A note on BSDEs and SDEs with time advanced and delayed coefficients

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by  Shiqiu Zheng

A note on BSDEs and SDEs with time advanced and
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1
delayed coefficients
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2

n
Shiqiu Zheng1,2†, Gaofeng Zong3‡
a
J

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(1, College of Applied Sciences, Beijing University of Technology, Beijing 100124,China)
1
(2, College of Sciences, North China University of Science and Technology, Tangshan 063009,China)

] (3, School of Mathematics and Quantitative Economics,
R
Shandong University of Finance and Economics, Jinan 250014,China)
P
.
h
t
a Abstract: This paper introduces a class of backward stochastic differential equations (BS-
m
DEs), whose coefficients not only depend on the value of its solutions of the present but also
[
the past and the future. For a sufficiently small time delay or a sufficiently small Lipschitz

1 constant, the existence and uniqueness of such BSDEs is obtained. As the adjoint process, a
v
class of stochastic differential equations (SDEs) is introduced, whose coefficients also depend
6
0 on the present, the past and the future of its solutions. The existence and uniqueness of such
1 SDEs is proved for a sufficiently small time advance or a sufficiently small Lipschitz constant.
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A duality between such BSDEs and SDEs is established.
0
.
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0 Keywords: Backwardstochasticdifferentialequation,stochasticdifferentialequation,time
7 delayed coefficients, time advanced coefficients, comparison theorem
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:
v
AMS Subject Classification: 60H10
i
X
r
a 1 Introduction
Sincethegeneralbackwardstochasticdifferentialequations(BSDEs)wasintroducedbyPardoux
and Peng [8], the theory of BSDEs has gone through rapid development in many different areas,
such as stochastic control, partial differential equations, mathematical finance, etc. Peng and
Yang [11] introduced a type of BSDEs, whose coefficients depend not only on the values of
solutions of the present but also the future. Such BSDEs is always called anticipated BSDEs,
and has been further studied in many literatures (see [12][14][15][18], etc.). Delong and Imkeller
[5, 6] introduced a type of BSDEs whose coefficients depend not only on the values of solutions
of the present but also the past. Such time delayed BSDEs has been further studied in many
literatures (see [1][2][4][17], etc.). The two types of BSDEs have been applied in many problems
arising from finance and stochastic control. It is natural to consider the BSDEs with time
∗The first authoris supported byNational Natural ScienceFoundation of China (No. 11571024) andNatural
Science Foundation of Beijing (No. 1132008). The second author is supported by the National Natural Science
Foundation of China (No. 11501325).
†E-mail: [email protected] (S.Zheng).
‡E-mail: gf [email protected] (G. Zong).
¯
1
advanced and delayed coefficients, i.e. the coefficients of such BSDEs not only depends on the
value of its solutions of the present butalso the past and the future. In fact, Cheridito and Nam
[3] had studied BSDEs whose coefficients can depend the whole path of the solutions in a very
general case. In this paper, we study BSDEs with time advanced and delayed coefficients in a
more concrete case. We consider the coefficient has the form: g(t, y , z ),
t+r −l≤r≤u t+r −l≤r≤u
{ } { }
where l 0 and u 0 are the time delayed parameter and the time advanced parameter
≥ ≥
respectively. For a sufficiently small time delay l or a sufficiently small Lipschitz constant, the
existence and uniqueness of solution of such BSDEs is obtained. Our result is independent on
the terminal time T, this is different from the result in Delong and Imkeller [5] for time delayed
BSDEs.
Stochastic differential equations (SDEs) with time delayed coefficients is classic in SDEs
theory (see Mao [7]). Recently, to study stochastic maximum principle, Chen and Huang [2]
introduces a type of SDEs with time advanced coefficients, whose coefficients depend on the
future of its solutions. In this paper, we introduce a type of SDEs, whose coefficients not only
depend on the value of its solutions of the present but also the past and the future. We consider
the coefficient with the form: b(t, X ) and σ(t, X ), where l 0 and u 0
t+r −l≤r≤u t+r −l≤r≤u
{ } { } ≥ ≥
are the time delayed parameter and the time advanced parameter respectively. For a sufficiently
small time advance u or a sufficiently small Lipschitz constant, the existence and uniqueness of
solution of such SDEs is proved. A duality between such BSDEs and SDEs with time advanced
and delayed coefficients is obtained.
