Lp solutions to backward stochastic differential equations with discontinuous generators

Statistics and Probability Letters 83 (2013) 503–510

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Lp solutions to backward stochastic differential equations with discontinuous generators✩ Dejian Tian a , Long Jiang b,∗ , Xuejun Shi b a

School of Management, China University of Mining and Technology, Xuzhou, 221116, PR China

b

School of Sciences, China University of Mining and Technology, Xuzhou, 221116, PR China

article

abstract

info

In this paper, we deal with the Lp (p > 1) solutions to one dimensional backward stochastic differential equations (BSDEs) with discontinuous generators. We obtain an existence theorem of Lp solutions for BSDEs whose generators satisfy a kind of discontinuous condition in y and are uniformly continuous in z. © 2012 Elsevier B.V. All rights reserved.

Article history: Received 26 May 2012 Received in revised form 22 October 2012 Accepted 26 October 2012 Available online 6 November 2012 MSC: 60H10 Keywords: Backward stochastic differential equations Existence theorem Lp solution Uniformly continuous condition

1. Introduction We consider the following one-dimensional backward stochastic differential equation (BSDE for short in the remaining): Yt = ξ +

T



g (s, Ys , Zs )ds − t

T



Zs · dBs ,

0 ≤ t ≤ T,

(1)

t

where T > 0 is a finite constant, ξ is a random variable called the terminal condition, the function g (ω, t , y, z ) : Ω × [0, T ] × R × Rd −→ R is progressively measurable for each (y, z ), called the generator of BSDE (1), and B is a ddimensional Brownian motion. The solution (Y· , Z· ) is a pair of adapted processes. BSDE (1) is denoted by BSDE(g , T , ξ ). Since the pioneering paper of Pardoux and Peng, who established an existence and uniqueness result for L2 solutions to multidimensional BSDEs with square integrable parameters under the Lipschitz assumption on the generator g, much attention has been paid to relaxing the Lipschitz hypothesis on the generator. For instance, Mao (1995), Lepeltier and San Martin (1997) and Fan et al. (2011) etc. Furthermore, many efforts have been made to obtain the Lp solutions to BSDE, such as El Karoui et al. (1997), Briand et al. (2003), Chen (2010) and Fan and Jiang (2012) etc. As for the discontinuous generators, to our knowledge, Jia (2006, 2008) first used the monotonic iteration technique, traced back to Heikkilä and Lakshmikantham (1994) for discontinuous nonlinear differential equations, to deal with the existence of L2 solutions for the BSDE under the assumptions that g is left Lipschitz and left continuous (or right continuous)

✩ Supported by the Fundamental Research Funds for the Central Universities (No. 2012LWB17), National Natural Science Foundation of China (No. 10971220, No. 11101422) and the FANEDD (No. 200919). ∗ Corresponding author. E-mail addresses: [email protected] (D. Tian), [email protected] (L. Jiang), [email protected] (X. Shi).

0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.10.026

504

D. Tian et al. / Statistics and Probability Letters 83 (2013) 503–510

in y, and Lipschitz in z. Zheng and Zhou (2008) and Lin (2010) then applied this method to settle the existence of L2 solutions for the reflected BSDEs and the backward doubly stochastic differential equations with discontinuous generators. The objective of this paper is to explore Lp (p > 1) solutions for BSDEs with discontinuous generators. We obtain the existence of Lp (p > 1) solutions for BSDEs whose generators satisfy a kind of discontinuous conditions in y which is weaker than the left Lipschitz condition, and satisfy the uniformly continuous condition in z. Our results generalize the results of Jia (2006, 2008). The key idea for dealing with the problem consists in constructing a monotonic sequence of solutions to BSDE and then passing to the limit with the help of the recent results of Fan and Jiang (2012). The paper is organized as follows. We introduce some preliminaries and lemmas in Section 2. We obtain the existence theorem for BSDEs with discontinuous generators in Section 3. 2. Preliminaries and lemmas Let (Ω , F , P ) be a complete probability space carrying a d-dimensional standard Brownian motion (Bt )t ≥0 . Let (Ft )t ≥0 denote the natural filtration generated by (Bt )t ≥0 , augmented by P-null sets of F . For any positive integer n, if z ∈ Rn , let |z | denote its Euclid norm. For what follows, let us fix two positive numbers T > 0 and p > 1. Let S p be the set of all continuous and adapted processes θ = (θt )t ∈[0,T ] with values in R such that

 ∥θ∥S p = E

sup |θt |p

1/p

< +∞.

