
yt ξ T
t
g s, ys , zs ds −
T
zs · dBs ,
t ∈ 0, T ,
1.2
t
where the random function gω, t, y, z : Ω × 0, T × R × Rd → R is progressively measurable
for each y, z in R × Rd , termed the generator of the BSDE1, and ξ is an FT -measurable
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random variables termed the terminal condition. The triple ξ, T, g is called the parameters
of the BSDE1. In this paper, for each ξ, T, g, by solution to the BSDE1 we mean a
pair of processes y· , z· in L2F 0, T ; R1
d which satisfies the BSDE1 and y is a continuous
process. Such equations, in nonlinear case, have been introduced by Pardoux and Peng 1;
they established an existence and uniqueness result of solutions of the BSDE1 under the
Lipschitz assumption of the generator g. Since then, these equations and their generalizations
have been the subject of a great number of investigations, such as 2, 3. Particularly, Lepeltier
and San Martin 4 obtained the following result when the generator g is only continuous
with a linear growth.
Proposition 1.1 see 4, Theorem 1. Assume that the generator g satisfies
H1 linear growth: there exists K < ∞, for all ω, t, y, z, |gω, t, y, z| ≤ K1 |y| |z|;
H2 for fixed ω, t, gω, t, ·, · is continuous.
Then, if ξ ∈ L2 Ω, FT , P , the BSDE(1) has a unique minimal solution Y · , Z · and a unique
maximal solution Y · , Z · , which means that both Y · , Z· and Y · , Z · are the solution of 1.2, and
for any other solution Y· , Z· of 1.2 one has Y · ≤ Y· ≤ Y · For convenience, for each t ∈ 0, T , one
g
g
denoted Y t by Et,T ξ, and Y t by Et,T ξ.
This paper will work on the assumptions H1 and H2 and investigate the monotonic
g
g
limit properties on the operators Et,T · and Et,T ·.
2. Main Results
In this section, we always assume that the generator g satisfies assumptions H1 and H2.
The following Theorem 2.1 and Remark 2.2 are the main results of this paper.
Theorem 2.1. Assume that the generator g satisfies assumptions (H1) and (H2). Let t ∈ 0, T ,
ξn ∈ L2 Ω, Ft , P , n ∈ N, and E|ξ|2 < ∞.
If ξn ↑ ξ, P − a.s., then for all s ∈ 0, t,
g
g
g
lim ↑ Es,t ξn Es,t lim ξn Es,t ξ,
n→∞
n→∞
P − a.s.
2.1
P − a.s.
2.2
If ξn ↓ ξ, P − a.s., then for all s ∈ 0, t,
g
g
g
lim ↓ Es,t ξn Es,t lim ξn Es,t ξ,
n→∞
n→∞
Remark 2.2. If the condition “ξn ↑ ξ” in Theorem 2.1 is replaced by “ξn ↓ ξ”, the conclusion
of the first part of Theorem 2.1 does not hold in general. Similarly, the condition “ξn ↓ ξ” in
Theorem 2.1 cannot be replaced by “ξn ↑ ξ” in general. For example, we consider the BSDE
with
g u, y, z 7y6/7 ,
ξ 0.
2.3
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3
It is easy to see that both
yr , zr
r∈0,t
Yr , Zr r∈0,t t − r7 , 0
0, 0r∈0,t ,
r∈0,t
2.4
are solutions of BSDE
yr t
r
7yu6/7 du
−
t
r ∈ 0, t.
zu dBu ,
2.5
r
For each n ∈ N, set ξn 1/n, then ξn ∈ L2 Ω, Ft , P , E|ξ|2 < ∞ and ξn ↓ ξ, P − a.s. However,
one can verify that for each s ∈ 0, t,
g
Es,t ξn 1
t−s
√
7
n
7
.
2.6
Consequently,
g
g
lim Es,t ξn Ys / Es,t ξ ≤ 0.
n→∞
2.7
So, the conclusion of the first part of Theorem 2.1 does not hold.
In order to prove Theorem 2.1, we need the following lemmas. Lemma 2.3 is actually
a direct corollary of Theorem 1.1 in 5.
Lemma 2.3. Assume that the generator g satisfies assumptions (H1) and (H2). Let t ∈ 0, T and
ξ, ξ ∈ L2 Ω, Ft , P . If ξ ≤ ξ , P − a.s., then
∀s ∈ 0, t,
∀s ∈ 0, t,
g
g Es,t ξ ≤ Es,t ξ ,
g
g Es,t ξ ≤ Es,t ξ ,
P − a.s.,
2.8
P − a.s.
From the procedure of the proof of Theorem 2.1 in 4, we can obtain the following
Lemma 2.4.
Lemma 2.4. If the function g satisfies (H1) and (H2), and one sets
g
m
t, y, z :
g m t, y, z :
inf
gt, u, v m y − u |z − v| ,
sup
gt, u, v − m y − u |z − v| ,
u,v∈Q1
d
u,v∈Q1
d
2.9
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then for any m > K, g
m
and g m are Lipschitz functions with constant m, that is, for any y1 , y2 ∈
R, z1 , z2 ∈ R and t ∈ 0, T ,
d
g t, y1 , z1 − g t, y2 , z2 ≤ m y1 − y2 |z1 − z2 | ,
m
m
2.10
g t, y1 , z1 − g t, y2 , z2 ≤ m y1 − y2 |z1 − z2 | .
m
m
m
m
m
Moreover, let t ∈ 0, T and ξ ∈ L2 Ω, Ft , P , and let Y m
r , Z r r∈0,t and Y r , Z r r∈0,t
be the unique solutions of the BSDEs with parameters ξ, t, g and ξ, t, g m , respectively. For
m
g
m
g
m
m
convenience, from now on, we denoted Y m
s by Es,t ξ, and Y s by Es,t ξ for each s ∈ 0, t,
then for each s ∈ 0, t, we have
g
g
lim ↑ Es,tm ξ Es,t ξ,
m→∞
g
g
lim ↓ Es,tm ξ Es,t ξ,
m→∞
P − a.s.,
2.11
P − a.s.