The purposeof this paper is to introduce BSDEs and SDEs with time advanced and delayed
coefficients, and find the duality between them. Our proofs of the existence and uniqueness
of such BSDEs and SDEs both use the contractive mapping method based on two estimates
for Itˆo’s type process, respectively, which is mainly inspired by Peng [10]. Following [10, 11],
we also obtain the continuous dependence property and comparison theorems for BSDEs and
SDEs with time advanced and delayed coefficients. From this paper, one can see an interesting
similarity between the results and arguments of such BSDEs and SDEs.
2 BSDEs with time advanced and delayed coefficients
In this paper, we consider a complete probability space (Ω, ,P) on which a d-dimensional
F
standardBrownianmotion(B ) isdefined. Let( ) denotethenaturalfiltrationgenerated
t t≥0 Ft t≥0
d
by (B ) , augmented by the P-null sets of . Let z denote its Euclidean norm, for z R .
t t≥0 F | | ∈
Let T > 0,l 0,u 0 be given constants. For s [0,l] and t [0,T + u], we define the
≥ ≥ ∈ ∈
following usual spaces:
([ s,t];Rd)= f(t) : Lebesgue measurable Rd-valued function defined on [ s,t] ;
L − { − }
([ s,t];Rd) = f(t): continuous function Rd-valued function defined on [ s,t] ;
C − { − }
L2( ;Rd) = ξ : -measurable Rd-valued random variable such that E ξ 2 < ;
t t
L2 (F0,t;Rd)={ ψF: Rd-valued predictable process such that E t ψ 2dt <| | ; ∞}
F { 0 | t| (cid:2)∞}(cid:3)
2(0,t;Rd) = ψ : continuous process in L2 (0,t;Rd) such that E[sup ψ 2] ;
SF { F R t∈[0,T]| t| ≤ ∞}
L2 ( s,t;Rd) = ψ : (ψ ) L2 (0,t;Rd) and (ψ ) ([ s,0];Rd) such that
F − { r r∈[0,t] ∈ F r r∈[−s,0] ∈ L −
0 f(r)2dr < ;
−s| | ∞}
R S2F(−s,t;Rd)= {ψ : (ψr)r∈[0,t] ∈SF2(0,t;Rd) and (ψr)r∈[−s,0] ∈ C([−s,0];Rd)}.
Note that when d = 1, we always denote L2( ;Rd) by L2( ) for convention and make the
t t
F F
same treatment for above notations of other spaces.
2
In this section, we consider a function g
g(ω,t,y,z) : Ω [0,T] ([ l,u]) ([ l,u];Rd) R,
× ×L − ×L − 7−→
suchthatforeach(y ,z ) L2 ( l,T+u) L2 ( l,T+u;Rd),(g(t, y , z ))
r r F F t+r −l≤r≤u t+r −l≤r≤u t∈[0,T]
∈ − × − { } { }
is a progressively measurable process. For the function g, we make the following assumptions:
(A1). There exist a constant K > 0 and a probability measure λ defined on [ l,u], such
• −
that for each t [0,T] and each (y ,z ),(y˜ ,z˜ ) L2 ( l,T +u) L2 ( l,T +u;Rd),
r r r r F F
∈ ∈ − × −
E g(t, y , z ) g(t, y˜ , z˜ )2
t+r −l≤r≤u t+r −l≤r≤u t+r −l≤r≤u t+r −l≤r≤u
| { } { } − { } { } |
u
KE (y y˜ 2+ z z˜ 2)λ(dr) ;
t+r t+r t+r t+r
≤ (cid:20)Z−l | − | | − | (cid:21)
(A2). g(t,0,0) L2 (0,T).