0≤t ≤T

And we denote M p be the set of all (Ft )-progressively measurable processes θ = (θt )t ∈[0,T ] with values in Rd such that



p/2 1/p

T 2

∥θ∥Mp = E

|θt | dt

< +∞.

0

Besides, let S be the set of all nondecreasing, continuous, sub-additive and linear growth functions ρ(·) from R+ to itself with ρ(0) = 0 and ρ(u) > 0 for u > 0. And in this paper by a Lp solution to BSDE (1), we mean a pair of progressively measurable processes (Yt , Zt )t ∈[0,T ] ∈ S p × M p and satisfies BSDE (1). We will use the following assumptions (H1)–(H3): (H1) g (t , ·, z ) is right continuous, and there exists a concave function φ ∈ S satisfying

1 dx 0+ φ(x)



z ∈ Rd , y1 , y2 ∈ R and y1 ≥ y2 , dP × dt-a.e.,

= +∞ such that for all

g (t , y1 , z ) − g (t , y2 , z ) ≥ −φ(y1 − y2 ). (H2) g (t , y, ·) satisfies the uniformly continuous condition, i.e., there exists a function ψ ∈ S such that for all y ∈ R, z1 , z2 ∈ Rd , dP × dt-a.e.,

|g (t , y, z1 ) − g (t , y, z2 )| ≤ ψ(|z1 − z2 |). (H3) There exist two generators g1 and g2 such that for all y ∈ R, z ∈ Rd , g1 (t , y, z ) ≤ g (t , y, z ) ≤ g2 (t , y, z ),

dP × dt-a.e.,

and for any ξ ∈ L (Ω , FT , P ), BSDE (gi , T , ξ ) (i = 1, 2) has at least one solution, denoted by (Y 0· , Z 0· ) and (Y¯·0 , Z¯·0 )     p

respectively. And P-a.s., ∀t ∈ [0, T ], Y 0t ≤ Y¯t0 and E

T 0

|g1 (s, Y 0s , Z 0s )|ds

p

+

T 0

|g2 (s, Y¯s0 , Z¯s0 )|ds

p

< +∞.

We list some useful lemmas. Lemma 2.2 is a special case of Proposition 1 in Fan and Jiang (2012). Lemma 2.1 (Fan and Jiang, 2012). Let the generator g satisfies the following conditions (A1) and (A2): (A1) E



T 0

|g (s, 0, 0)|ds

p 

< +∞.

(A2) There exists a function ψ ∈ S and a concave function φ ∈ S with y1 , y2 , z1 , z2 ,

1 dx 0+ φ(x)



= +∞ such that dP × dt-a.e., for each

|g (t , y1 , z1 ) − g (t , y2 , z2 )| ≤ φ(|y1 − y2 |) + ψ(|z1 − z2 |). Then for each ξ ∈ Lp (Ω , FT , P ), BSDE (ξ , T , g ) has a unique Lp solution. Lemma 2.2 (Fan and Jiang, 2012). Let ξ ∈ Lp (Ω , F , P ), g and g ′ be two generators of BSDEs. And let (y· , z· ) and (y′· , z·′ ) be, respectively, Lp solutions to BSDE (g , T , ξ ) and BSDE (g ′ , T , ξ ). If g (resp. g ′ ) satisfies (A1) and (A2) and dP × dt-a.e., g (t , y′t , zt′ ) ≤ g ′ (t , y′t , zt′ ) (resp. g (t , yt , zt ) ≤ g ′ (t , yt , zt )), then we have P-a.s., ∀t ∈ [0, T ], yt ≤ y′t .