Finally, the following Lemma 2.5 can be easily obtained by 6, Lemma 1.
Lemma 2.5. Let t ∈ 0, T , ξn ∈ L2 Ω, Ft , P , n ∈ N, and E|ξ|2 < ∞, and let the generators g, g
m
and g m be defined as that in Lemma 2.4.
If ξn ↑ ξ, P − a.s., then for all s ∈ 0, t and each m > K,
g
g
lim ↑ Es,t ξ Es,t lim ξ
m
n
m
n→∞
n→∞
n
g
Es,tm ξ,
P − a.s.
2.12
P − a.s.
2.13
If ξn ↓ ξ, P − a.s., then for all s ∈ 0, t and each m > K,
g
g
g
lim ↓ Es,tm ξn Es,tm lim ξn Es,tm ξ,
n→∞
n→∞
Now, we are in the position to prove Theorem 2.1.
Proof of Theorem 2.1. We only prove the first part of this theorem, in the same way, one can
complete the proof of the second part.
Since ξn ∈ L2 Ω, Ft , P and limn → ∞ ↑ ξn ξ, P − a.s., one knows that ξ ∈ Ft . Thus, by
g
2
E|ξ| < ∞, we have ξ ∈ L2 Ω, Ft , P , then by Proposition 1.1, for each s ∈ 0, t, both Es,t ξn g
g
and Es,t ξ are well defined. Moreover, according to Lemma 2.3, Es,t ξn is nondecreasing with
g
respect to n and bounded by Es,t ξ from above, so in the sense of “almost surely,” the limit
g
of the sequence Es,t ξn must exist. Thus, in order to complete the proof of Theorem 2.1, we
g
need only to prove that this limit is just Es,t ξ.
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5
Let functions g and g m be defined for each m > K as that in Lemma 2.4, then from
m
Lemmas 2.3 and 2.4 one deduce that for each n ∈ N, m > K and s ∈ 0, t,
g
g
g
g
g
g
g
g
g
g
0 ≥ Es,t ξn − Es,t ξ Es,t ξn − Es,tm ξn Es,tm ξn − Es,tm ξ Es,tm ξ − Es,t ξ
g
g
≥ Es,tm ξn − Es,tm ξ Es,tm ξ − Es,t ξ,
2.14
P − a.s.
Letting n → ∞ in 2.14, from Lemma 2.5 we get that for each m > K,
g
g
g
g
0 ≥ lim Es,t ξn − Es,t ξ ≥ Es,tm ξ − Es,t ξ,
n→∞
P − a.s.
2.15
Furthermore, letting m → ∞ in 2.15, from Lemma 2.4 we can easily deduce that for each
s ∈ 0, t,
g
lim E ξn n → ∞ s,t
g
Es,t ξ,
P − a.s.
2.16
The proof of Theorem 2.1 is completed.
According to Theorem 2.1, we can obtain the following theorem.
Theorem 2.6. Assume that the generator g satisfies assumptions (H1) and (H2). Let t ∈ 0, T and
2
ξn ∈ L2 Ω, Ft , P , n ∈ N, Let Eη < ∞ and E|ζ|2 < ∞.
2
n
n
If ξ ≥ ζ, P − a.s. n ∈ N with E lim ξ < ∞, then for all s ∈ 0, t,
n→∞
g
Es,t
lim ξ
g
≤ lim Es,t ξn ,
n
n→∞
n→∞
P − a.s.
2.17
2
If ξn ≤ η, P − a.s. n ∈ N with E lim ξn < ∞, then for all s ∈ 0, t,
n→∞
g
g
Es,t lim ξn ≥ lim Es,t ξn ,
n→∞
n→∞
P − a.s.
2.18
Proof. We only prove the first part of this theorem, the proof of the second part is similar.
Let us fix s ∈ 0, t. Since ξn ∈ L2 Ω, Ft , P , then limn → ∞ ξn ∈ Ft , and by the assumption
of this theorem one knows that
lim ξn ∈ L2 Ω, Ft , P ,
n→∞
g
thus by Proposition 1.1, Es,t limn → ∞ ξn is well defined.
2.19
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International Journal of Mathematics and Mathematical Sciences
We set Yn infξk , then Yn ∈ Ft , and ζ ≤ Yn ↑ limn → ∞ ξn , P − a.s. So for each n ∈ N,
k≥n
2 n
E|Yn | ≤ max E|ζ| , E lim ξ < ∞.
n→∞
2
2
2.20
Thus,
2.21
Yn ∈ L2 Ω, Ft , P ,
g
g
then Es,t Yn is also well defined. Since Yn ≤ ξn , P −a.s., by Lemma 2.3 we know that Es,t Yn ≤
g
Es,t ξn , P − a.s., then applying Theorem 2.1 to the random variable sequence {Yn }, we get
g
g
g
g
Es,t lim ξn Es,t lim Yn lim Es,t Yn ≤ lim Es,t ξn ,
n→∞
n→∞
n→∞
n→∞
P − a.s.
2.22
The proof is completed.
Acknowledgments
This work is supported by the National Natural Science Foundation of China no. 10971220
and Youth Foundation of China University of Mining and Technology no. 2007A029.
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