• ∈ F
Remark 1
Clearly, the following assumption (A1’) implies (A1):
•
– (A1’). There exist a constant K > 0 and a probability measure λ defined on [ l,u],
−
2 2
such that for each t [0,T] and each (y ,z ),(y˜ ,z˜ ) L ( l,T +u;) L ( l,T +
r r r r F F
∈ ∈ − × −
u;Rd),
g(t, y , z ) g(t, y˜ , z˜ )2
t+r −l≤r≤u t+r −l≤r≤u t+r −l≤r≤u t+r −l≤r≤u
| { } { } − { } { } |
u
KE (y y˜ 2+ z z˜ 2)λ(dr) ;
t+r t+r t+r t+r t
≤ (cid:20)Z−l | − | | − | |F (cid:21)
2
Is there a function g satisfying our setting? In fact, for each (y ,z ) L ( l,T +u)
r r F
• ∈ − ×
L2 ( l,T +u;Rd), let
F
−
g (t, y , z ) = (E[y ]+z ), t [0,T],
1 t+r −l≤r≤u t+r −l≤r≤u t+u t t−l
{ } { } |F ∈
and
u
g (t, y , z )= E (y +z )dr , t [0,T].
2 t+r −l≤r≤u t+r −l≤r≤u t+r t+r t
{ } { } (cid:20)Z−l |F (cid:21) ∈
By a predictable projection theorem (see Jacod and Shiryaev [7, page 23]), we can check
for each (y ,z ) L2 ( l,T +u) L2 ( l,T +u;Rd), g and g are both progressively
r r F F 1 2
∈ − × −
measurable and satisfy (A1) and (A2).
Given ξ S2(T,T +u),η L2 (T,T +u;Rd), in this section, we consider the following
t ∈ F t ∈ F
BSDE:
T T
Y = ξ + g(s, Y , Z )ds Z dB , t [0,T);
t T t { s+r}−l≤r≤u { s+r}−l≤r≤u − t s s ∈
 (Yt,Zt) = (Rξt,ηt), t ∈ [T,T +u] and (Yt,Zt)= (Y0R,0), t ∈ [−l,0); (1)
 (Yt,Zt) 2F( l,T +u) L2F( l,T +u;Rd).
∈ S − × −
The solution of BSDE (1) is a pair (Y ,Z ) satisfying (1). The coefficient g(s, , ) of BSDE (1)
t t
· ·
dependsonthevalueof solution on[s l,s+u].Itisclear thatthesolution of BSDE(1) depends
−
3
only on the parameter (g,T,ξ,η,l,u).
From Peng [10, Lemma 3.1], we can get the following Lemma 2.1.
Lemma 2.1 Suppose that ξ S2(T,T +u),η L2 (T,T +u;Rd) and g (s) L2 (0,T). Then
t ∈ F t ∈ F 0 ∈ F
the BSDE (1) with coefficient g (s) has a unique solution (Y,Z), and the following estimate
0
T β 2 T
Y 2e−βl+E ( Y 2+ Z 2)eβsds E ξ 2eβT + E g (s)2eβsds , (2)
0 s s 0
| | "Z−l 2| | | | # ≤ h| | i β "Z0 | | #
holds true for an arbitrary constant β > 0. We also have
T
E sup Y 2 CE ξ 2+ g (s)2ds , (3)
t T 0
"0≤t≤T | | # ≤ "| | Z0 | | #
where C > 0 is a constant depending only on T.
Proof. By Peng [10, Lemma 3.1], such BSDE has a unique solution, and for an arbitrary
constant β > 0 and t [0,T], we have
∈
T β 2 T
Y 2+E ( Y 2+ Z 2)eβ(s−t)ds E ξ 2eβT + E g (s)2eβ(s−t)ds .
t s s t t 0 t
| | " t 2| | | | |F # ≤ | | |F β " t | | |F #
Z h i Z
(4)
Since Y = Y , s [ l,0), we have
s 0
∈ −
0
Y 2 = Y 2e−βl+β Y 2eβsds. (5)
0 0 s
| | | | −l| |
Z
By the fact Z = 0, s [ l,0), (4) and (5), we can get (2). By BDG inequality and (2), we can
s
∈ −
get (3). This proof is complete. ✷
The following is the main result of this section.
Theorem 2.2 Suppose that ξ S2(T,T +u),η L2 (T,T +u;Rd), and g satisfies (A1) and
t ∈ F t ∈ F
(A2). If there exists a constant β 2 such that 2Keβl < 1, then BSDE (1) has a unique solution.