D. Tian et al. / Statistics and Probability Letters 83 (2013) 503–510

505

Lemma 2.3 (Briand et al., 2003). If (Y· , Z· ) is a solution of the BSDE (1), γ ≥ 1, c (γ ) = γ [(γ − 1) ∧ 1]/2 and 0 ≤ t ≤ u ≤ T , then

|Yt |γ + c (γ )

u



|Ys |γ −2 IYs ̸=0 ds ≤ |Yu |γ + γ

u



|Ys |γ −1 Yˆs g (s, Ys , Zs )ds − γ

u

|Ys |γ −1 Yˆs Zs · dBs ,

t

t

t



where xˆ := |x|−1 Ix̸=0 for any x ∈ R. 3. Existence theorem for BSDEs with discontinuous generators The main result of this paper is as follows. Theorem 3.1. Let the assumptions (H1)–(H3) hold for g. Then for any ξ ∈ Lp (Ω , FT , P ), there exists at least one Lp solution (y· , z· ) to BSDE (g , T , ξ ). We give an example which we do not know whether BSDE (g , T , ξ ) has a Lp solution or not by the existing references. But by our Theorem 3.1 we know that BSDE (g , T , ξ ) has at least one Lp solution. Example 3.1. Let g (t , y, z ) = p(y) + q(|y|) + by M and

 q(x) =

√

|z | + |Bt |, where p(·) is a nondecreasing right continuous function bounded

0 < x ≤ δ; x > δ; other cases

−xlnx q′ (δ−)(x − δ) + q(δ) 0

with δ > 0 small enough. √ √ √ Setting ψ(x) = x, φ(x) = q(x), g1 (t , y, z ) = −M + q(|y|) + |z | + |Bt | and g2 (t , y, z ) = M + q(|y|) + |z | + |Bt |. It is not hard to verify that  g satisfies (H2) and (H3). g also satisfies (H1). In fact, since we have known that q(·) is a concave function in S satisfying 0+ φ(1x) dx = +∞ by Mao (1995), thus for all z ∈ Rd , y1 , y2 ∈ R and y1 ≥ y2 , dP × dt-a.e., we have g (t , y1 , z ) − g (t , y2 , z ) ≥ p(y1 ) − p(y2 ) − q(y1 − y2 ) ≥ −φ(y1 − y2 ). Thus for ξ ∈ Lp (Ω , FT , P ), BSDE (g , T , ξ ) has a Lp solution by Theorem 3.1. In order to prove Theorem 3.1, inspired by Jia (2006), we construct a sequence of BSDEs as follows: i = 0, 1, 2, . . . Y¯t0 = ξ +

T



g2 (s, Y¯s0 , Z¯s0 )ds − t

Y¯ti+1 = ξ +

T



Z¯s0 · dBs ,

0 ≤ t ≤ T,



T

 g (s, Y¯si , Z¯si ) + φ(|Y¯si − Y¯si+1 |) + ψ(|Z¯si − Z¯si+1 |) ds −



t

Y 0t = ξ +

T



Z¯si+1 · dBs ,

0 ≤ t ≤ T,

(3)

t

T



(2)

t

g1 (s, Y 0s , Z 0s )ds −

T



t

Z 0s · dBs ,

0 ≤ t ≤ T.

(4)

t

Remark 3.1. In view of (H3), (Y 0· , Z 0· ) and (Y¯·0 , Z¯·0 ) are Lp solutions of BSDE (4) and BSDE (2) respectively. From the proof of the following Lemma 3.1, we can see that BSDE (3) has a unique Lp solution for any i ≥ 0. Besides, we denote the constant of linear growth for φ and ψ by K , i.e., 0 ≤ φ(x) + ψ(x) ≤ K (1 + x), ∀x ∈ R+ . Lemma 3.1. Let (H1)–(H3) hold for g and let ξ ∈ Lp (Ω , FT , P ), then for any i ≥ 0, we have (i) E



T 0

|g (s, Y¯si , Z¯si )|ds

p 

< +∞;

(ii) BSDE (3) has a unique L solution, denoted by (Y¯·i+1 , Z¯·i+1 ); p

(iii) P-a.s., ∀t ∈ [0, T ], Y 0t ≤ Y¯ti+1 ≤ Y¯ti ≤ Y¯t0 . Proof. We use induction to prove this Lemma 3.1. For i = 0, by (H1), (H2) and Y 0t ≤ Y¯t0 , we have g (t , Y¯t0 , Z¯t0 ) − g (t , Y 0t , Z 0t ) ≥ −φ(Y¯t0 − Y 0t ) − ψ(|Z¯t0 − Z 0t |). Then from (H3), we obtain g1 (s, Y 0s , Z 0s ) − φ(Y¯s0 − Y 0s ) − ψ(|Z¯s0 − Z 0s |) ≤ g (s, Y¯s0 , Z¯s0 ) ≤ g2 (s, Y¯s0 , Z¯s0 ).

