- Email: [email protected]
Reflected backward stochastic differential equations with time delayed generators
Statistics and Probability Letters 82 (2012) 979–990
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Reflected backward stochastic differential equations with time delayed generators✩ Qing Zhou a , Yong Ren b,∗ a
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
b
Department of Mathematics, Anhui Normal University, Wuhu 241000, China
article
info
Article history: Received 10 September 2011 Received in revised form 29 January 2012 Accepted 13 February 2012 Available online 21 February 2012
abstract In this paper, we establish the existence and uniqueness of the solution for a class of reflected backward stochastic differential equations with time delayed generator (RBSDEs with time delayed generator, in short) for a sufficiently small Lipschitz constant of the generator. © 2012 Elsevier B.V. All rights reserved.
MSC: 60H10 60G40 Keywords: Reflected backward stochastic differential equation Time delayed generator Fixed point theorem
1. Introduction The theory of nonlinear backward stochastic differential equations (BSDEs, in short) was developed by Pardoux and Peng (1990). Precisely, given a data (ξ , f ) consisting of a progressively measurable process f , so-called the generator, and a square integrable random variable ξ , they proved the existence and uniqueness of an adapted process (Y , Z ) solution to the following BSDE Y (t ) = ξ +
T
f (s, Y (s), Z (s))ds − t
T
Z (s)dW (s),
0 ≤ t ≤ T.
t
These equations have attracted great interest due to their connections with mathematical finance (see e.g. El Karoui et al., 1997a; El Karoui and Quenez, 1997b), stochastic control and stochastic games (see e.g. Hamadène, 2003; Hamadène and Lepeltier, 1995a,b). Furthermore, it was shown in various papers that BSDEs give the probabilistic representation for the solution (at least in the viscosity sense) of a large class of systems of semi-linear parabolic partial differential equations (PDEs in short) (see e.g. Pardoux, 1998, 1999; Pardoux and Peng, 1992; Peng, 1991).
✩ The work of Qing Zhou is supported by the National Natural Science Foundation of China (Nos 11001029 and 10971220) and the Fundamental Research Funds for the Central Universities (No. BUPT2009RC0705). The work of Yong Ren is supported by the National Natural Science Foundation of China (Nos 10901003 and 11126238), the Distinguished Young Scholars of Anhui Province (No. 1108085J08), the Key Project of Chinese Ministry of Education (No. 211077) and the Anhui Provincial Natural Science Foundation (No. 10040606Q30). ∗ Corresponding author. E-mail addresses: [email protected] (Q. Zhou), [email protected], [email protected] (Y. Ren).
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.02.012
980
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
Among the BSDEs, El Karoui et al. (1997) introduced a special class of reflected BSDEs (RBSDEs, in short), which is a BSDE but the solution is forced to stay above a lower barrier. In detail, a solution of such equations is a triple of processes (Y , Z , K ) satisfying that Y (t ) = ξ +
T
f (s, Y (s), Z (s))ds + K (T ) − K (t ) −
T
Z (s)dW (s),
Y (t ) ≥ S (t ),
(1.1)
t
t
where S, so-called the barrier or obstacle, is a given stochastic process. The role of the continuous non-decreasing process K Tis to push the state process upward with the minimal energy, in order to keep it above S; in this sense, it satisfies that (Y (t ) − S (t ))dK (t ) = 0. RBSDEs have been proven to be powerful tools in mathematical finance (see e.g. El Karoui and 0 Quenez, 1997b; Crépey and Moutoussi, 2008), the mixed game problems (see e.g. Cvitanic and Karatzas, 1996; Hamadène and Lepeltier, 2000), providing a probabilistic formula for the viscosity solution of an obstacle problem for a class of parabolic PDEs (see e.g. Cvitanic and Ma, 2001; El Karoui et al., 1997; Hamadène and Popier, 2011; Ren and Xia, 2006) and so on. On other interesting results on RBSDEs driven by a Brownian motion with different barrier conditions, one can see Hamadène (2002), Lepeltier and Xu (2005); Peng and Xu (2005). Recently, Delong and Imkeller (2010a) introduced the following BSDE Y (t ) = ξ +
T
f (s, Ys , Zs )ds − t
T
Z (s)dW (s),
0 ≤ t ≤ T,
t
where the generator f at time s depends arbitrary on the past values of a solution (Ys , Zs ) = (Y (s + u), Z (s + u))−T ≤u≤0 . This class of BSDEs is called as BSDEs with time delayed generators. In Delong and Imkeller (2010a), the authors proved the existence and uniqueness of a solution for the above equation. Moreover, in Delong and Imkeller (2010b), they established the existence and uniqueness result for BSDEs with time delayed generators driven by Brownian motions and Poisson random measures. In addition, they studied the Malliavin’s differentiability of the solution. For the applications of BSDEs with time delayed generators in insurance and finance, one can see Delong (2010). Very recently, Reis et al. (2011) extended the results of Delong and Imkeller (2010a,b) in Lp -spaces. Unlike Delong and Imkeller (2010a,b) and Reis et al. (2011), Peng and Yang (2009) introduce a new type of BSDEs, called as anticipated BSDEs, where the authors established the existence, uniqueness and comparison theorem of the solutions and the duality relations between them and stochastic differential delay equations. To our best knowledge, there is no result on RBSDEs with time delayed generators. To close the gap and as the fundamental step, in this paper, we aim to derive the existence and uniqueness of the solutions for RBSDEs with time delayed generators. We expect to give its applications in finance and study some other properties of the solution in our further study. Even for BSDEs with time delayed generators, as shown in Delong and Imkeller (2010a,b), the general existence and uniqueness results do not hold for arbitrary Lipschitz condition and arbitrary time horizon. Our results hold also under the condition of a sufficiently small Lipschitz constant of the generator. The paper is organized as follows. In Section 2, we propose some preliminaries and notations. The main results are given in Section 3. 2. Preliminaries and notations Let (Ω , F , P) be a probability space equipped with a standard Brownian motion W . For a fixed real number T > 0, we consider the filtration F := (Ft )0≤t ≤T which is generated by W and augmented by all P-null sets. The filtered probability space (Ω , F , F, P) satisfies the usual conditions. We shall work with the following spaces.
• Let L2−T (R) denote the space of measurable functions z : [−T , 0] → R satisfying that 0 |z (t )|2 dt < ∞. −T
• Let
S∞ −T (R)
denote the space of bounded, measurable functions y : [−T , 0] → R satisfying that
sup |y(t )|2 < ∞.
t ∈[−T ,0]
• Let L2 (R) denote the space of FT -measurable random variable ξ : Ω → R satisfying that E|ξ |2 < ∞.
• Let H2T (R) denote the space of F-predictable processes Z : Ω × [0, T ] → R satisfying that T E |Z (t )|2 dt < ∞. 0
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
981
• Finally, let S2T (R) denote the space of F-adapted, product measurable processes Y : Ω × [0, T ] → R satisfying that 2 E sup |Y (t )| < ∞. t ∈[0,T ]
The spaces H2T (R) and S2T (R) are respectively endowed with the norms
∥Z ∥
2 H2T
T
βt
e |Z (t )| dt
=E
2
and ∥Y ∥
0
2 S2T
=E
βt
sup e |Y (t )|
2
t ∈[0,T ]
,
with some β > 0. As usual, by λ we denote Lebesgue measure on ([−T , 0], B ([−T , 0])), where B ([−T , 0]) stands for the Borel sets of [−T , 0]. In the sequel, let us simply write S2 (R) × H2 (R) for S2T (R) × H2T (R). Now, we give three objects. The first one is the terminal value ξ . (i) ξ ∈ L2 (R). The second one is the ‘‘coefficient’’ f , 2 f : Ω × [0, T ] × S∞ −T (R) × L−T (R) → R,
which is a product measurable, F-adapted map such that
T
(ii) E 0 |f (t , 0, 0)|2 dt < ∞, (iii) for a probability measure α on ([−T , 0], B ([−T , 0])) and a positive constant L
|f (t , yt , zt ) − f (t , y˜ t , z˜t )|2 ≤ L
0
|y(t + u) − y˜ (t + u)|2 α(du) + −T
0
|z (t + u) − z˜ (t + u)|2 α(du) ,
−T
S∞ −T (R)
holds for P × λ-a.e. (ω, t ) ∈ Ω × [0, T ] and for any (yt , zt ), (˜yt , z˜t ) ∈ × L2−T (R), (iv) f (t , ·, ·) = 0 for t < 0. The third one is the ‘‘obstacle’’ {S (t ), 0 ≤ t ≤ T }, which is a continuous progressively measurable R-valued process satisfying that (v) E(sup0≤t ≤T eβ t (S (t )+ )2 ) < ∞, for some β > 0. We shall always assume that S (T ) ≤ ξ a.s. Let us now introduce RBSDE with time delayed generator. Definition 2.1. The solution of our RBSDE with time delayed generator is a triple {(Y (t ), Z (t ), K (t )), 0 ≤ t ≤ T } of
Ft -progressively measurable processes taking values in R, R and R+ , respectively, and satisfying that (1) Z ∈ H2T (R), Y ∈ S2T (R) and K (T ) ∈ L2 (R);
T
(2) Y (t ) = ξ + t f (s, Ys , Zs )ds + K (T ) − K (t ) − (3) Y (t ) ≥ S (t ), 0 ≤ t ≤ T ;
T t
Z (s)dW (s), 0 ≤ t ≤ T ;
(4) K (t ) is adapted, continuous and non-decreasing, K0 = 0 and
T 0
eβ t (Y (t ) − S (t ))dK (t ) = 0, for some β > 0.
We remark that the generator f at time s ∈ [0, T ] depends on the past values of the solution denoted by Ys := (Y (s + u))−T ≤u≤0 and Zs := (Z (s + u))−T ≤u≤0 .f (t , 0, 0) in (ii) should be understood as a value of the generator f (t , yt , zt ) at y(t + u) = z (t + u) = 0, −T ≤ u ≤ 0. Note that for (Y , Z ) ∈ S2 (R) × H2 (R) the generator f is well-defined and P-a.s. integrable as T
|f (t , Yt , Zt )| dt ≤ 2 2
T
0
|f (t , 0, 0)| dt + 2L 2
0
|f (t , 0, 0)|2 dt + 2L
=2
0
−T
0 T
0
|Y (t + u)| α(du)dt + 2
−T T +u
0 T
T
|Y (v)|2 dvα(du) + 2L
≤2 0
< ∞.
2
0≤t ≤T
0
|Z (t + u)| α(du)dt
−T 0 T +u
−T
u
2
0
|f (t , 0, 0)| dt + 2L T sup |Y (t )| + 2
T
T
|Z (t )| dt 2
|Z (v)|2 dvα(du)
u
0
(2.1)
To justify this, we apply the Fubini theorem, change the variables, use the assumption that Z (t ) = 0 and Y (t ) = Y (0) for t < 0 and the fact that the probability measure α integrates to 1. Note that from (2) and (4), it follows that {Y (t ), 0 ≤ t ≤ T } is continuous. Intuitively, dK (t )/dt represents the amount of ‘‘push upwards’’ that we add to −(dY (t )/dt ), so that the constraint (3) is satisfied. Condition (4) says that the push is minimal, in the sense that we push only when the constraint is saturated, that is, when Y (t ) = S (t ).