≥ β
2
Proof. We will prove this theorem using a contractive method. Clearly, L ( l,T + u)
F
− ×
L2 ( l,T +u;Rd) is a Banach space with the norm:
F
−
1
T+u 2
(, ) = E ( 2+ 2)eβsds ,
β
k · · k " −l |·| |·| #
Z
2
where β > 0 is a constant. By Fubini’s theorem, (A1) and (A2), for each (y ,z ) L ( l,T +
t t F
∈ −
u) L2 ( l,T +u;Rd), we have
F
× −
T
E g(s, y , z )2ds
s+r −l≤r≤u s+r −l≤r≤u
"Z0 | { } { } | #
4
T T u
2E g(s,0,0)2ds +2KE (y 2+ z 2)λ(dr)ds
s+r s+r
≤ "Z0 | | # "Z0 Z−l | | | | #
T u T+r
2E g(s,0,0)2ds +2KE (y 2+ z 2)dvλ(dr)
v v
≤ "Z0 | | # "Z−lZr | | | | #
T T+u
2E g(s,0,0)2ds +2KE (y 2+ z 2)dv
v v
≤ "Z0 | | # "Z−l | | | | #
< .
∞
Thus by Lemma 2.1, we can define a mapping φ from L2 ( l,T +u) L2 ( l,T +u;Rd) into
F F
− × −
itself by setting (Y,Z) := φ(y,z), where (Y,Z) is the solution of the BSDE (1) with coefficient
g(s, y , z ). For any (y1,z1),(y2,z2) L2 ( l,T +u) L2 ( l,T +u;Rd),
s+r −l≤r≤u s+r −l≤r≤u t t t t F F
{ } { } ∈ − × −
let
(Yi,Zi) := φ(yi,zi), i= 1,2.
We set (Yˆ,Zˆ ):= (Y1 Y2,Z1 Z2) and (yˆ,zˆ) := (y1 y2,z1 z2). By Lemma 3.1, Fubini’s
t t t t t t t t t t t t
− − − −
theorem and (A1), we can deduce, for an arbitrary constant β > 0,
T+u β
E ( Yˆ 2+ Zˆ 2)eβsds
s s
" −l 2| | | | #
Z
2 T
E g(s, y1 , z1 ) g(s, y2 , z2 )2eβsds
≤ β "Z0 | { s+r}−l≤r≤u { s+r}−l≤r≤u − { s+r}−l≤r≤u { s+r}−l≤r≤u | #
2 T u
E K (yˆ 2+ zˆ 2)λ(dr)eβsds
s+r s+r
≤ β " Z0 Z−l | | | | #
2 u T
= E K (yˆ 2+ zˆ 2)eβsdsλ(dr)
s+r s+r
β " Z−lZ0 | | | | #
2 u T+r
= E K e−r (yˆ 2+ zˆ 2)eβvdvλ(dr)
v v
β " −l r | | | | #
Z Z
2K T+u
eβlE (yˆ 2+ zˆ 2)eβvdv .
v v
≤ β " −l | | | | #
Z
Thus,ifthereexistsaconstantβ 2suchthat 2Keβl < 1,thenthereexistsaconstant0 < γ < 1,
≥ β
such that
T+u T+u
E (Yˆ 2+ Zˆ 2)eβsds γE (yˆ 2+ zˆ 2)eβsds .
s s s s
" −l | | | | #≤ " −l | | | | #
Z Z
Thus, (Yˆs,Zˆs) β < √γ (yˆs,zˆs) β. The proof is complete. ✷
k k k k
Remark 2
FromTheorem2.2,itisfollowsthatforasufficientlysmalllorasufficientlysmallK,BSDE
•
(1) has a unique solution. But the uniqueness and existence of solution of BSDE (1) may
benot true for arbitrary K and l (see Delong and Imkeller [5, Example 3.1]). Theorem 2.2
is an extension of Delong and Imkeller [5, Theorem 2.1], in which the coefficient depends
only on the values of the solution of past, but not on the future. It also generalize the
5
corresponding results in Peng and Yang [11] in some way, which introduce the BSDEs
whose coefficients depend only on the values of the solution of future, but not on the past,
under some different conditions.