(5)

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D. Tian et al. / Statistics and Probability Letters 83 (2013) 503–510

Due to the fact that (Y 0· , Z 0· ) and (Y¯·0 , Z¯·0 ) are Lp solutions of BSDE (4) and BSDE (2), by the linear growth of φ and ψ , we deduce that



T

φ( ¯ − Ys0

E

Y 0s

)ds

p 



p

≤ K E

0

T

(1 + ¯ − Ys0

Y 0s

)ds

p 

0 p

p



≤ K T E

 0 p 0 ¯ sup |1 + Ys − Y s | < ∞,

(6)

0≤s≤T



p 

T

ψ(| ¯ − Zs0

E

Z 0s

|)ds

p 2

p



T

(1 + | ¯ − Zs0

≤K T E

0

Z 0s

2

 2p 

|) ds

< ∞.

(7)

0

In view of the integrability of g1 and g2 in (H3), from (5)–(7), we get



T

|g (s, ¯ , ¯ )|ds Ys0

E

Zs0

p 

< +∞.

(8)

0

Thus BSDE (3) has a unique Lp solution (Y¯·1 , Z¯·1 ) by Lemma 2.1. Then by Lemma 2.2, in view of (5), we have P-a.s., ∀t ∈ [0, T ],

Y 0t ≤ Y¯t1 ≤ Y¯t0 .

For i = 1, since Y 0t ≤ Y¯t1 ≤ Y¯t0 , then by (H1)–(H3), we obtain g1 (s, Y 0s , Z 0s ) − φ(Y¯s1 − Y 0s ) − ψ(|Z¯s1 − Z 0s |) ≤ g (s, Y¯s1 , Z¯s1 ),

(9)

g (s, Y¯s1 , Z¯s1 ) ≤ g (s, Y¯s0 , Z¯s0 ) + φ(Y¯s0 − Y¯s1 ) + ψ(|Z¯s0 − Z¯s1 |).

In view of (8) and the integrability of g1 in (H3), we derive, using the similar estimate results of (6), (7) and by the above inequalities (9), that



T

|g (s, Y¯s1 , Z¯s1 )|ds

E

p 

< +∞.

0

BSDE (3) therefore has a unique Lp solution (Y¯·2 , Z¯·2 ) by Lemma 2.1. Then by the inequalities (9) and Lemma 2.2, we have P-a.s., ∀t ∈ [0, T ],

Y 0t ≤ Y¯t2 ≤ Y¯t1 .

Supposethe assertions in Lemma   3.1 hold for i

T

(Y¯·k , Z¯·k ), E

0

|g (s, Y¯sk−1 , Z¯sk−1 )|ds

p

k − 1, i.e., BSDE (3) has a unique Lp solution

=

< +∞ and P-a.s., ∀t ∈ [0, T ], Y 0t ≤ Y¯tk ≤ Y¯tk−1 . Then by the similar argument

as the case i = 1, we get g1 (s, Y 0s , Z 0s ) − φ(Y¯sk − Y 0s ) − ψ(|Z¯sk − Z 0s |) ≤ g (s, Y¯sk , Z¯sk ), g (s, Y¯sk , Z¯sk ) ≤ g (s, Y¯sk−1 , Z¯sk−1 ) + φ(Y¯sk−1 − Y¯sk ) + ψ(|Z¯sk−1 − Z¯sk |). By induction hypothesis we can deduce



T

p 

|g (s, ¯ , ¯ )|ds Ysk

E

Zsk

< +∞.

0

Thus, from Lemma 2.1, BSDE (3) has a unique Lp solution (Y¯·k+1 , Z¯·k+1 ). Then by Lemma 2.2, we have P-a.s., ∀t ∈ [0, T ],

Y 0t ≤ Y¯tk+1 ≤ Y¯tk .

The proof of Lemma 3.1 is complete.