982
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
3. Main results on RBSDEs with time delayed generators In this section, we give the priori estimates on the solution and the difference of two solutions for RBSDEs with time delayed generators, which play an important role in proving existence and uniqueness of the solution, which is the main result of this section and this article. 3.1. Priori estimates We now give a priori estimate on the solution. Proposition 3.1. Let {(Y (t ), Z (t ), K (t )), 0 ≤ t ≤ T } be a solution of the RBSDE with time delayed generator (1)–(4). If the Lipschitz constant L of the generator f is small enough, then there exist two positive constants β and γ satisfying that D1 := β − γ > 0,
D2 := 1 −
2Lα˜
γ
> 0,
0 and a positive constant C = C (β, γ , α, ˜ L, T ) depending on β, γ , α˜ (α as in (iii) and α˜ = −T e−β u α(du)), L and T such that
sup eβ t |Y (t )|2 +
E
T
0≤t ≤T
eβ t |Z (t )|2 dt + eβ T |K (T )|2
0
T
≤ C E eβ T |ξ |2 +
eβ t |f (t , 0, 0)|2 dt + sup eβ t (S (t )+ )2
.
0≤t ≤T
0
Proof. Applying the Itô formula to eβ t |Y (t )|2 yields that βt
T
e |Y (t )| + 2
βs
β e |Y (s)| ds + 2
t
βT
T
eβ s |Z (s)|2 ds
t T
2
βs
e Y (s)f (s, Ys , Zs )ds + 2
= e |ξ | + 2 t
≤ eβ T |ξ |2 + γ
1
eβ s |Y (s)|2 ds +
T
eβ s Y (s)dK (s) − 2
T
t
T
γ
t
+2
βs
e Y (s)dK (s) − 2
T
t T
T
eβ s Y (s)Z (s)dW (s)
t
eβ s |f (s, Ys , Zs )|2 ds
t
eβ s Y (s)Z (s)dW (s),
(3.1)
t
where the last inequality result is derived by using the Young inequality for some γ > 0. By Eq. (2.1) and reorganizing (3.1), we get eβ t |Y (t )|2 + (β − γ )
T
eβ s |Y (s)|2 ds +
t
βT
1
2
≤ e |ξ | + βT
γ 2
2
≤ e |ξ | + T
γ
T
βs
e |f (s, Ys , Zs )| ds + 2 2
e |f (s, Ys , Zs ) − f (s, 0, 0)| ds + t T
eβ s Y (s)Z (s)dW (s)
t 2
T
T
2
γ
eβ s |f (s, 0, 0)|2 ds
t
eβ s Y (s)Z (s)dW (s)
t
T
γ
e Y (s)dK (s) − 2
βs
βs
≤ eβ T |ξ |2 + +
βs
t T
T
t
t
2
eβ s |Z (s)|2 ds
t
e Y (s)dK (s) − 2
+2
T
2L
γ βs
T
t
0
eβ s |Y (s + u)|2 α(du) ds +
T
−T
e |f (s, 0, 0)| ds + 2 t
T
βs
e Y (s)dK (s) − 2 t
T
eβ s |Z (s + u)|2 α(du) ds
−T
t
2
0
βs
e Y (s)Z (s)dW (s). t
By a change of integration order argument, for φ = Z , we obtain T
t
0
eβ s |φ(s + u)|2 α(du) ds =
−T
T
0
eβ(s+u) e−β u I{s+u≥0} |φ(s + u)|2 α(du) ds
−T
t
0
T +u
= −T
(t +u)∨0
eβ r e−β u I{r ≥0} |φ(r )|2 dr α(du)
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
(r −t )∧0
T
=
eβ r e−β u I{r ≥0} |φ(r )|2 α(du)dr
r −T
0 T
eβ r |φ(r )|2
=
0
e−β u α(du) dr −T
0 T
983
α˜ eβ r |φ(r )|2 dr ,
≤
(3.2)
0 0 with α˜ = −T e−β u α(du). Continuing the inequality from above, we get
eβ t |Y (t )|2 + (β − γ )
T
eβ s |Y (s)|2 ds +
2Lα˜
2
≤ e |ξ | + +
2
T
γ
γ
eβ s |Z (s)|2 ds
t
t
βT
T
βt
T sup e |Y (t )| + 2
T
0≤t ≤T
βs
e |Z (s)| ds 2
0
eβ s |f (s, 0, 0)|2 ds + 2
T
eβ s Y (s)dK (s) − 2
eβ s Y (s)Z (s)dW (s).
(3.3)
t
t
t
T
Step 1. We claim that T
βs
e |Z (s)| ds ≤ C˜ E
E
2
T
0
βs
βt
e |f (s, 0, 0)| ds + sup e |Y (t )| 2
2
,
(3.4)
0≤t ≤T
0
where C˜ > 0 depends on β, γ , α, ˜ L and T . The estimate of (3.4) can be deduced as follows. Since D1 = β − γ > 0, in (3.3) putting t = 0 yields that
T T 2Lα˜ 2Lα˜ 1 2 |Y (0)|2 + 1 − eβ s |Z (s)|2 ds ≤ eβ T |ξ |2 + T+ sup eβ t |Y (t )|2 + eβ s |f (s, 0, 0)|2 ds γ γ ε 0≤t ≤T γ 0 0 T + ε eβ T |K (T )|2 − 2 eβ s Y (s)Z (s)dW (s), t
where ε > 0 is a constant. From the equation K (T ) = Y (0) − ξ −
T
f (t , Yt , Zt )dt +
0
T
Z (t )dW (t ), 0
there exists a constant C1 depending on L and T such that
e
βT
|K (T )| ≤ C1 |Y (0)| + ξ + 2
2
2
T
|f (s, 0, 0)| ds + sup |Y (t )| + 2
2
0 ≤t ≤T
0
T
|Z (s)| ds + 2
0
0
T
2 Z (s)dW (s) .
Plugging this last inequality in the previous one, we get 2Lα˜
T
βs
T
(1 − ε C1 )|Y (0)| + 1 − e |Z (s)| ds − ε C1 |Z (s)|2 ds γ 0 0 2Lα˜ 1 T+ sup eβ t |Y (t )|2 + ε C1 sup |Y (t )|2 ≤ (eβ T + ε C1 )|ξ |2 + γ ε 0≤t ≤T 0≤t ≤T T T 2 T 2 + ε C1 + ε C1 eβ s |f (s, 0, 0)|2 ds + 2 eβ s Y (s)Z (s)dW (s) . Z (s)dW (s) + γ 0 0 0 2
2
Choosing now ε small enough and γ such that D2 := 1 − 2Lγα˜ > 0, we obtain T
E
eβ s |Z (s)|2 ds ≤ C2 E
T
0
0
0≤t ≤T
T
eβ s |f (s, 0, 0)|2 ds + sup eβ t |Y (t )|2 + C2 E
0
eβ s Y (s)Z (s)dW (s) ,
where C2 depends on β, γ , α, ˜ L and T . Next thanks to the B–D–G inequality (see e.g. Dellacherie and Meyer, 1980), we have T
E
0
eβ s Y (s)Z (s)dW (s) ≤ C3 E
T
0
eβ s |Y (s)|2 |Z (s)|2 ds
1/2
984
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
βt
sup e |Y (t )|
≤ C3 E
2
1/2
0≤t ≤T
C32
≤
2η
βs
e |Z (s)| ds 2
1/2
0
sup eβ t |Y (t )|2
E
T
η
+
2
0≤t ≤T
T
E
eβ s |Z (s)|2 ds,
0
where η > 0 is a constant. Finally, plugging the last inequality in the previous one, choosing η small enough and using the Fatou Lemma, we obtain that there exists a real constant C˜ depending on β, γ , α, ˜ L and T such that T
E
eβ s |Z (s)|2 ds ≤ C˜ E
T
eβ s |f (s, 0, 0)|2 ds + sup eβ t |Y (t )|2 . 0≤t ≤T
0
0
Step 2. We claim that
E
βt
sup e |Y (t )|
2
T
βT 2 ˆ ≤ C E e |ξ | +
0≤t ≤T
βs
βt
e |f (s, 0, 0)| ds + sup e (S (t ) ) 2
+ 2
(3.5)
0≤t ≤T
0
holds for a positive constant Cˆ depending on β, γ , α, ˜ L and T . To prove (3.5), going back to (3.3), using the identity
T 0
eβ t (Y (t ) − S (t ))dK (t ) = 0, we obtain T
eβ t |Y (t )|2 + (β − γ )
eβ s |Y (s)|2 ds +
t
≤ eβ T |ξ |2 + +
2
γ
T
1
δ
T sup eβ t |Y (t )|2 +
γ
eβ s |Z (s)|2 ds
t
0 ≤t ≤T
T
|Z (s)|2 ds
0 T
eβ s |f (s, 0, 0)|2 ds + 2
t
≤ eβ T |ξ |2 +
+
2Lα˜
T
eβ s S (s)dK (s) − 2
t
2Lα˜
eβ s Y (s)Z (s)dW (s)
t
T sup eβ t |Y (t )|2 +
γ
T
0 ≤t ≤T
T
eβ s |Z (s)|2 ds
2
+
γ
0 T
sup eβ t (S (t )+ )2 + δ
e
0≤t ≤T
βs 2
dK (s)
2
T
−2
t
T
eβ s |f (s, 0, 0)|2 ds
t
eβ s Y (s)Z (s)dW (s),
T
where δ > 0 is a constant. Next, we focus on the control of the term t e property (see e.g. Dellacherie and Meyer, 1980), for all 0 ≤ t ≤ T , we have
E[(K (T ) − K (t ))2 ] = E
T
(3.6)
t
2(K (T ) − K (s))dK (s) = 2E
2
dK (s). Using the predictable dual projection
T
t
βs
E[K (T ) − K (s)|Fs ]dK (s). t
Recall that K (T ) − K (t ) = Y (t ) − ξ −
T
f (s, Ys , Zs )dt +
T
t
Z (s)dW (s); t
then,
E[(K (T ) − K (t ))2 ] = 2E
T
E Y (s) − ξ − t
E 2 sup |Y (u)| +
≤ 2E
t ≤u≤T
t
≤
1 2
f (u, Yu , Zu )du|Fs dK (s) s
T
T
T
|f (u, Yu , Zu )|du|Fs dK (s)
s
E[(K (T ) − K (t ))2 ] + 2E sup
E 2 sup |Y (u)| +
t ≤s≤T
T
t ≤u≤T
|f (u, Yu , Zu )|du|Fs t
Thus, using the Doob maximal inequality, we obtain 1 2
E[(K (T ) − K (t )) ] ≤ 8 sup E E 2 sup |Y (u)| + 2
t ≤s≤T
≤ C4 E
t ≤u≤T
sup |Y (u)| +
t ≤u≤T
2
T
|f (u, Yu , Zu )|du|Fs t
T
|f (u, Yu , Zu )| du 2
t
2
2
.
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
sup |Y (u)|2 + 2
≤ C4 E
t ≤u ≤T
|f (u, 0, 0)|2 du + 2L T sup |Y (u)|2 +
0≤u≤T
|f (u, 0, 0)|2 du +
T
T
t ≤u ≤T
T
985
t
sup |Y (u)|2 +
≤ C5 E
T
|Z (u)|2 du
0
|Z (u)|2 du .
0
t
Then, by Step 1, we have eβ T E[(K (T ) − K (t ))2 ] ≤ C¯ E
sup eβ t |Y (t )|2 +
T
0 ≤t ≤T
eβ t |f (t , 0, 0)|2 dt
(3.7)
0
holds for a positive constant C¯ depending on β, γ , α, ˜ L and T . Now taking expectation in (3.6) and taking into account of (3.7), we obtain
1−
2Lα˜
T
e |Z (s)| ds ≤ Ee
E
γ
βs
2
βT
2
|ξ | +
2Lα˜
γ
0
1
+ E δ
βt 2 ¯ T + δ C E sup e |Y (t )| 0 ≤t ≤T
sup eβ t (S (t )+ )2 +
γ
0≤t ≤T
T
+ δ C¯ E
2
eβ s |f (s, 0, 0)|2 ds.