From Theorem 2.2, we know the uniqueness and existence of solution of BSDE (1) is not
•
dependent on time horizon T, this is different from the result in Delong and Imkeller [5]
for delay BSDEs. Such phenomena had also been discovered in [4] for forward BSDE with
time delayed coefficient.
We give a simple example of BSDEs with time advanced and delayed coefficients.
Example 2.3 Let ξ L2( ), the time delay parameter l = T and Lipchitz constant K < 1 ,
∈ FT e2T
we consider
Y = ξ + T K(Y +E[Z ] Z )ds T Z dB , t [0,T);
t T t s−T s+u|Fs − s − t s s ∈
 (Yt,Zt) = (Rξ+Bt2−t,2Bt), t ∈ [T,T +u] andR (Yt,Zt) = (Y0,0), t ∈[−l,0); (6)
 (Yt,Zt) 2F( l,T +u) L2F( l,T +u;Rd).
∈S − × −
We firstly consider the delayed BSDE:
T T
Y1 = ξ + KY1 ds Z1dB , t [0,T]. (7)
t s−T s s
t − t ∈
Z Z
By Example 3.1 in Delong and Imkeller [5], we can let Z1 L2 (0,T;Rd) be the process ap-
s ∈ F
pearing in the following martingale representation of ξ:
T
ξ = E[ξ]+ Z1dB ,
s s
Z0
and
1 tK t
Y1 = − E[ξ]+ Z1dB , t [0,T],
t 1 TK s s ∈
− Z0
then (Y1,Z1) is a solution of BSDE (7). Now, we consider the advanced BSDE
t t
T T
Y2 = B2 T + K(E[Z2 ] Z2)ds Z2dB , t [0,T]. (8)
t T − t s+u|Fs − s − t s s ∈
Z Z
We can check (Y2,Z2) = (B2 t,2B ) is a solution of BSDE (8). Set (Y ,Z ) := (Y1,Z1)+
t t t t t t t t
−
(Y2,Z2), t [0,T], thenby Theorem2.2, wecan get (Y ,Z )is theuniquesolution of BSDE(6).
t t t t
∈
The following is a continuous dependence property of the BSDE (1).
Proposition 2.4 Let ξ ,ξ′ S2(T,T +u),η ,η′ L2 (T,T +u;Rd),ϕ ,ϕ′ L2 (0,T;R) and
t t ∈ F t t ∈ F t t ∈ F
g satisfies (A1) and (A2). Let f(s, , ) := g(s, , )+ϕ and f′(s, , ) := g(s, , )+ϕ′. Suppose
t t
· · · · · · · ·
that there exists a constant β 2 such that 4Keβl < 1. Let (Y,Z) and (Y′,Z′) be the solutions
≥ β
of BSDE (1) with coefficients f and f′, respectively, then for each t [0,T], we have
∈
T
Y Y′ 2+E (Y Y′ 2+ Z Z′ 2)ds
t t s s s s t
| − | " t | − | | − | |F #
Z
6
T T+u
CE ξ ξ′ 2+ ϕ ϕ′ 2ds+ (ξ ξ′ 2+ η η′)ds
T T s s s s s s t
≤ "| − | t | − | T | − | | − |F #
Z Z
t
+C (Y Y′ 2+ Z Z′ 2)ds, (9)
s s s s
t−l | − | | − |
Z
where C > 0 is a constant depending only on T,K,µ and l.
Proof. Set Yˆ = Y Y′, Zˆ = Z Z′, ξˆ = ξ ξ′ and ηˆ = η η′. By (4), (A1’) and Fu-
− − − −
bini’s theorem, we deduce
T β
Yˆ 2+E ( Yˆ 2+ Zˆ 2)eβ(s−t)ds
t s s t
| | " t 2| | | | |F #
Z
4 T
E ξ ξ′ 2eβT + E ϕ ϕ′ 2eβ(s−t)ds
≤ | T − T| |Ft β " t | s− s| |Ft#
h i Z
4 T
+ E g(s, Y , Z ) g(s, Y′ , Z′ )2eβ(s−t)ds
β " t | { s+r}−l≤r≤u { s+r}−l≤r≤u − { s+r}−l≤r≤u { s+r}−l≤r≤u | |Ft#
Z
4 T
E ξ ξ′ 2eβT + E ϕ ϕ′ 2eβ(s−t)ds
≤ | T − T| |Ft β " t | s− s| |Ft#
h i Z
4Keβl T+u
+ E (Yˆ 2+ Zˆ 2)eβ(s−t)ds .