Lemma 3.2. Let (H1)–(H3) hold for g and let ξ ∈ Lp (Ω , FT , P ), then we have supi E

 

T 0

|Z¯si |2 ds

p/2 

< +∞.

Proof. For i ≥ 0, setting Y∗i = supt ∈[0,T ] |Y¯ti | and Y 0∗ = supt ∈[0,T ] |Y 0t |. Since Y¯ i is decreasing by Lemma 3.1, we get that

 



 i p ¯ E sup |Y∗ | ≤ E [|Y 0∗ |p ] + E [|Y¯∗0 |p ] < +∞,

(10)

i

P-a.s., ∀t ∈ [0, T ],

Y¯ (t ) := lim Y¯ i (t ), i→+∞

 E

sup |Y¯t |p 0≤t ≤T



< +∞.

(11)

D. Tian et al. / Statistics and Probability Letters 83 (2013) 503–510

507

Applying Itô’s formula to |Y¯ti+1 |2 , we can get

|Y¯0i+1 |2 +

T



T



|Z¯si+1 |2 ds = |ξ |2 + 2

Y¯si+1 Z¯si+1 · dBs , 0

0

0

T



Y¯si+1 ϕsi+1 ds − 2

where ϕsi+1 = g (s, Y¯si , Z¯si ) + φ(Y¯si − Y¯si+1 ) + ψ(|Z¯si − Z¯si+1 |). On the other hand, since Y 0t ≤ Y¯ti ≤ Y¯t0 , then by (H1)–(H3) we obtain

    |g (s, Y¯si , Z¯si )| ≤ g1 (s, Y 0s , Z 0s ) − φ(Y¯si − Y 0s ) − ψ(|Z¯si − Z 0s |) + g2 (s, Y¯s0 , Z¯s0 ) + φ(Y¯s0 − Y¯si ) + ψ(|Z¯s0 − Z¯si |) .

(12)

Thus, setting fs := |g1 (s, Y 0s , Z 0s )| + |g2 (s, Y¯s0 , Z¯s0 )|, in view of the linear growth of functions φ and ψ , we deduce that 2Y¯si+1 ϕsi+1 ≤ 2|Y¯si+1 |fs + 2K |Y¯si+1 | 3 + |Y 0s | + |Y¯s0 | + |Y¯si+1 | + 3|Y¯si |





  + 2K |Y¯si+1 | 3 + |Z 0s | + |Z¯s0 | + |Z¯si+1 | + 3|Z¯si | . Furthermore, thanks to (iii) of Lemma 3.1 and the inequality 2ab ≤ 2a2 + b2 /2, we have 2Y¯si+1 ϕsi+1 ≤ 2|Y¯si+1 |fs + 2K |Y¯si+1 | 6 + 5|Y 0s | + 5|Y¯s0 | + (K + 2K 2 )|Y¯si+1 |2





+ K (|Z 0s | + |Z¯s0 |)2 + 6K |Y¯si+1 ∥ Z¯si | + |Z¯si+1 |2 /2.

(13)

Thus, by inequality (13), we deduce that 1

T



2

|Z¯si+1 |2 ds ≤ |ξ |2 + 2|Y¯∗i+1 |

T



0

(fs + 6K ) ds + 10KT |Y¯∗i+1 |(|Y 0∗ | + |Y¯∗0 |) 0

+ (K + 2K 2 )T |Y¯∗i+1 |2 + K

T

 0

T



+ 6K |Y¯∗i+1 |

0

  i ¯ |Zs |ds + 2 

0

(|Z 0s | + |Z¯s0 |)2 ds  T  ¯Ysi+1 Z¯si+1 · dBs  . 

Choosing a suitable constant C , it follows that



T



Zsi+1 2 ds

|¯

|

≤ C 1 + |ξ |2 + |Y¯∗i+1 |2 + |Y 0∗ |2 + |Y¯∗0 |2 +

2

T



fs ds 0

0 T



(|Z 0s | + |Z¯s0 |)2 ds + |Y¯∗i+1 |

+

T



|Z¯si |2 ds

1/2

0

0

  + 

T

0

  Y¯si+1 Z¯si+1 · dBs  ,

and thus that T



Zsi+1 2 ds

|¯

p/2

|

 ≤ cp 1 + |ξ |p + |Y¯∗i+1 |p + |Y 0∗ |p + |Y¯∗0 |p +

p

T



fs ds 0

0 T



(|Z 0s | + |Z¯s0 |)2 ds

+

p/2 

+ cp |Y¯∗i+1 |p/2

0

  + cp 

|Z¯si |2 ds

p/2  Y¯si+1 Z¯si+1 · dBs  .