0
Thus, T
E
1 −1 βT 2 eβ s |Z (s)|2 ds ≤ D− 2 E[e |ξ | ] + D2
2Lα˜
γ
0 1 + D− 2
2
γ
T
+ δ C¯ E
T + δ C¯
E
11 sup eβ t |Y (t )|2 + D− E 2
δ
0≤t ≤T
sup eβ t (S (t )+ )2
0≤t ≤T
eβ s |f (s, 0, 0)|2 ds.
(3.8)
t
Next, going back to (3.6) and taking into account of (3.7), we get
E
sup eβ t |Y (t )|2
2Lα˜
≤ E eβ T |ξ |2 +
γ
0≤t ≤T
+
1
δ
T sup eβ t |Y (t )|2 + 0≤t ≤T
βt
eβ s |Z (s)|2 ds
+
0
sup e (S (t ) ) + δ sup + 2
0≤t ≤T
T
T
e
0≤t ≤T
βs 2
dK (s)
t
2
2
γ
+ 2 sup 0≤t ≤T
T
eβ s |f (s, 0, 0)|2 ds
0 T
t
e Y (s)Z (s)dW (s) . (3.9) βs
Next, using the B–D–G inequality, there exists a constant c > 0 such that
2E
0 ≤t ≤T
T
sup
t
eβ s Y (s)Z (s)dW (s) ≤ c E ρ sup eβ t |Y (t )|2 + 0≤t ≤T
1
ρ
T
eβ s |Z (s)|2 ds ,
0
where ρ > 0 is a constant. Plugging this inequality in (3.9), we obtain
E
sup eβ t |Y (t )|2
≤ Eeβ T |ξ |2 +
0≤t ≤T
1
2Lα˜
γ
T + δ C¯ + c ρ
E
sup eβ t |Y (t )|2 +
0 ≤t ≤T
2
2Lα˜
γ
+
c
ρ
T
eβ s |Z (s)|2 ds
0
T
+ δ C¯ E + E sup eβ t (S (t )+ )2 + eβ s |f (s, 0, 0)|2 ds δ 0 ≤t ≤T γ t c ˜ −1 2Lα ≤ 1 + D2 + Eeβ T |ξ |2 γ ρ c 2Lα˜ 1 1 βt + 2 + E sup e ( S ( t ) ) + 1 + D− 2 γ ρ δ 0≤t ≤T T 2Lα˜ c 2 1 ¯ + 1 + D− + + δ C E eβ s |f (s, 0, 0)|2 ds 2 γ ρ γ t ˜ c 2Lα˜ −1 2Lα + 1 + D2 + T + δ C¯ + c ρ E sup eβ t |Y (t )|2 . γ ρ γ 0≤t ≤T 1 The second inequality comes from (3.8). Finally it is enough to choose ρ = 2c and δ small enough to obtain (3.5). From the above inequalities (3.4), (3.5) and (3.7), we obtain that there exists a positive constant C = C (β, γ , α, ˜ L, T ) depending on β, γ , α, ˜ L and T such that
986
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
E
T
sup eβ t |Y (t )|2 + 0≤t ≤T
eβ t |Z (t )|2 dt + eβ T |K (T )|2
0
T
≤ C E eβ T |ξ |2 +
eβ t |f (t , 0, 0)|2 dt + sup eβ t (S (t )+ )2
.
0≤t ≤T
0
We can now estimate the variation in the solution induced by a variation in the data. Proposition 3.2. Let (ξ , f , S ) and (ξ ′ , f ′ , S ′ ) be two triplets satisfying the above assumptions (i)–(v). Suppose that (Y , Z , K ) is a solution of the RBSDE with time delayed generator (ξ , f , S ) and (Y ′ , Z ′ , K ′ ) is a solution of the RBSDE with time delayed generator (ξ ′ , f ′ , S ′ ). Define
1ξ = ξ − ξ ′ , 1Y = Y − Y ′ ,
1f = f − f ′ , 1Z = Z − Z ′ ,
1S = S − S ′ ; 1K = K − K ′ .
If the Lipschitz constant L of the generator f is small enough, then there exist two positive constants β and γ satisfying that D1 := β − γ > 0,
D2 := 1 −
2Lα˜
>0
γ
and a positive constant D = D(β, γ , α, ˜ L, T ) depending on β, γ , α, ˜ L and T such that
E
sup eβ t |1Y (t )|2 +
T
0≤t ≤T
eβ t |1Z (t )|2 dt
0
T
≤ DE eβ T |1ξ |2 +
eβ t |1f (t , Yt , Zt )|2 dt
1/2 1/2 + D E sup eβ t |1S (t )|2 ΨT ,
(3.10)
0 ≤t ≤T
0
where
T
ΨT = E eβ T |ξ |2 +
eβ t |f (t , 0, 0)|2 dt + sup eβ t (S (t )+ )2 0≤t ≤T
0 T
+ eβ T |ξ ′ |2 +
eβ t |f ′ (t , 0, 0)|2 dt + sup eβ t (S (t )′+ )2 . 0≤t ≤T
0
Proof. The computations are similar to those in the previous proof, so we shall only sketch the arguments. Similar to (3.1), applying the Itô formula to eβ t |1Y (t )|2 yields that eβ t |1Y (t )|2 + β
T
eβ s |1Y (s)|2 ds +
t
= eβ T |1ξ |2 + 2
T
eβ s |1Z (s)|2 ds
t T
eβ s 1f (s, Ys , Zs )1Y (s)ds + 2
t T
+2
T
eβ s 1Y (s)d1K (s)
t
eβ s [f ′ (s, Ys , Zs ) − f ′ (s, Ys′ , Zs′ )]1Y (s)ds − 2
βT
T
≤ e |1ξ | + 2 2
βs
e |1f (s, Ys , Zs )1Y (s)|ds + 2
T
γ
βs
e |f (s, Ys , Zs ) − f (s, Ys , Zs )| ds + γ ′
′
′
′
2
t T
+
γ
T
eβ s |1f (s, Ys , Zs )1Y (s)|ds + 2
T t
2
T sup eβ t |1Y (t )|2 + 0≤t ≤T
T
eβ s 1Y (s)1Z (s)dW (s)
t T
eβ s |1S (s)|d1K (s)
t T
0
eβ s (1Y (s) − 1S (s))d1K (s) ≤ 0
eβ s |1Z (s)|2 ds + γ
for some γ > 0, where we have used the relation
βs
e |1Y (s)| ds − 2
t
eβ s 1S (s)d1K (s)
t
t
≤ eβ T |1ξ |2 + 2 Lα˜
T
t
+
eβ s 1Y (s)1Z (s)dW (s)
t
t
1
T
T
t
eβ s |1Y (s)|2 ds − 2
T
t
eβ s 1Y (s)1Z (s)dW (s),
(3.11)
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
987
and (3.2) for φ = 1Z . With t = 0 and taking the expectation, we have T
(β − γ )E
βs
e |1Y (s)| ds + 2
1−
Lα˜
βT
≤ Ee |1ξ | + 2E 2
T
eβ s |1Z (s)|2 ds
E
γ
0
T
0
eβ s |1f (s, Ys , Zs )1Y (s)|ds
0 T
Lα˜ T
βs
e |1S (s)|d1K (s) +
+ 2E
γ
0
βt
sup e |1Y (t )|
E
2
.
0≤t ≤T
Recall that D1 := β − γ > 0 and D2 := 1 − 2Lγα˜ > 0. Now with the Hölder inequality T
eβ s |1S (s)|d1K (s) ≤ E
E
sup eβ t |1S (t )|2
1/2
E eβ T |1K (T )|2
1/2
0 ≤t ≤T
0
and since E|1K (T )|2 ≤ C6 (E|K (T )|2 + E|K ′ (T )|2 ), using inequalities (3.7) and Proposition 3.1, we deduce that
Eeβ T |1K (T )|2 ≤ C7 ΨT , where C7 > 0 depends on β, γ , α, ˜ L and T . Therefore, we obtain T
1 eβ s |1Z (s)|2 ds ≤ D− 2
E
Eeβ T |1ξ |2 + 2E
T
0
eβ s |1f (s, Ys , Zs )1Y (s)|ds
0
+
T γ
Lα˜
−1
E
βt
sup e |1Y (t )|
2
+ C8 E
0≤t ≤T
βt
sup e |1S (t )|
2
1/2
(ΨT )1/2 ,
(3.12)
0≤t ≤T
where C8 > 0 depends on β, γ , α, ˜ L and T . But T
2
eβ s |1f (s, Ys , Zs )1Y (s)|ds ≤ 2 sup e
βt 2
|1Y (t )|
0≤t ≤T
0
T
e
βs 2
|1f (s, Ys , Zs )|ds
0
≤ λ sup eβ t |1Y (t )|2 + 0≤t ≤T
1
T
λ
eβ s |1f (s, Ys , Zs )|2 ds,
0
where λ > 0 is a constant. Next with (3.11), the B–D–G inequality, and the two previous inequalities, we obtain
1−
Lα˜ T
E
γ
sup eβ t |1Y (t )|2
0≤t ≤T
≤ E eβ T |1ξ |2 + λ sup eβ t |1Y (t )|2 + 0≤t ≤T
1/2
1
T
λ
eβ s |1f (s, Ys , Zs )|2 ds +
0
Lα˜
γ
T
eβ s |1Z (s)|2 ds
0
+ C8 E sup e |1S (t )| (ΨT ) e 1Y (s)1Z (s)dW (s) 0 ≤t ≤T t T 1 ≤ E eβ T |1ξ |2 + λ sup eβ t |1Y (t )|2 + eβ s |1f (s, Ys , Zs )|2 ds λ 0 0≤t ≤T 1/2 + C8 E sup eβ t |1S (t )|2 (ΨT )1/2
βt
2
1/2
+ 2 sup 0≤t ≤T
T
βs
0≤t ≤T
+ E c ρ sup eβ t |1Y (t )|2 + 0 ≤t ≤T
c
ρ
T
0
eβ s |1Z (s)|2 ds +
Lα˜
γ
T
eβ s |1Z (s)|2 ds ,
0
where c , ρ > 0 are constants. By (3.12) and (3.13), we have
Lα˜ c −1 βt 2 + D λ + − λ − c ρ E sup e | 1 Y ( t )| γ 2 γ −1 γ ρ 0≤t ≤T Lα˜ 1/2 c c ˜ Lα˜ −1 Lα βT 2 βt 2 ≤ 1 + D2 + Ee |1ξ | + C8 1 + + E sup e |1S (t )| (ΨT )1/2 γ ρ γ ρ 0≤t ≤T T Lα˜ c 1 1 + E eβ s |1f (s, Ys , Zs )|2 ds. + 1 + D− 2 γ ρ λ t
1−
Lα˜ T
−
T
(3.13)
988
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
1 Finally, it is enough to choose ρ = 2c and λ small enough in the above inequalities (3.12) and (3.13) to obtain (3.10) (L is small enough). The desired result of the proposition follows.
We deduce immediately the following uniqueness result from Proposition 3.2 with ξ ′ = ξ , f ′ = f and S ′ = S . Corollary 3.1. Under assumptions (i)–(v), if the Lipschitz constant L of the generator f is small enough and for two positive constants β and γ the conditions of Proposition 3.2 are satisfied, then there exists at most one solution of the RBSDE with time delayed generators (1)–(4). Proof. Using the previous Proposition 3.2, we obtain immediately Y = Y ′ and Z = Z ′ . Therefore we have K = K ′ , whence uniqueness of the solution of the reflected BSDE with time delayed generator associated with (ξ , f , S ). 3.2. Existence and uniqueness of the solution To begin with, let us first assume that f does not depend on (y, z ), that is, it is a given Ft -progressively measurable process satisfying that ′
(ii ) E ′
T 0
|f (t )|2 dt < ∞.