s s t
β " t−l | | | | |F #
Z
Since there exists a constant β 2 such that 4Keβl < 1, from the above inequality, we can get
≥ β
(9). This proof is complete. ✷
In general, the comparison theorem of BSDE (1) may not true (see Delong and Imkeller [5]
and Peng and Yang [11]). But under some restrict condition, we have the following Proposition
2.4. For simplicity, this comparison theorem is established only for BSDE (1), whose coeffi-
cents are not independent on z. In fact, this method also works for BSDE (1) with coefficent
g(t, y ,z ) depending on the present value of the solution z.
t+r −l≤r≤u t
{ }
Proposition 2.5 Let ξ ,ξ′ S2(T,T + u),η ,η′ L2 (T,T + u;Rd) and g,g′ are both in-
t t ∈ F t t ∈ F
dependent on z and satisfy (A1) and (A2). Suppose that
(i) ξ ξ′ for each t [T,T +u];
t t
≥ ∈
(ii)g(s, y ) g′(s, y )foreachy ([ l,T+u]), andg′(s, y )
s+r −l≤r≤u s+r −l≤r≤u s+r −l≤r≤u
{ } ≥ { } ∈ C − { } ≥
g′(s, y′ ), if y y′ for each r [ l,T +u];
{ s+r}−l≤r≤u r ≥ r ∈ −
(iii) there exists a constant β 2 such that 2Keβl < 1.
≥ β
Let (Y,Z) and (Y′,Z′) be the solutions of BSDEs (1) with parameters (g,ξ,η) and (g′,ξ′,η′),
respectively. Then for each t [ l,T +u], we have Y Y′.
t t
∈ − ≥
Proof. We use an iteration method from Peng and Yang [10]. By Theorem 2.2, the BSDE
(1) with coefficient (g′(s, Y ),ξ′,η′) has a unique solution (Y1,Z1). By (ii) and com-
s+r −l≤r≤u
{ }
parison theorem (see Peng [10]), we have Y Y1. By Theorem 2.2, the BSDE (1) with
t t
≥
coefficient (g′(s, Y1 ),ξ′,η′) has a unique solution (Y2,Z2). Since Y Y1, by (ii)
{ s+r}−l≤r≤u t ≥ t
and comparison theorem, we have Y1 Y2. Similarly, for n > 2, the BSDE (1) with co-
t t
≥
efficient (g′(s, Yn−1 ),ξ′,η′) has a unique solution (Yn,Zn) and Yn−1 Yn. Setting
{ s+r }−l≤r≤u t ≥ t
7
Yˆn = Yn Yn−1, Zˆn = Zn Zn−1, by Lemma 2.1 and the same treatment in the proof of
− −
Theorem 2.2, we have
T+u β
E ( Yˆn 2+ Zˆn 2)eβsds
s s
" −l 2| | | | #
Z
2 T
E g′(s, Yn−1 ) g′(s, Yn−2 )2eβsds
≤ β "Z0 | { s+r }−l≤r≤u − { s+r }−l≤r≤u | #
2K T+u
eβlE Yˆn−1 2eβsds ,
≤ β " −l | s | #
Z
By (iii) and the above inequality, we can deduce that there exists a constant 0 < γ < 1, such
that
T+u T+u
E (Yˆn 2+ Zˆn 2)eβsds γn−1E (Yˆ1 2+ Zˆ1 2)eβsds
s s s s
" −l | | | | # ≤ " −l | | | | #
Z Z
By this inequality, we get that (Yn,Zn) is a Cauchy sequence in L2 ( l,T +u) L2 ( l,T +
n≥1 F F
− × −
u;Rd). Then by (A1), we can easily show the limit of (Yn,Zn) is the solution of BSDEs
n≥1
(1) with parameter (g′,ξ′,η′). Thus Yn Y′, dt dP a.s. In view of Y Y1, we have
t t
ց × − ≥
Y Y′, dt dP a.s. The proof is complete. ✷
≥ × −
Now, we give a slightly generalization of Theorem 2.2. We consider the function g :
g(ω,t,y,z) : Ω [0,T] ([ l,T +u]) ([ l,T +u];Rd) R,
× ×L − ×L − 7−→
suchthatforeach(y ,z ) L2 ( l,T+u) L2 ( l,T+u;Rd),(g(t, y , z ))