(14)

But by the BDG inequality, we get

  cp E 

T

Ysi+1 Zsi+1

¯

¯

0

 p/2  p/4  T  i + 1 2 i + 1 2 · dBs  ≤ dp E |Y¯s | |Z¯s | ds 0   T p/4  i+1 p/2 i+1 2 ¯ ¯ ≤ dp E |Y∗ | |Zs | ds , 0

and thus

  cp E 

0

T

Ysi+1 Zsi+1

¯

¯

p/4

0 T

0

T



 p/2   1 dp i + 1 p · dBs  ≤ E [|Y¯∗ | ] + E 2 2

T

Zsi+1 2 ds

|¯ 0

|

p/2 

.

508

D. Tian et al. / Statistics and Probability Letters 83 (2013) 503–510

Coming back to the estimate (14), in view of estimate (10) we get



T

Zsi+1 2 ds

|¯

E

p/2 

|

 0 p

0 p

≤ ep E 1 + |Y ∗ | + |Y¯∗ | +

p

T



+ ep E |Y¯∗i+1 |p/2

T



|Z¯si |2 ds

Zs0

2

p/2 

0

0



(| | + | ¯ |) ds Z 0s

+

fs ds

0

T



p/4 

.

0

Using the inequality 2ab ≤ 2a2 + b2 /2 and the estimate (10), we obtain,



T

|Z¯si+1 |2 ds

E

p/2 

 ≤ kp E 1 + |Y 0∗ |p + |Y¯∗0 |p +

fs ds 0

0 T



(| | + | ¯ |) ds Z 0s

+

Zs0

2

p/2 





T f ds 0 s

Setting Cp = kp E 1 + |Y 0∗ |p + |Y¯∗0 |p + T

|Z¯si+1 |2 ds

E

p/2 

p



1

2

Thus we finish the proof of Lemma 3.2.



2

(|Z 0s | + |Z¯s0 |)2 ds 0

|Z¯si |2 ds

T

p/2 

p/2 

Zsi 2 ds

|¯ |

.

0

 T

T

≤ Cp + E

0

+

1

+ E

0



p

T



p/2 

, thus

.

0



Lemma 3.3. Let (H1)–(H3) hold for g and let ξ ∈ Lp (Ω , FT , P ), then (Y¯·i , Z¯·i )i≥1 , constructed by BSDEs (3), is a Cauchy sequence in S p × M p . Proof. Firstly, we show that (Y¯·i ) is a Cauchy sequence in S p . For all i ≥ 0, let

ϕsi+1 = g (s, Y¯si , Z¯si ) + φ(Y¯si − Y¯si+1 ) + ψ(|Z¯si − Z¯si+1 |). j

For any i, j ≥ 1, applying Lemma 2.3 to |Y¯ti − Y¯t |p , we have

|Y¯ti − Y¯tj |p + c (p)

T

 t

|Y¯si − Y¯sj |p−2 IY¯ i −Y¯ j ̸=0 ds ≤ p s

s

T



 |Y¯si − Y¯sj |p−1 (Y¯si − Y¯sj )(ϕsi − ϕsj )ds t T



 |Y¯si − Y¯sj |p−1 (Y¯si − Y¯sj )(Z¯si − Z¯sj ) · dBs .

−p t

Using the standard estimate methods in the Proposition 3.2 of Briand et al. (2003) or in the Lemma 4 of Chen (2010), we get

 E

T

  i j p ¯ ¯ sup |Ys − Ys | ≤ cp E

0≤s≤T

 i j p−1 i j ¯ ¯ |Ys − Ys | |ϕs − ϕs |ds .