A solution to the backward reflection problem (BRP, in short) is a triple (Y , Z , K ) which satisfies (1), (3), (4) and
(2 ) Y (t ) = ξ +
T t
f (s)ds + K (T ) − K (t ) −
T t
Z (s)dW (s), 0 ≤ t ≤ T .
The following proposition is from El Karoui et al. (1997) Proposition 5.1. ′
′
Proposition 3.3. Under assumptions (i), (ii ) and (v), the BRP (1), (2 ), (3), (4) has a unique solution {(Y (t ), Z (t ), K (t )); 0 ≤ t ≤ T }. We now deal with the general case of generator i.e. f depends on (y, z ). Theorem 3.1. Assume assumptions (i)–(v) hold. If the Lipschitz constant L of the generator f is small enough and for two positive constants β and γ the conditions of Proposition 3.2 are satisfied, then the RBSDE with time delayed generator (1)–(4) has a unique solution {(Y (t ), Z (t ), K (t )); 0 ≤ t ≤ T }. Proof. Denote by L the space of progressively measurable {(Y (t ), Z (t )); 0 ≤ t ≤ T } with values in R × R which satisfy (1) and (3). We define a mapping Φ from L into itself as follows. Given (U , V ) ∈ L, we define (Y , Z ) = Φ (U , V ) where (Y , Z , K ) is the solution of the RBSDE associated with (ξ , f (t , Ut , Vt ), S ), i.e.,
(Y , Z ) ∈ L, Y (t ) = ξ +
K ∈ S2T (R); T
f (s, Us , Vs )ds + K (T ) − K (t ) −
T
t
Z (s)dW (s),
0 ≤ t ≤ T;
t
Y (t ) ≥ S (t ) and
T
eβ t (Y (t ) − S (t ))dK (t ) = 0,
for some β > 0.
0
The solution of this equation exists and is unique thanks to Proposition 3.3. Now for (U ′ , V ′ ) in L, we define in the same way (Y ′ , Z ′ ) = Φ (U ′ , V ′ ) and U¯ = U − U ′ ,
V¯ = V − V ′ ,
Y¯ = Y − Y ′ ,
Z¯ = Z − Z ′ .
We are now going to prove that there exists a real constant c ∈ R such that the mapping Φ is a strict contraction on L equipped with the norm
1/2 ∥(Y , Z )∥β = E sup eβ t |Y (t )|2 + E 0≤t ≤T
T
eβ t |Z (t )|2 dt
1/2
.
0
In fact, it follows from the arguments similar to those in the proofs of Propositions 3.1 and 3.2 that eβ t |Y¯ (t )|2 +
T
eβ s (β|Y¯ (s)|2 + |Z¯ (s)|2 )ds = 2
t
T
eβ s Y¯ (s)[f (s, Us , Vs ) − f (s, Us′ , Vs′ )]ds
t T
+2
eβ s Y¯ (s)dK¯ (s) − 2
t
≤γ
T
t
T
eβ s Y¯ (s)Z¯ (s)dW (s)
t
eβ s |Y¯ (s)|2 ds +
1
γ
T
t
eβ s |f (s, Us , Vs )
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
− f (s, Us′ , Vs′ )|2 ds − 2
T
989
eβ s Y¯ (s)Z¯ (s)dW (s)
t T
≤γ
Lα˜
eβ s |Y¯ (s)|2 ds + T
−2
T sup eβ t |U¯ (t )|2 +
γ
t
0≤t ≤T
T
eβ s |V¯ (s)|2 ds
0
eβ s Y¯ (s)Z¯ (s)dW (s),
(3.14)
t
for any γ > 0, where we have used the inequality have eβ t E|Y¯ (t )|2 + (β − γ ) E
T
eβ s Y¯ (s)dK¯ (s) ≤ 0. Reorganizing (3.14) and taking the expectation we
T t
eβ s |Y¯ (s)|2 ds + E
Lα˜
γ
sup eβ t |U¯ (t )|2 +
E
0≤t ≤T
eβ s |Z¯ (s)|2 ds
t
t
≤
T
T
eβ s |V¯ (s)|2 ds .
0
By choosing β such that β − γ > 0, we obtain T
E
eβ s |Z¯ (s)|2 ds ≤
0
≤
Lα˜
γ Lα˜
γ
E T sup eβ t |U¯ (t )|2 + 0≤t ≤T
max{1, T }E
T
eβ s |V¯ (s)|2 ds
0
βt
sup e |U¯ (t )|2 +
T
0≤t ≤T
2 ¯ e |V (s)| ds . βs
(3.15)
0
Coming back to (3.14), taking supremum and then the expectation, there exists a real number c˜ such that
T Lα˜ 2 βt ¯ 2 βs ¯ 2 ¯ E sup e |Y (t )| ≤ max{1, T }E sup e |U (t )| + e |V (s)| ds γ 0≤t ≤T 0≤t ≤T 0 T eβ s Y¯ (s)Z¯ (s)dW (s) + 2E sup 0 ≤t ≤T t T Lα˜ ≤ max{1, T }E sup eβ t |U¯ (t )|2 + eβ s |V¯ (s)|2 ds γ 0≤t ≤T 0 T 1 eβ s |Z¯ (s)|2 ds. + E sup eβ t |Y¯ (t )|2 + c˜ E
βt
2
0 ≤t ≤T
(3.16)
0
For the last inequality we have used the B–D–G inequality. Plugging now the inequality in (3.15) and (3.16), we obtain
E
sup eβ t |Y¯ (t )|2 +
T
0≤t ≤T
eβ s |Z¯ (s)|2 ds ≤
0
Lα˜
γ
max{1, T } 2c˜ + 3 E
sup eβ t |U¯ (t )|2 +
0 ≤t ≤T
T
eβ s |V¯ (s)|2 ds .
0
So if we choose γ = 2Lα˜ max{1, T }(2c˜ + 3), since L is small enough and β − γ > 0, we deduce that
E
sup eβ t |Y¯ (t )|2 +
0 ≤t ≤T
T
0
eβ s |Z¯ (s)|2 ds
≤
1 2
E
sup eβ t |U¯ (t )|2 + 0≤t ≤T
T
eβ s |V¯ (s)|2 ds .
0
Hence, the mapping Φ is a strict contraction on L and therefore it has a unique fixed point, which in combination with the associated K is the unique solution of the RBSDE with time delayed generator. Acknowledgments The authors are deeply grateful to the anonymous referee and the editor Prof. Yimin Xiao for the careful reading, valuable comments and correcting some errors, which have greatly improved the quality of the paper. The work was partially done while the second author was visiting Shandong University. He would like to thank Prof. Shige Peng for providing a stimulating working environment. References Crépey, S., Moutoussi, A., 2008. Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison. Ann. Appl. Probab. 18, 2041–2069. Cvitanic, J., Karatzas, I., 1996. Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24, 2024–2056. Cvitanic, J., Ma, J., 2001. Reflected forward–backward SDEs and obstacle problems with boundary conditions. J. Appl. Math. Stochastic Anal. 14, 113–138.
990
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
Dellacherie, C., Meyer, P.A., 1980. Probabilités et Potentiel V–VIII. Hermann, Paris. Delong, Ł, 2010. Applications of time-delayed backward stochastic differential equations to pricing, hedging and management of insurance and financial risks, Preprint. arXiv:1005.4417. Delong, Ł, Imkeller, P., 2010a. Backward stochastic differential equations with time delayed generators-results and counterexamples. Ann. Appl. Probab. 20, 1512–1536. Delong, Ł, Imkeller, P., 2010b. On Malliavinś differentiability of BSDE with time delayed generators driven by Brownian motions and Poisson random measures. Stochastic Process. Appl. 120, 1748–1775. El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, P., Quenez, M.C., 1997. Reflected solution of backward SDE’s, and related obstacle problem for PDE’s. Ann. Probab. 25, 702–737. El Karoui, N., Peng, S., Quenez, M.C., 1997a. Backward stochastic differential equations in finance. Math. Fina. 7, 1–71. El Karoui, N., Quenez, M.C., 1997b. Non-linear pricing theory and backward stochastic differential equations. In: Runggaldier, W.J. (Ed.), Financial Mathematics. In: Lecture Notes in Math, vol. 1656. Springer-Verlag, Berlin, pp. 191–246. Hamadène, S., 2003. BSDEs and risk sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Stochastic Process. Appl. 107, 145–169. Hamadène, S., 2002. Reflectd BSDE’s with discontinuous barrier and application. Stochastics Stochastics Rep. 74, 571–596. Hamadène, S., Lepeltier, J.P., 1995a. Zero-sum stochastic differential games and BSDEs. Systems Control Lett. 24, 259–263. Hamadène, S., Lepeltier, J.P., 1995b. Backward equations, stochastic control and zero-sum stochastic differential games. Stochastics Stochastics Rep. 54, 221–231. Hamadène, S., Lepeltier, J.P., 2000. Reflected BSDEs and mixed game problem. Stochastic Process. Appl. 85, 177–188. Hamadène, S., Popier, A., 2011. Lp -solutions for reflected backward stochastic differential equations. Stochast. Dyna. doi:10.1142/S0219493712003651. Lepeltier, J.P., Xu, M., 2005. Penalization method for reflected backward stochastic differential equations with one rcll barrer. Statist. Probab. Lett. 75, 58–66. Pardoux, E., 1998. Backward stochastic differential equations and viscosity solutions of systems of semi-linear parabolic and elliptic PDEs of second order. In: Decreuefond, L., Gjerd, J., Øksendal, B., Üstünel, A.S. (Eds.), Stochastic analysis and related topics VI (The Gelio Workshop, 1996). In: Progr. Probab., vol. 42. Birkhäuser, Boston, Boston, MA., pp. 79–127. Pardoux, E., 1999. BSDEs, weak convergence and homogenization of semilinear PDEs, nonlinear analysis, differential equations and control. (Montreal, QC, 1998). In: NATO Sci. Ser. C Math. Phys. Sci., vol. 528. Kluwer Acad. Publ., Dordrecht, pp. 503–549. Pardoux, E., Peng, S., 1990. Adapted solution of backward stochastic differential equation. Systems Control Lett. 4, 55–61. Pardoux, E., Peng, S., 1992. Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Rozovskii, B.L., Sowers, R.B. (Eds.), Stochastic Partial Differential Equations and their Applications (Charlotte, NC, 1991). In: Lecture Note in Control and Inform. Sci., vol. 176. pp. 200–217. Peng, S., 1991. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics Stochastics Rep. 37, 61–74. Peng, S., Xu, M., 2005. The smallest g-supermartingale and reflected BSDE with single and double L2 obstacles. Ann. Inst. H. Poincaré Probab. Statist. 41, 605–630. Peng, S., Yang, Z., 2009. Anticipated backward stochastic differential equations. Ann. Probab. 37, 877–902. Reis, G., Réveillac, A., Zhang, J., 2011. FBSDEs with time delayed generators: Lp -solutions, differentiability, representation formulas and path regularity. Stochastic Process Appl. 121, 2114–2150. Ren, Y., Xia, N., 2006. Generalized reflected BSDEs and an obstacle problem for PDEs with a nonlinear Neumann boundary condition. Stochastic Anal. Appl. 24, 1013–1033.