r r F F r −l≤r≤T+u r −l≤r≤T+u t∈[0,T]
∈ − × − { } { }
is a progressively measurable process. We make the following assumptions:
2
(A1”). There exists a constant K > 0, such that (y ,z ),(y˜ ,z˜ ) L ( l,T + u)
0 r r r r F
• ∀ ∈ − ×
L2 ( l,T +u;Rd) :
F
−
T
E g(t, y , z ) g(t, y˜ , z˜ )2dt
r −l≤r≤T+u r −l≤r≤T+u r −l≤r≤T+u r −l≤r≤T+u
"Z0 | { } { } − { } { } | #
T+u
K E y y˜ 2+ z z˜ 2dt ;
0 t t t t
≤ " −l | − | | − | #
Z
(A2”). g(t,0,0) L2 (0,T);
• ∈ F
Clearly, If g satisfies (A1) and (A2), then g satisfies (A1”) and (A2”), but the converse is
not ture. For examples, (y ,z ) L2 ( l,T +u) L2 ( l,T +u;Rd), let
r r F F
∀ ∈ − × −
T+u
g (t, y , z ) = E (y +z )dr , t [0,T].
3 t+r −l≤r≤u t+r −l≤r≤u r r t
{ } { } " −l |F # ∈
Z
Clearly, g is a martingale on [0,T], thus has a progressive measurable version. It satisfies (A1”)
3
and (A2”), but does not satisfies (A1).
8
Given ξ S2(T,T +u),η L2 (T,T +u;Rd), we consider the following BSDE:
t ∈ F t ∈ F
T T
Y = ξ + g(s, Y , Z )ds Z dB , t [0,T);
t T t { r}−l≤r≤T+u { r}−l≤r≤T+u − t s s ∈
 (Yt,Zt) = (Rξt,ηt), t ∈ [T,T +u], and (Yt,Zt) = (Y0R,0), t ∈ [−l,0); (10)
 (Yt,Zt) 2F( l,T +u) L2F( l,T +u;Rd).
∈ S − × −
The similar proof as Theorem 2.2 can give the following Theorem 2.6. We omit its proof here.
Theorem 2.6 Suppose that ξ S2(T,T + u),η L2 (T,T + u;Rd), and g satisfies (A1”)
t ∈ F t ∈ F
and (A2”). If there exists a constant β 2 such that 2K0eβ(T+l) < 1, then BSDE (10) has a
≥ β
unique solution.
Remark 3
From Theorem 2.6, it is follows that for a sufficiently small T +l or for a sufficiently small
•
K , BSDE (2) has a unique solution. It generalizes the corresponding results in Yang and
0
Elliott [14] in some way, which introduce the BSDEs whose coefficients depend only on
the values of the solution of future, but not on the past, under some different conditions.
Cheridito and Nam [3] introduced a BSDE whose coefficient depending the whole path
of solution on [0,T] in a very general case. Our result is for a BSDE whose coefficient
depending the whole path of solution [ l,T +u].
−
3 SDEs with time advanced and delayed coefficients
In this section, we consider the functions b and σ :
b(ω,t,x) : Ω [0,T] ([ l,u]) R,
× ×L − 7−→
σ(ω,t,x) : Ω [0,T] ([ l,u]) Rd,
× ×L − 7−→
2
such that for each x L ( l,T +u), (b(t, x )) and (σ(t, x ))
t F t+r −l≤r≤u t∈[0,T] t+r −l≤r≤u t∈[0,T]
∈ − { } { }
are both progressively measurable processes. we make the following assumptions:
(B1). There exist a constant K > 0 and a probability measure λ defined on [ l,u], such
1 1
• −
2
that for each t [0,T] and each x ,x˜ L ( l,T +u),
r r F
∈ ∈ −
E b(t, x ) b(t, x˜ )2+ σ(t, x ) σ(t, x˜ )2
t+r −l≤r≤u t+r −l≤r≤u t+r −l≤r≤u t+r −l≤r≤u
| { } − { } | | { } − { } |
h u i
K E x x˜ 2λ (dr) ;
1 t+r t+r 1
≤ (cid:20)Z−l| − | (cid:21)
(B2). b(t,0) L2 (0,T) and σ(t,0) L2 (0,T;Rd).