(15)

0

Taking into account the estimate (12) and the (iii) of Lemma 3.1, we can choose a suitable constant C such that

|ϕsi − ϕsj | ≤ C (1 + fs + |Y 0s | + |Y¯s0 | + |Z 0s | + |Z¯s0 | + |Z¯si−1 | + |Z¯si | + |Z¯sj−1 | + |Z¯sj |),

(16)

where fs = |g1 (s, Y 0s , Z 0s )| + |g2 (s, Y¯s0 , Z¯s0 )|. Combining (15) and (16), we get

 E

sup |Y¯si − Y¯sj |p



T



|Y¯si − Y¯sj |p−1 (1 + |fs | + |Y 0s | + |Y¯s0 |)ds

≤ dp E

0≤s≤T



0 T

 + dp E

 |Y¯si − Y¯sj |p−1 (|Z 0s | + |Z¯s0 | + |Z¯si−1 | + |Z¯si | + |Z¯sj−1 | + |Z¯sj |)ds .

(17)

0

Due to the fact that (11), (10) and (H3), Lebesgue’s dominated convergence theorem yields that the first term of the right side of inequality (17) converges to zero as i, j → +∞. As for the second term of the right side of inequality (17), by Hölder’s inequality, it yields that T



|Y¯si − Y¯sj |p−1 (|Z 0s | + |Z¯s0 | + |Z¯si−1 | + |Z¯si | + |Z¯sj−1 | + |Z¯sj |)ds

E 0



D. Tian et al. / Statistics and Probability Letters 83 (2013) 503–510

  ≤E

T

|Y¯si − Y¯sj |2(p−1) ds

 21 

0



509

 1

T

(|Z 0s | + |Z¯s0 | + |Z¯si−1 | + |Z¯si | + |Z¯sj−1 | + |Z¯sj |)2 ds

2



0

T

|Y¯si − Y¯sj |2(p−1) ds

≤E

 2(pp−1) 

p−1 p

1



T

(|Z 0s | + |Z¯s0 | + |Z¯si−1 | + |Z¯si | + |Z¯sj−1 | + |Z¯sj |)2 ds

×E

0

 2p  p

0



T

Ysj 2(p−1) ds

Ysi

|¯ − ¯ |

≤ Cp E

 2(pp−1) 

p−1 p



(| ¯ | + | | )ds Zsi 2

sup E i ≥0

0

p/2  1p

T

Z 0s 2

.

0

By (11), (10), (H3) and Lemma 3.2, Lebesgue’s dominated convergence theorem yields that the second term of the right side of inequality (17) converges to zero when i, j → +∞. Thus (Y¯·i ) is a Cauchy sequence in S p . Next, we claim that (Z¯·i ) is a Cauchy sequence in M p . j In fact, for any i, j ≥ 1, using Itô’s lemma to |Y¯ti − Y¯t |2 , we can get

|Y¯0i − Y¯0j |2 +

T



T



T



(Y¯si − Y¯sj )(Z¯si − Z¯sj ) · dBs  t   T   ≤2 |Y¯si − Y¯sj ||ϕsi − ϕsj |ds + 2 sup  (Y¯si − Y¯sj )(Z¯si − Z¯sj ) · dBs  . 0≤s≤T

|Z¯si − Z¯sj |2 ds = 2

(Y¯si − Y¯sj )(ϕsi − ϕsj )ds − 2

0

0

0

0

0

Then, using the standard estimate method in Lemma 3.1 of Briand et al. (2003) or Lemma 4 in Chen (2010), we can obtain that



T

Zsi

Zsj 2 ds

|¯ − ¯ |

E

 2p 



Ysj p

sup | ¯ − ¯ |

≤ Cp E

0≤s≤T

0

Ysi



 + Cp E

Ysi

Ysj p

sup | ¯ − ¯ |

0≤s≤T

 12



T

|ϕ − ϕ |ds i s

E

j s

p  12

.

0

From the estimate (16) and the fact that (Y¯·i ) is a Cauchy sequence in S p , we deduce that



T

|Z¯si − Z¯sj |2 ds

E

 2p 

→ 0,

i, j → +∞.

0

The proof is complete.



Now we can prove the existence theorem. Proof of Theorem 3.1. We follow the arguments in Lepeltier and San Martin (1997). We denote by Z¯ the limit of Z¯ i in M p , that is to say,



T

Zsi

| ¯ − Z¯s | ds

lim E

i→+∞

2

 2p 

= 0.