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Reflected backward stochastic differential equations with time delayed generators✩ Qing Zhou a , Yong Ren b,∗ a
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
b
Department of Mathematics, Anhui Normal University, Wuhu 241000, China
article
info
Article history: Received 10 September 2011 Received in revised form 29 January 2012 Accepted 13 February 2012 Available online 21 February 2012
abstract In this paper, we establish the existence and uniqueness of the solution for a class of reflected backward stochastic differential equations with time delayed generator (RBSDEs with time delayed generator, in short) for a sufficiently small Lipschitz constant of the generator. © 2012 Elsevier B.V. All rights reserved.
MSC: 60H10 60G40 Keywords: Reflected backward stochastic differential equation Time delayed generator Fixed point theorem
1. Introduction The theory of nonlinear backward stochastic differential equations (BSDEs, in short) was developed by Pardoux and Peng (1990). Precisely, given a data (ξ , f ) consisting of a progressively measurable process f , so-called the generator, and a square integrable random variable ξ , they proved the existence and uniqueness of an adapted process (Y , Z ) solution to the following BSDE Y (t ) = ξ +
T
f (s, Y (s), Z (s))ds − t
T
Z (s)dW (s),
0 ≤ t ≤ T.
t
These equations have attracted great interest due to their connections with mathematical finance (see e.g. El Karoui et al., 1997a; El Karoui and Quenez, 1997b), stochastic control and stochastic games (see e.g. Hamadène, 2003; Hamadène and Lepeltier, 1995a,b). Furthermore, it was shown in various papers that BSDEs give the probabilistic representation for the solution (at least in the viscosity sense) of a large class of systems of semi-linear parabolic partial differential equations (PDEs in short) (see e.g. Pardoux, 1998, 1999; Pardoux and Peng, 1992; Peng, 1991).
✩ The work of Qing Zhou is supported by the National Natural Science Foundation of China (Nos 11001029 and 10971220) and the Fundamental Research Funds for the Central Universities (No. BUPT2009RC0705). The work of Yong Ren is supported by the National Natural Science Foundation of China (Nos 10901003 and 11126238), the Distinguished Young Scholars of Anhui Province (No. 1108085J08), the Key Project of Chinese Ministry of Education (No. 211077) and the Anhui Provincial Natural Science Foundation (No. 10040606Q30). ∗ Corresponding author. E-mail addresses: [email protected] (Q. Zhou), [email protected], [email protected] (Y. Ren).
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.02.012
980
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
Among the BSDEs, El Karoui et al. (1997) introduced a special class of reflected BSDEs (RBSDEs, in short), which is a BSDE but the solution is forced to stay above a lower barrier. In detail, a solution of such equations is a triple of processes (Y , Z , K ) satisfying that Y (t ) = ξ +
T
f (s, Y (s), Z (s))ds + K (T ) − K (t ) −
T
Z (s)dW (s),
Y (t ) ≥ S (t ),
(1.1)
t
t
where S, so-called the barrier or obstacle, is a given stochastic process. The role of the continuous non-decreasing process K Tis to push the state process upward with the minimal energy, in order to keep it above S; in this sense, it satisfies that (Y (t ) − S (t ))dK (t ) = 0. RBSDEs have been proven to be powerful tools in mathematical finance (see e.g. El Karoui and 0 Quenez, 1997b; Crépey and Moutoussi, 2008), the mixed game problems (see e.g. Cvitanic and Karatzas, 1996; Hamadène and Lepeltier, 2000), providing a probabilistic formula for the viscosity solution of an obstacle problem for a class of parabolic PDEs (see e.g. Cvitanic and Ma, 2001; El Karoui et al., 1997; Hamadène and Popier, 2011; Ren and Xia, 2006) and so on. On other interesting results on RBSDEs driven by a Brownian motion with different barrier conditions, one can see Hamadène (2002), Lepeltier and Xu (2005); Peng and Xu (2005). Recently, Delong and Imkeller (2010a) introduced the following BSDE Y (t ) = ξ +
T
f (s, Ys , Zs )ds − t
T
Z (s)dW (s),
0 ≤ t ≤ T,
t
where the generator f at time s depends arbitrary on the past values of a solution (Ys , Zs ) = (Y (s + u), Z (s + u))−T ≤u≤0 . This class of BSDEs is called as BSDEs with time delayed generators. In Delong and Imkeller (2010a), the authors proved the existence and uniqueness of a solution for the above equation. Moreover, in Delong and Imkeller (2010b), they established the existence and uniqueness result for BSDEs with time delayed generators driven by Brownian motions and Poisson random measures. In addition, they studied the Malliavin’s differentiability of the solution. For the applications of BSDEs with time delayed generators in insurance and finance, one can see Delong (2010). Very recently, Reis et al. (2011) extended the results of Delong and Imkeller (2010a,b) in Lp -spaces. Unlike Delong and Imkeller (2010a,b) and Reis et al. (2011), Peng and Yang (2009) introduce a new type of BSDEs, called as anticipated BSDEs, where the authors established the existence, uniqueness and comparison theorem of the solutions and the duality relations between them and stochastic differential delay equations. To our best knowledge, there is no result on RBSDEs with time delayed generators. To close the gap and as the fundamental step, in this paper, we aim to derive the existence and uniqueness of the solutions for RBSDEs with time delayed generators. We expect to give its applications in finance and study some other properties of the solution in our further study. Even for BSDEs with time delayed generators, as shown in Delong and Imkeller (2010a,b), the general existence and uniqueness results do not hold for arbitrary Lipschitz condition and arbitrary time horizon. Our results hold also under the condition of a sufficiently small Lipschitz constant of the generator. The paper is organized as follows. In Section 2, we propose some preliminaries and notations. The main results are given in Section 3. 2. Preliminaries and notations Let (Ω , F , P) be a probability space equipped with a standard Brownian motion W . For a fixed real number T > 0, we consider the filtration F := (Ft )0≤t ≤T which is generated by W and augmented by all P-null sets. The filtered probability space (Ω , F , F, P) satisfies the usual conditions. We shall work with the following spaces.
• Let L2−T (R) denote the space of measurable functions z : [−T , 0] → R satisfying that 0 |z (t )|2 dt < ∞. −T
• Let
S∞ −T (R)
denote the space of bounded, measurable functions y : [−T , 0] → R satisfying that
sup |y(t )|2 < ∞.
t ∈[−T ,0]
• Let L2 (R) denote the space of FT -measurable random variable ξ : Ω → R satisfying that E|ξ |2 < ∞.
• Let H2T (R) denote the space of F-predictable processes Z : Ω × [0, T ] → R satisfying that T E |Z (t )|2 dt < ∞. 0
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
981
• Finally, let S2T (R) denote the space of F-adapted, product measurable processes Y : Ω × [0, T ] → R satisfying that 2 E sup |Y (t )| < ∞. t ∈[0,T ]
The spaces H2T (R) and S2T (R) are respectively endowed with the norms
∥Z ∥
2 H2T
T
βt
e |Z (t )| dt
=E
2
and ∥Y ∥
0
2 S2T
=E
βt
sup e |Y (t )|
2
t ∈[0,T ]
,
with some β > 0. As usual, by λ we denote Lebesgue measure on ([−T , 0], B ([−T , 0])), where B ([−T , 0]) stands for the Borel sets of [−T , 0]. In the sequel, let us simply write S2 (R) × H2 (R) for S2T (R) × H2T (R). Now, we give three objects. The first one is the terminal value ξ . (i) ξ ∈ L2 (R). The second one is the ‘‘coefficient’’ f , 2 f : Ω × [0, T ] × S∞ −T (R) × L−T (R) → R,
which is a product measurable, F-adapted map such that
T
(ii) E 0 |f (t , 0, 0)|2 dt < ∞, (iii) for a probability measure α on ([−T , 0], B ([−T , 0])) and a positive constant L
|f (t , yt , zt ) − f (t , y˜ t , z˜t )|2 ≤ L
0
|y(t + u) − y˜ (t + u)|2 α(du) + −T
0
|z (t + u) − z˜ (t + u)|2 α(du) ,
−T
S∞ −T (R)
holds for P × λ-a.e. (ω, t ) ∈ Ω × [0, T ] and for any (yt , zt ), (˜yt , z˜t ) ∈ × L2−T (R), (iv) f (t , ·, ·) = 0 for t < 0. The third one is the ‘‘obstacle’’ {S (t ), 0 ≤ t ≤ T }, which is a continuous progressively measurable R-valued process satisfying that (v) E(sup0≤t ≤T eβ t (S (t )+ )2 ) < ∞, for some β > 0. We shall always assume that S (T ) ≤ ξ a.s. Let us now introduce RBSDE with time delayed generator. Definition 2.1. The solution of our RBSDE with time delayed generator is a triple {(Y (t ), Z (t ), K (t )), 0 ≤ t ≤ T } of
Ft -progressively measurable processes taking values in R, R and R+ , respectively, and satisfying that (1) Z ∈ H2T (R), Y ∈ S2T (R) and K (T ) ∈ L2 (R);
T
(2) Y (t ) = ξ + t f (s, Ys , Zs )ds + K (T ) − K (t ) − (3) Y (t ) ≥ S (t ), 0 ≤ t ≤ T ;
T t
Z (s)dW (s), 0 ≤ t ≤ T ;
(4) K (t ) is adapted, continuous and non-decreasing, K0 = 0 and
T 0
eβ t (Y (t ) − S (t ))dK (t ) = 0, for some β > 0.
We remark that the generator f at time s ∈ [0, T ] depends on the past values of the solution denoted by Ys := (Y (s + u))−T ≤u≤0 and Zs := (Z (s + u))−T ≤u≤0 .f (t , 0, 0) in (ii) should be understood as a value of the generator f (t , yt , zt ) at y(t + u) = z (t + u) = 0, −T ≤ u ≤ 0. Note that for (Y , Z ) ∈ S2 (R) × H2 (R) the generator f is well-defined and P-a.s. integrable as T
|f (t , Yt , Zt )| dt ≤ 2 2
T
0
|f (t , 0, 0)| dt + 2L 2
0
|f (t , 0, 0)|2 dt + 2L
=2
0
−T
0 T
0
|Y (t + u)| α(du)dt + 2
−T T +u
0 T
T
|Y (v)|2 dvα(du) + 2L
≤2 0
< ∞.
2
0≤t ≤T
0
|Z (t + u)| α(du)dt
−T 0 T +u
−T
u
2
0
|f (t , 0, 0)| dt + 2L T sup |Y (t )| + 2
T
T
|Z (t )| dt 2
|Z (v)|2 dvα(du)
u
0
(2.1)
To justify this, we apply the Fubini theorem, change the variables, use the assumption that Z (t ) = 0 and Y (t ) = Y (0) for t < 0 and the fact that the probability measure α integrates to 1. Note that from (2) and (4), it follows that {Y (t ), 0 ≤ t ≤ T } is continuous. Intuitively, dK (t )/dt represents the amount of ‘‘push upwards’’ that we add to −(dY (t )/dt ), so that the constraint (3) is satisfied. Condition (4) says that the push is minimal, in the sense that we push only when the constraint is saturated, that is, when Y (t ) = S (t ).