• ∈ F ∈ F
2
Given t [0,T] and θ (t l,t ), we consider the following SDE:
0 t F 0 0
∈ ∈ S −
t t
X = X + b(s, X )ds+ σ(s, X )dB , t (t ,T];
t t0 t0 { s+r}−l≤r≤u t0 { s+r}−l≤r≤u s ∈ 0
X = θ , t [t l,t ] and X = X , t (T,T +u]; (11)
 t t ∈R 0− 0 t T R ∈
 Xt ∈ SF2(t0,T +u),

9
whose solution is X satisfying (11). The coefficients b(t, ) and σ(t, ) of SDE (11)
{ t}t0−l≤t≤T+u · ·
both depend on the values of solution on [t l,t+u]. It is clear that the solution of SDE (11)
−
depends only on parameter (b,σ,T,θ,l,u).
Motivated by Peng [10, Lemma 3.1], we have the following Lemma 3.1.
Lemma 3.1 Given t [0,T] and θ 2 (t l,t ). Suppose that b (s),σ (s) L2 (0,T).
0 ∈ t ∈ SF 0 − 0 0 0 ∈ F
Then the SDE with coefficients b (s) and σ (s) has a unique solution X, and the following
0 0
estimate
β T+u
E[X 2e−β(T+u) ]+ E X 2e−βsds
T t s t
| | |F0 2 " t | | |F0#
Z0
T 2
X 2e−βt0 +E ( b (s)2+ σ (s)2)e−βsds , (12)
≤ | t0| " t β| 0 | | 0 | |Ft0#
Z0
holds true for arbitrary constant β > 0. We also have
T
E sup X 2 CE X 2+ (b (s)2+ σ (s)2)ds , (13)
"t0≤t≤T| t| #≤ "| t0| Zt0 | 0 | | 0 | #
where C > 0 is a constant depending only on T.
Proof. By the classic SDEs theory, this SDE has a unique solution. For an arbitrary constant
β > 0, applying Itˆo’s formula for X 2e−βs for s [t ,t], we can deduce for t [t ,T],
s 0 0
| | ∈ ∈
t
X 2e−βt0 +E ( β X 2+ σ (s)2)e−βsds
| t0| (cid:20)Zt0 − | s| | 0 | |Ft0(cid:21)
t
= E[X 2e−βt ] 2E X b (s)e−βsds
| t| |Ft0 − (cid:20)Zt0 s 0 |Ft0(cid:21)
β t 2 t
E X 2e−βt E X 2e−βsds E b (s)2e−βsds .
≥ h| t| |Ft0i− 2 (cid:20)Zt0 | s| |Ft0(cid:21)− β (cid:20)Zt0 | 0 | |Ft0(cid:21)
Then we have for t [t ,T],
0
∈
β t t 2
E[X 2e−βt ]+ E X 2e−βsds X 2e−βt0+E ( b (s)2 + σ (s)2)e−βsds ,
| t| |Ft0 2 (cid:20)Zt0 | s| |Ft0(cid:21) ≤| t0| (cid:20)Zt0 β| 0 | | 0 | |Ft0(cid:21)
(14)
Since X = X ,t [T,T +u], we have
t T
∈
T+u
X 2e−β(T+u) = X 2e−βT β X 2e−βsds. (15)
T T s
| | | | − T | |
Z
By (14) and (15), we can get (12). By BDG inequality and (12), we can get (13). ✷
The following is the main result of this section.
2
Theorem 3.2 Given t [0,T] and θ (t l,t ). Suppose that b and σ both satisfy
0 t F 0 0
∈ ∈ S −
(B1) and (B2). If there exists a constant β > 0 such that 2K1eβu 1+ 2 < 1, then SDE (11)
β β
(cid:16) (cid:17)
10