(18)

0

By the similar arguments of Lemma 2.5 in Kobylanski (2000) we can assume, passing a sequence if necessary, that T Z¯·i → Z¯· , dP × dt-a.e., and E [ 0 supi |Z¯si |ds] < +∞. Moreover, from (11), (iii) of Lemmas 3.1 and 3.3, we get Y¯·i → Y¯· , dP × dt-a.e., E



T 0



< +∞ and Y¯· ∈ S p . Due to the fact that g is right-continuous in y and continuous in z and the continuity of functions φ and ψ and φ(0) = ψ(0) = 0, we get that as i → +∞, dP × dt-a.e., ϕti+1 = g (t , Y¯ti , Z¯ti ) + φ(|Y¯ti − Y¯ti+1 |) + ψ(|Z¯ti − Z¯ti+1 |) → g (t , Y¯t , Z¯t ). supi |Y¯si |ds

On the other hand, from (12), we can choose a suitable constant C such that

  |ϕsi+1 | ≤ C 1 + |g1 (s, Y 0s , Z 0s )| + |g2 (s, Y¯s0 , Z¯s0 )| + |Y 0s | + |Z 0s | + sup |Y¯si | + sup |Z¯si | ∈ L1 (0, T ; dt ). i

i

Thus, for almost all ω and uniformly in t, T



|ϕsi+1 |ds → t

T



|g (s, Y¯s , Z¯s )|ds,

as i → +∞.

t

By virtue of the BDG inequality we know

  T p    i E sup  (Z¯s − Z¯s )dBs  ≤ Cp E 0 ≤t ≤T 

t

0

T

Zsi

2

| ¯ − Z¯s | ds

 2p 

→ 0,

as i → +∞.

510

D. Tian et al. / Statistics and Probability Letters 83 (2013) 503–510

Choosing a subsequence if necessary, we can get that for almost all ω

 T    i  ¯ ¯ sup  (Zs − Zs )dBs  → 0, 0≤t ≤T

as i → +∞.

t

Hence, (Y¯· , Z¯· ) ∈ S p × M p is a solution to BSDE (1). The proof is complete.



Acknowledgments The authors would like to thank the anonymous referees for their careful reading and helpful suggestions. Thanks are due to Dr. Fan S for his help. References Briand, Ph., Delyon, B., Hu, Y., Pardoux, E., Stoica, L., 2003. Lp solutions of backward stochastic differential equations. Stochastic Process. Appl. 108, 109–129. Chen, S., 2010. Lp solutions of one-dimensional backward stochastic differential equations with continuous coefficients. Stoch. Anal. Appl. 28, 820–841. El Karoui, N., Peng, S., Quenez, M.C., 1997. Backward stochastic differential equations in finance. Math. Finance 7, 1–71. Fan, S., Jiang, L., 2012. Lp (p > 1) solutions for one-dimensional BSDEs with linear-growth generators. J. Appl. Math. Comput. 38, 295–304. Fan, S., Jiang, L., Tian, D., 2011. One-dimensional BSDEs with finite and infinite time horizons. Stochastic Process. Appl. 121, 427–440. Heikkilä, S., Lakshmikantham, V., 1994. Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations. Marcel Dekker Inc., New York. Jia, G., 2006. A generalized existence theorem of BSDEs. C. R. Acad. Sci. Paris, Ser. I 342 (9), 685–688. Jia, G., 2008. A class of backward stochastic differential equations with discontinuous coefficients. Statist. Probab. Lett. 78, 231–287. Kobylanski, M., 2000. Backward stochastic differential equations and partial equations with quadratic growth. Ann. Probab. 28, 259–276. Lepeltier, J.P., San Martin, J., 1997. Backward stochastic differential equations with continuous coefficients. Statist. Probab. Lett. 34, 425–430. Lin, Q., 2010. A generalized existence theorem of backward doubly stochastic differential equations. Acta Math. Sin. (Engl. Ser.) 26, 1525–1534. Mao, X., 1995. Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients. Stochastic Process. Appl. 58, 281–292. Zheng, S., Zhou, S., 2008. A generalized existence theorem of reflected BSDEs with double obstacles. Statist. Probab. Lett. 78, 528–536.