982
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
3. Main results on RBSDEs with time delayed generators In this section, we give the priori estimates on the solution and the difference of two solutions for RBSDEs with time delayed generators, which play an important role in proving existence and uniqueness of the solution, which is the main result of this section and this article. 3.1. Priori estimates We now give a priori estimate on the solution. Proposition 3.1. Let {(Y (t ), Z (t ), K (t )), 0 ≤ t ≤ T } be a solution of the RBSDE with time delayed generator (1)–(4). If the Lipschitz constant L of the generator f is small enough, then there exist two positive constants β and γ satisfying that D1 := β − γ > 0,
D2 := 1 −
2Lα˜
γ
> 0,
0 and a positive constant C = C (β, γ , α, ˜ L, T ) depending on β, γ , α˜ (α as in (iii) and α˜ = −T e−β u α(du)), L and T such that
sup eβ t |Y (t )|2 +
E
T
0≤t ≤T
eβ t |Z (t )|2 dt + eβ T |K (T )|2
0
T
≤ C E eβ T |ξ |2 +
eβ t |f (t , 0, 0)|2 dt + sup eβ t (S (t )+ )2
.
0≤t ≤T
0
Proof. Applying the Itô formula to eβ t |Y (t )|2 yields that βt
T
e |Y (t )| + 2
βs
β e |Y (s)| ds + 2
t
βT
T
eβ s |Z (s)|2 ds
t T
2
βs
e Y (s)f (s, Ys , Zs )ds + 2
= e |ξ | + 2 t
≤ eβ T |ξ |2 + γ
1
eβ s |Y (s)|2 ds +
T
eβ s Y (s)dK (s) − 2
T
t
T
γ
t
+2
βs
e Y (s)dK (s) − 2
T
t T
T
eβ s Y (s)Z (s)dW (s)
t
eβ s |f (s, Ys , Zs )|2 ds
t
eβ s Y (s)Z (s)dW (s),
(3.1)
t
where the last inequality result is derived by using the Young inequality for some γ > 0. By Eq. (2.1) and reorganizing (3.1), we get eβ t |Y (t )|2 + (β − γ )
T
eβ s |Y (s)|2 ds +
t
βT
1
2
≤ e |ξ | + βT
γ 2
2
≤ e |ξ | + T
γ
T
βs
e |f (s, Ys , Zs )| ds + 2 2
e |f (s, Ys , Zs ) − f (s, 0, 0)| ds + t T
eβ s Y (s)Z (s)dW (s)
t 2
T
T
2
γ
eβ s |f (s, 0, 0)|2 ds
t
eβ s Y (s)Z (s)dW (s)
t
T
γ
e Y (s)dK (s) − 2
βs
βs
≤ eβ T |ξ |2 + +
βs
t T
T
t
t
2
eβ s |Z (s)|2 ds
t
e Y (s)dK (s) − 2
+2
T
2L
γ βs
T
t
0
eβ s |Y (s + u)|2 α(du) ds +
T
−T
e |f (s, 0, 0)| ds + 2 t
T
βs
e Y (s)dK (s) − 2 t
T
eβ s |Z (s + u)|2 α(du) ds
−T
t
2
0
βs
e Y (s)Z (s)dW (s). t
By a change of integration order argument, for φ = Z , we obtain T
t
0
eβ s |φ(s + u)|2 α(du) ds =
−T
T
0
eβ(s+u) e−β u I{s+u≥0} |φ(s + u)|2 α(du) ds
−T
t
0
T +u
= −T
(t +u)∨0
eβ r e−β u I{r ≥0} |φ(r )|2 dr α(du)
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
(r −t )∧0
T
=
eβ r e−β u I{r ≥0} |φ(r )|2 α(du)dr
r −T
0 T
eβ r |φ(r )|2
=
0
e−β u α(du) dr −T
0 T
983
α˜ eβ r |φ(r )|2 dr ,
≤
(3.2)
0 0 with α˜ = −T e−β u α(du). Continuing the inequality from above, we get
eβ t |Y (t )|2 + (β − γ )
T
eβ s |Y (s)|2 ds +
2Lα˜
2
≤ e |ξ | + +
2
T
γ
γ
eβ s |Z (s)|2 ds
t
t
βT
T
βt
T sup e |Y (t )| + 2
T
0≤t ≤T
βs
e |Z (s)| ds 2
0
eβ s |f (s, 0, 0)|2 ds + 2
T
eβ s Y (s)dK (s) − 2
eβ s Y (s)Z (s)dW (s).
(3.3)
t
t
t
T
Step 1. We claim that T
βs
e |Z (s)| ds ≤ C˜ E
E
2
T
0
βs
βt
e |f (s, 0, 0)| ds + sup e |Y (t )| 2
2
,
(3.4)
0≤t ≤T
0
where C˜ > 0 depends on β, γ , α, ˜ L and T . The estimate of (3.4) can be deduced as follows. Since D1 = β − γ > 0, in (3.3) putting t = 0 yields that
T T 2Lα˜ 2Lα˜ 1 2 |Y (0)|2 + 1 − eβ s |Z (s)|2 ds ≤ eβ T |ξ |2 + T+ sup eβ t |Y (t )|2 + eβ s |f (s, 0, 0)|2 ds γ γ ε 0≤t ≤T γ 0 0 T + ε eβ T |K (T )|2 − 2 eβ s Y (s)Z (s)dW (s), t
where ε > 0 is a constant. From the equation K (T ) = Y (0) − ξ −
T
f (t , Yt , Zt )dt +
0
T
Z (t )dW (t ), 0
there exists a constant C1 depending on L and T such that
e
βT
|K (T )| ≤ C1 |Y (0)| + ξ + 2
2
2
T
|f (s, 0, 0)| ds + sup |Y (t )| + 2
2
0 ≤t ≤T
0
T
|Z (s)| ds + 2
0
0
T
2 Z (s)dW (s) .
Plugging this last inequality in the previous one, we get 2Lα˜
T
βs
T
(1 − ε C1 )|Y (0)| + 1 − e |Z (s)| ds − ε C1 |Z (s)|2 ds γ 0 0 2Lα˜ 1 T+ sup eβ t |Y (t )|2 + ε C1 sup |Y (t )|2 ≤ (eβ T + ε C1 )|ξ |2 + γ ε 0≤t ≤T 0≤t ≤T T T 2 T 2 + ε C1 + ε C1 eβ s |f (s, 0, 0)|2 ds + 2 eβ s Y (s)Z (s)dW (s) . Z (s)dW (s) + γ 0 0 0 2
2
Choosing now ε small enough and γ such that D2 := 1 − 2Lγα˜ > 0, we obtain T
E
eβ s |Z (s)|2 ds ≤ C2 E
T
0
0
0≤t ≤T
T
eβ s |f (s, 0, 0)|2 ds + sup eβ t |Y (t )|2 + C2 E
0
eβ s Y (s)Z (s)dW (s) ,
where C2 depends on β, γ , α, ˜ L and T . Next thanks to the B–D–G inequality (see e.g. Dellacherie and Meyer, 1980), we have T
E
0
eβ s Y (s)Z (s)dW (s) ≤ C3 E
T
0
eβ s |Y (s)|2 |Z (s)|2 ds
1/2
984
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
βt
sup e |Y (t )|
≤ C3 E
2
1/2
0≤t ≤T
C32
≤
2η
βs
e |Z (s)| ds 2
1/2
0
sup eβ t |Y (t )|2
E
T
η
+
2
0≤t ≤T
T
E
eβ s |Z (s)|2 ds,
0
where η > 0 is a constant. Finally, plugging the last inequality in the previous one, choosing η small enough and using the Fatou Lemma, we obtain that there exists a real constant C˜ depending on β, γ , α, ˜ L and T such that T
E
eβ s |Z (s)|2 ds ≤ C˜ E
T
eβ s |f (s, 0, 0)|2 ds + sup eβ t |Y (t )|2 . 0≤t ≤T
0
0
Step 2. We claim that
E
βt
sup e |Y (t )|
2
T
βT 2 ˆ ≤ C E e |ξ | +
0≤t ≤T
βs
βt
e |f (s, 0, 0)| ds + sup e (S (t ) ) 2
+ 2
(3.5)
0≤t ≤T
0
holds for a positive constant Cˆ depending on β, γ , α, ˜ L and T . To prove (3.5), going back to (3.3), using the identity
T 0
eβ t (Y (t ) − S (t ))dK (t ) = 0, we obtain T
eβ t |Y (t )|2 + (β − γ )
eβ s |Y (s)|2 ds +
t
≤ eβ T |ξ |2 + +
2
γ
T
1
δ
T sup eβ t |Y (t )|2 +
γ
eβ s |Z (s)|2 ds
t
0 ≤t ≤T
T
|Z (s)|2 ds
0 T
eβ s |f (s, 0, 0)|2 ds + 2
t
≤ eβ T |ξ |2 +
+
2Lα˜
T
eβ s S (s)dK (s) − 2
t
2Lα˜
eβ s Y (s)Z (s)dW (s)
t
T sup eβ t |Y (t )|2 +
γ
T
0 ≤t ≤T
T
eβ s |Z (s)|2 ds
2
+
γ
0 T
sup eβ t (S (t )+ )2 + δ
e
0≤t ≤T
βs 2
dK (s)
2
T
−2
t
T
eβ s |f (s, 0, 0)|2 ds
t
eβ s Y (s)Z (s)dW (s),
T
where δ > 0 is a constant. Next, we focus on the control of the term t e property (see e.g. Dellacherie and Meyer, 1980), for all 0 ≤ t ≤ T , we have
E[(K (T ) − K (t ))2 ] = E
T
(3.6)
t
2(K (T ) − K (s))dK (s) = 2E
2
dK (s). Using the predictable dual projection
T
t
βs
E[K (T ) − K (s)|Fs ]dK (s). t
Recall that K (T ) − K (t ) = Y (t ) − ξ −
T
f (s, Ys , Zs )dt +
T
t
Z (s)dW (s); t
then,
E[(K (T ) − K (t ))2 ] = 2E
T
E Y (s) − ξ − t
E 2 sup |Y (u)| +
≤ 2E
t ≤u≤T
t
≤
1 2
f (u, Yu , Zu )du|Fs dK (s) s
T
T
T
|f (u, Yu , Zu )|du|Fs dK (s)
s
E[(K (T ) − K (t ))2 ] + 2E sup
E 2 sup |Y (u)| +
t ≤s≤T
T
t ≤u≤T
|f (u, Yu , Zu )|du|Fs t
Thus, using the Doob maximal inequality, we obtain 1 2
E[(K (T ) − K (t )) ] ≤ 8 sup E E 2 sup |Y (u)| + 2
t ≤s≤T
≤ C4 E
t ≤u≤T
sup |Y (u)| +
t ≤u≤T
2
T
|f (u, Yu , Zu )|du|Fs t
T
|f (u, Yu , Zu )| du 2
t
2
2
.
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
sup |Y (u)|2 + 2
≤ C4 E
t ≤u ≤T
|f (u, 0, 0)|2 du + 2L T sup |Y (u)|2 +
0≤u≤T
|f (u, 0, 0)|2 du +
T
T
t ≤u ≤T
T
985
t
sup |Y (u)|2 +
≤ C5 E
T
|Z (u)|2 du
0
|Z (u)|2 du .
0
t
Then, by Step 1, we have eβ T E[(K (T ) − K (t ))2 ] ≤ C¯ E
sup eβ t |Y (t )|2 +
T
0 ≤t ≤T
eβ t |f (t , 0, 0)|2 dt
(3.7)
0
holds for a positive constant C¯ depending on β, γ , α, ˜ L and T . Now taking expectation in (3.6) and taking into account of (3.7), we obtain
1−
2Lα˜
T
e |Z (s)| ds ≤ Ee
E
γ
βs
2
βT
2
|ξ | +
2Lα˜
γ
0
1
+ E δ
βt 2 ¯ T + δ C E sup e |Y (t )| 0 ≤t ≤T
sup eβ t (S (t )+ )2 +
γ
0≤t ≤T
T
+ δ C¯ E
2
eβ s |f (s, 0, 0)|2 ds.
0
Thus, T
E
1 −1 βT 2 eβ s |Z (s)|2 ds ≤ D− 2 E[e |ξ | ] + D2
2Lα˜
γ
0 1 + D− 2
2
γ
T
+ δ C¯ E
T + δ C¯
E
11 sup eβ t |Y (t )|2 + D− E 2
δ
0≤t ≤T
sup eβ t (S (t )+ )2
0≤t ≤T
eβ s |f (s, 0, 0)|2 ds.
(3.8)
t
Next, going back to (3.6) and taking into account of (3.7), we get
E
sup eβ t |Y (t )|2
2Lα˜
≤ E eβ T |ξ |2 +
γ
0≤t ≤T
+
1
δ
T sup eβ t |Y (t )|2 + 0≤t ≤T
βt
eβ s |Z (s)|2 ds
+
0
sup e (S (t ) ) + δ sup + 2
0≤t ≤T
T
T
e
0≤t ≤T
βs 2
dK (s)
t
2
2
γ
+ 2 sup 0≤t ≤T
T
eβ s |f (s, 0, 0)|2 ds
0 T
t
e Y (s)Z (s)dW (s) . (3.9) βs
Next, using the B–D–G inequality, there exists a constant c > 0 such that
2E
0 ≤t ≤T
T
sup
t
eβ s Y (s)Z (s)dW (s) ≤ c E ρ sup eβ t |Y (t )|2 + 0≤t ≤T
1
ρ
T
eβ s |Z (s)|2 ds ,
0
where ρ > 0 is a constant. Plugging this inequality in (3.9), we obtain
E
sup eβ t |Y (t )|2
≤ Eeβ T |ξ |2 +
0≤t ≤T
1
2Lα˜
γ
T + δ C¯ + c ρ
E
sup eβ t |Y (t )|2 +
0 ≤t ≤T
2
2Lα˜
γ
+
c
ρ
T
eβ s |Z (s)|2 ds
0
T
+ δ C¯ E + E sup eβ t (S (t )+ )2 + eβ s |f (s, 0, 0)|2 ds δ 0 ≤t ≤T γ t c ˜ −1 2Lα ≤ 1 + D2 + Eeβ T |ξ |2 γ ρ c 2Lα˜ 1 1 βt + 2 + E sup e ( S ( t ) ) + 1 + D− 2 γ ρ δ 0≤t ≤T T 2Lα˜ c 2 1 ¯ + 1 + D− + + δ C E eβ s |f (s, 0, 0)|2 ds 2 γ ρ γ t ˜ c 2Lα˜ −1 2Lα + 1 + D2 + T + δ C¯ + c ρ E sup eβ t |Y (t )|2 . γ ρ γ 0≤t ≤T 1 The second inequality comes from (3.8). Finally it is enough to choose ρ = 2c and δ small enough to obtain (3.5). From the above inequalities (3.4), (3.5) and (3.7), we obtain that there exists a positive constant C = C (β, γ , α, ˜ L, T ) depending on β, γ , α, ˜ L and T such that
986
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
E
T
sup eβ t |Y (t )|2 + 0≤t ≤T
eβ t |Z (t )|2 dt + eβ T |K (T )|2
0
T
≤ C E eβ T |ξ |2 +
eβ t |f (t , 0, 0)|2 dt + sup eβ t (S (t )+ )2
.
0≤t ≤T
0
We can now estimate the variation in the solution induced by a variation in the data. Proposition 3.2. Let (ξ , f , S ) and (ξ ′ , f ′ , S ′ ) be two triplets satisfying the above assumptions (i)–(v). Suppose that (Y , Z , K ) is a solution of the RBSDE with time delayed generator (ξ , f , S ) and (Y ′ , Z ′ , K ′ ) is a solution of the RBSDE with time delayed generator (ξ ′ , f ′ , S ′ ). Define
1ξ = ξ − ξ ′ , 1Y = Y − Y ′ ,
1f = f − f ′ , 1Z = Z − Z ′ ,
1S = S − S ′ ; 1K = K − K ′ .
If the Lipschitz constant L of the generator f is small enough, then there exist two positive constants β and γ satisfying that D1 := β − γ > 0,
D2 := 1 −
2Lα˜
>0
γ
and a positive constant D = D(β, γ , α, ˜ L, T ) depending on β, γ , α, ˜ L and T such that
E
sup eβ t |1Y (t )|2 +
T
0≤t ≤T
eβ t |1Z (t )|2 dt
0
T
≤ DE eβ T |1ξ |2 +
eβ t |1f (t , Yt , Zt )|2 dt
1/2 1/2 + D E sup eβ t |1S (t )|2 ΨT ,
(3.10)
0 ≤t ≤T
0
where
T
ΨT = E eβ T |ξ |2 +
eβ t |f (t , 0, 0)|2 dt + sup eβ t (S (t )+ )2 0≤t ≤T
0 T
+ eβ T |ξ ′ |2 +
eβ t |f ′ (t , 0, 0)|2 dt + sup eβ t (S (t )′+ )2 . 0≤t ≤T
0
Proof. The computations are similar to those in the previous proof, so we shall only sketch the arguments. Similar to (3.1), applying the Itô formula to eβ t |1Y (t )|2 yields that eβ t |1Y (t )|2 + β
T
eβ s |1Y (s)|2 ds +
t
= eβ T |1ξ |2 + 2
T
eβ s |1Z (s)|2 ds
t T
eβ s 1f (s, Ys , Zs )1Y (s)ds + 2
t T
+2
T
eβ s 1Y (s)d1K (s)
t
eβ s [f ′ (s, Ys , Zs ) − f ′ (s, Ys′ , Zs′ )]1Y (s)ds − 2
βT
T
≤ e |1ξ | + 2 2
βs
e |1f (s, Ys , Zs )1Y (s)|ds + 2
T
γ
βs
e |f (s, Ys , Zs ) − f (s, Ys , Zs )| ds + γ ′
′
′
′
2
t T
+
γ
T
eβ s |1f (s, Ys , Zs )1Y (s)|ds + 2
T t
2
T sup eβ t |1Y (t )|2 + 0≤t ≤T
T
eβ s 1Y (s)1Z (s)dW (s)
t T
eβ s |1S (s)|d1K (s)
t T
0
eβ s (1Y (s) − 1S (s))d1K (s) ≤ 0
eβ s |1Z (s)|2 ds + γ
for some γ > 0, where we have used the relation
βs
e |1Y (s)| ds − 2
t
eβ s 1S (s)d1K (s)
t
t
≤ eβ T |1ξ |2 + 2 Lα˜
T
t
+
eβ s 1Y (s)1Z (s)dW (s)
t
t
1
T
T
t
eβ s |1Y (s)|2 ds − 2
T
t
eβ s 1Y (s)1Z (s)dW (s),
(3.11)
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
987
and (3.2) for φ = 1Z . With t = 0 and taking the expectation, we have T
(β − γ )E
βs
e |1Y (s)| ds + 2
1−
Lα˜
βT
≤ Ee |1ξ | + 2E 2
T
eβ s |1Z (s)|2 ds
E
γ
0
T
0
eβ s |1f (s, Ys , Zs )1Y (s)|ds
0 T
Lα˜ T
βs
e |1S (s)|d1K (s) +
+ 2E
γ
0
βt
sup e |1Y (t )|
E
2
.
0≤t ≤T
Recall that D1 := β − γ > 0 and D2 := 1 − 2Lγα˜ > 0. Now with the Hölder inequality T
eβ s |1S (s)|d1K (s) ≤ E
E
sup eβ t |1S (t )|2
1/2
E eβ T |1K (T )|2
1/2
0 ≤t ≤T
0
and since E|1K (T )|2 ≤ C6 (E|K (T )|2 + E|K ′ (T )|2 ), using inequalities (3.7) and Proposition 3.1, we deduce that
Eeβ T |1K (T )|2 ≤ C7 ΨT , where C7 > 0 depends on β, γ , α, ˜ L and T . Therefore, we obtain T
1 eβ s |1Z (s)|2 ds ≤ D− 2
E
Eeβ T |1ξ |2 + 2E
T
0
eβ s |1f (s, Ys , Zs )1Y (s)|ds
0
+
T γ
Lα˜
−1
E
βt
sup e |1Y (t )|
2
+ C8 E
0≤t ≤T
βt
sup e |1S (t )|
2
1/2
(ΨT )1/2 ,
(3.12)
0≤t ≤T
where C8 > 0 depends on β, γ , α, ˜ L and T . But T
2
eβ s |1f (s, Ys , Zs )1Y (s)|ds ≤ 2 sup e
βt 2
|1Y (t )|
0≤t ≤T
0
T
e
βs 2
|1f (s, Ys , Zs )|ds
0
≤ λ sup eβ t |1Y (t )|2 + 0≤t ≤T
1
T
λ
eβ s |1f (s, Ys , Zs )|2 ds,
0
where λ > 0 is a constant. Next with (3.11), the B–D–G inequality, and the two previous inequalities, we obtain
1−
Lα˜ T
E
γ
sup eβ t |1Y (t )|2
0≤t ≤T
≤ E eβ T |1ξ |2 + λ sup eβ t |1Y (t )|2 + 0≤t ≤T
1/2
1
T
λ
eβ s |1f (s, Ys , Zs )|2 ds +
0
Lα˜
γ
T
eβ s |1Z (s)|2 ds
0
+ C8 E sup e |1S (t )| (ΨT ) e 1Y (s)1Z (s)dW (s) 0 ≤t ≤T t T 1 ≤ E eβ T |1ξ |2 + λ sup eβ t |1Y (t )|2 + eβ s |1f (s, Ys , Zs )|2 ds λ 0 0≤t ≤T 1/2 + C8 E sup eβ t |1S (t )|2 (ΨT )1/2
βt
2
1/2
+ 2 sup 0≤t ≤T
T
βs
0≤t ≤T
+ E c ρ sup eβ t |1Y (t )|2 + 0 ≤t ≤T
c
ρ
T
0
eβ s |1Z (s)|2 ds +
Lα˜
γ
T
eβ s |1Z (s)|2 ds ,
0
where c , ρ > 0 are constants. By (3.12) and (3.13), we have
Lα˜ c −1 βt 2 + D λ + − λ − c ρ E sup e | 1 Y ( t )| γ 2 γ −1 γ ρ 0≤t ≤T Lα˜ 1/2 c c ˜ Lα˜ −1 Lα βT 2 βt 2 ≤ 1 + D2 + Ee |1ξ | + C8 1 + + E sup e |1S (t )| (ΨT )1/2 γ ρ γ ρ 0≤t ≤T T Lα˜ c 1 1 + E eβ s |1f (s, Ys , Zs )|2 ds. + 1 + D− 2 γ ρ λ t
1−
Lα˜ T
−
T
(3.13)
988
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
1 Finally, it is enough to choose ρ = 2c and λ small enough in the above inequalities (3.12) and (3.13) to obtain (3.10) (L is small enough). The desired result of the proposition follows.
We deduce immediately the following uniqueness result from Proposition 3.2 with ξ ′ = ξ , f ′ = f and S ′ = S . Corollary 3.1. Under assumptions (i)–(v), if the Lipschitz constant L of the generator f is small enough and for two positive constants β and γ the conditions of Proposition 3.2 are satisfied, then there exists at most one solution of the RBSDE with time delayed generators (1)–(4). Proof. Using the previous Proposition 3.2, we obtain immediately Y = Y ′ and Z = Z ′ . Therefore we have K = K ′ , whence uniqueness of the solution of the reflected BSDE with time delayed generator associated with (ξ , f , S ). 3.2. Existence and uniqueness of the solution To begin with, let us first assume that f does not depend on (y, z ), that is, it is a given Ft -progressively measurable process satisfying that ′
(ii ) E ′
T 0
|f (t )|2 dt < ∞.
A solution to the backward reflection problem (BRP, in short) is a triple (Y , Z , K ) which satisfies (1), (3), (4) and
(2 ) Y (t ) = ξ +
T t
f (s)ds + K (T ) − K (t ) −
T t
Z (s)dW (s), 0 ≤ t ≤ T .
The following proposition is from El Karoui et al. (1997) Proposition 5.1. ′
′
Proposition 3.3. Under assumptions (i), (ii ) and (v), the BRP (1), (2 ), (3), (4) has a unique solution {(Y (t ), Z (t ), K (t )); 0 ≤ t ≤ T }. We now deal with the general case of generator i.e. f depends on (y, z ). Theorem 3.1. Assume assumptions (i)–(v) hold. If the Lipschitz constant L of the generator f is small enough and for two positive constants β and γ the conditions of Proposition 3.2 are satisfied, then the RBSDE with time delayed generator (1)–(4) has a unique solution {(Y (t ), Z (t ), K (t )); 0 ≤ t ≤ T }. Proof. Denote by L the space of progressively measurable {(Y (t ), Z (t )); 0 ≤ t ≤ T } with values in R × R which satisfy (1) and (3). We define a mapping Φ from L into itself as follows. Given (U , V ) ∈ L, we define (Y , Z ) = Φ (U , V ) where (Y , Z , K ) is the solution of the RBSDE associated with (ξ , f (t , Ut , Vt ), S ), i.e.,
(Y , Z ) ∈ L, Y (t ) = ξ +
K ∈ S2T (R); T
f (s, Us , Vs )ds + K (T ) − K (t ) −
T
t
Z (s)dW (s),
0 ≤ t ≤ T;
t
Y (t ) ≥ S (t ) and
T
eβ t (Y (t ) − S (t ))dK (t ) = 0,
for some β > 0.
0
The solution of this equation exists and is unique thanks to Proposition 3.3. Now for (U ′ , V ′ ) in L, we define in the same way (Y ′ , Z ′ ) = Φ (U ′ , V ′ ) and U¯ = U − U ′ ,
V¯ = V − V ′ ,
Y¯ = Y − Y ′ ,
Z¯ = Z − Z ′ .
We are now going to prove that there exists a real constant c ∈ R such that the mapping Φ is a strict contraction on L equipped with the norm
1/2 ∥(Y , Z )∥β = E sup eβ t |Y (t )|2 + E 0≤t ≤T
T
eβ t |Z (t )|2 dt
1/2
.
0
In fact, it follows from the arguments similar to those in the proofs of Propositions 3.1 and 3.2 that eβ t |Y¯ (t )|2 +
T
eβ s (β|Y¯ (s)|2 + |Z¯ (s)|2 )ds = 2
t
T
eβ s Y¯ (s)[f (s, Us , Vs ) − f (s, Us′ , Vs′ )]ds
t T
+2
eβ s Y¯ (s)dK¯ (s) − 2
t
≤γ
T
t
T
eβ s Y¯ (s)Z¯ (s)dW (s)
t
eβ s |Y¯ (s)|2 ds +
1
γ
T
t
eβ s |f (s, Us , Vs )
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
− f (s, Us′ , Vs′ )|2 ds − 2
T
989
eβ s Y¯ (s)Z¯ (s)dW (s)
t T
≤γ
Lα˜
eβ s |Y¯ (s)|2 ds + T
−2
T sup eβ t |U¯ (t )|2 +
γ
t
0≤t ≤T
T
eβ s |V¯ (s)|2 ds
0
eβ s Y¯ (s)Z¯ (s)dW (s),
(3.14)
t
for any γ > 0, where we have used the inequality have eβ t E|Y¯ (t )|2 + (β − γ ) E
T
eβ s Y¯ (s)dK¯ (s) ≤ 0. Reorganizing (3.14) and taking the expectation we
T t
eβ s |Y¯ (s)|2 ds + E
Lα˜
γ
sup eβ t |U¯ (t )|2 +
E
0≤t ≤T
eβ s |Z¯ (s)|2 ds
t
t
≤
T
T
eβ s |V¯ (s)|2 ds .
0
By choosing β such that β − γ > 0, we obtain T
E
eβ s |Z¯ (s)|2 ds ≤
0
≤
Lα˜
γ Lα˜
γ
E T sup eβ t |U¯ (t )|2 + 0≤t ≤T
max{1, T }E
T
eβ s |V¯ (s)|2 ds
0
βt
sup e |U¯ (t )|2 +
T
0≤t ≤T
2 ¯ e |V (s)| ds . βs
(3.15)
0
Coming back to (3.14), taking supremum and then the expectation, there exists a real number c˜ such that
T Lα˜ 2 βt ¯ 2 βs ¯ 2 ¯ E sup e |Y (t )| ≤ max{1, T }E sup e |U (t )| + e |V (s)| ds γ 0≤t ≤T 0≤t ≤T 0 T eβ s Y¯ (s)Z¯ (s)dW (s) + 2E sup 0 ≤t ≤T t T Lα˜ ≤ max{1, T }E sup eβ t |U¯ (t )|2 + eβ s |V¯ (s)|2 ds γ 0≤t ≤T 0 T 1 eβ s |Z¯ (s)|2 ds. + E sup eβ t |Y¯ (t )|2 + c˜ E
βt
2
0 ≤t ≤T
(3.16)
0
For the last inequality we have used the B–D–G inequality. Plugging now the inequality in (3.15) and (3.16), we obtain
E
sup eβ t |Y¯ (t )|2 +
T
0≤t ≤T
eβ s |Z¯ (s)|2 ds ≤
0
Lα˜
γ
max{1, T } 2c˜ + 3 E
sup eβ t |U¯ (t )|2 +
0 ≤t ≤T
T
eβ s |V¯ (s)|2 ds .
0
So if we choose γ = 2Lα˜ max{1, T }(2c˜ + 3), since L is small enough and β − γ > 0, we deduce that
E
sup eβ t |Y¯ (t )|2 +
0 ≤t ≤T
T
0
eβ s |Z¯ (s)|2 ds
≤
1 2
E
sup eβ t |U¯ (t )|2 + 0≤t ≤T
T
eβ s |V¯ (s)|2 ds .
0
Hence, the mapping Φ is a strict contraction on L and therefore it has a unique fixed point, which in combination with the associated K is the unique solution of the RBSDE with time delayed generator. Acknowledgments The authors are deeply grateful to the anonymous referee and the editor Prof. Yimin Xiao for the careful reading, valuable comments and correcting some errors, which have greatly improved the quality of the paper. The work was partially done while the second author was visiting Shandong University. He would like to thank Prof. Shige Peng for providing a stimulating working environment. References Crépey, S., Moutoussi, A., 2008. Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison. Ann. Appl. Probab. 18, 2041–2069. Cvitanic, J., Karatzas, I., 1996. Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24, 2024–2056. Cvitanic, J., Ma, J., 2001. Reflected forward–backward SDEs and obstacle problems with boundary conditions. J. Appl. Math. Stochastic Anal. 14, 113–138.
990
Q. Zhou, Y. Ren / Statistics and Probability Letters 82 (2012) 979–990
Dellacherie, C., Meyer, P.A., 1980. Probabilités et Potentiel V–VIII. Hermann, Paris. Delong, Ł, 2010. Applications of time-delayed backward stochastic differential equations to pricing, hedging and management of insurance and financial risks, Preprint. arXiv:1005.4417. Delong, Ł, Imkeller, P., 2010a. Backward stochastic differential equations with time delayed generators-results and counterexamples. Ann. Appl. Probab. 20, 1512–1536. Delong, Ł, Imkeller, P., 2010b. On Malliavinś differentiability of BSDE with time delayed generators driven by Brownian motions and Poisson random measures. Stochastic Process. Appl. 120, 1748–1775. El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, P., Quenez, M.C., 1997. Reflected solution of backward SDE’s, and related obstacle problem for PDE’s. Ann. Probab. 25, 702–737. El Karoui, N., Peng, S., Quenez, M.C., 1997a. Backward stochastic differential equations in finance. Math. Fina. 7, 1–71. El Karoui, N., Quenez, M.C., 1997b. Non-linear pricing theory and backward stochastic differential equations. In: Runggaldier, W.J. (Ed.), Financial Mathematics. In: Lecture Notes in Math, vol. 1656. Springer-Verlag, Berlin, pp. 191–246. Hamadène, S., 2003. BSDEs and risk sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Stochastic Process. Appl. 107, 145–169. Hamadène, S., 2002. Reflectd BSDE’s with discontinuous barrier and application. Stochastics Stochastics Rep. 74, 571–596. Hamadène, S., Lepeltier, J.P., 1995a. Zero-sum stochastic differential games and BSDEs. Systems Control Lett. 24, 259–263. Hamadène, S., Lepeltier, J.P., 1995b. Backward equations, stochastic control and zero-sum stochastic differential games. Stochastics Stochastics Rep. 54, 221–231. Hamadène, S., Lepeltier, J.P., 2000. Reflected BSDEs and mixed game problem. Stochastic Process. Appl. 85, 177–188. Hamadène, S., Popier, A., 2011. Lp -solutions for reflected backward stochastic differential equations. Stochast. Dyna. doi:10.1142/S0219493712003651. Lepeltier, J.P., Xu, M., 2005. Penalization method for reflected backward stochastic differential equations with one rcll barrer. Statist. Probab. Lett. 75, 58–66. Pardoux, E., 1998. Backward stochastic differential equations and viscosity solutions of systems of semi-linear parabolic and elliptic PDEs of second order. In: Decreuefond, L., Gjerd, J., Øksendal, B., Üstünel, A.S. (Eds.), Stochastic analysis and related topics VI (The Gelio Workshop, 1996). In: Progr. Probab., vol. 42. Birkhäuser, Boston, Boston, MA., pp. 79–127. Pardoux, E., 1999. BSDEs, weak convergence and homogenization of semilinear PDEs, nonlinear analysis, differential equations and control. (Montreal, QC, 1998). In: NATO Sci. Ser. C Math. Phys. Sci., vol. 528. Kluwer Acad. Publ., Dordrecht, pp. 503–549. Pardoux, E., Peng, S., 1990. Adapted solution of backward stochastic differential equation. Systems Control Lett. 4, 55–61. Pardoux, E., Peng, S., 1992. Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Rozovskii, B.L., Sowers, R.B. (Eds.), Stochastic Partial Differential Equations and their Applications (Charlotte, NC, 1991). In: Lecture Note in Control and Inform. Sci., vol. 176. pp. 200–217. Peng, S., 1991. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics Stochastics Rep. 37, 61–74. Peng, S., Xu, M., 2005. The smallest g-supermartingale and reflected BSDE with single and double L2 obstacles. Ann. Inst. H. Poincaré Probab. Statist. 41, 605–630. Peng, S., Yang, Z., 2009. Anticipated backward stochastic differential equations. Ann. Probab. 37, 877–902. Reis, G., Réveillac, A., Zhang, J., 2011. FBSDEs with time delayed generators: Lp -solutions, differentiability, representation formulas and path regularity. Stochastic Process Appl. 121, 2114–2150. Ren, Y., Xia, N., 2006. Generalized reflected BSDEs and an obstacle problem for PDEs with a nonlinear Neumann boundary condition. Stochastic Anal. Appl. 24, 1013–1033.