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A class of BSDE with integrable parameters
Statistics and Probability Letters 80 (2010) 2024–2031
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A class of BSDE with integrable parameters✩ ShengJun Fan ∗ , DeQun Liu College of Sciences, China University of Mining & Technology, Xuzhou, Jiangsu 221116, PR China
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abstract
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Article history: Received 22 March 2010 Received in revised form 13 September 2010 Accepted 13 September 2010 Available online 19 September 2010
In this paper, we establish an existence and uniqueness result for solutions to onedimensional backward stochastic differential equations (BSDEs) with only integrable parameters, where the generator g is α -Hölder (0 < α < 1) continuous in z. © 2010 Elsevier B.V. All rights reserved.
MSC: primary 60H10 Keywords: Backward stochastic differential equation Existence and uniqueness Integrable parameters Hölder continuous
1. Introduction In this paper, 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 ,
t ∈ [0, T ],
(1)
t
where ξ is a random variable called the terminal condition, the random function g (ω, t , y, z ) : Ω × [0, T ] × R × Rd → R is progressively measurable for each (y, z ), called the generator of the BSDE (1), and B is a d-dimensional Brownian motion. The solution (y· , z· ) is a pair of adapted processes. The triple (ξ , T , g ) is called the parameters of the BSDE (1). Such equations, in the nonlinear case, were firstly introduced in Pardoux and Peng (1990), who established an existence and uniqueness result for solutions to multidimensional BSDEs with square integrable parameters under the Lipschitz assumption of the generator g. Since then, much attention has been paid towards relaxing the Lipschitz hypothesis on g, for instance, some details can be found in Mao (1995), Lepeltier and San Martín (1997), Kobylanski (2000), Jia (2008) and Briand and Hu (2008) etc., most of which dealt with BSDEs with square-integrable parameters. Peng (1997) introduced the notion of g-martingales by solutions to BSDEs, which can be viewed, in some sense, as nonlinear martingales. Since the classical theory of martingales is carried in the integrable space, the question of solving a BSDE with only integrable parameters comes up naturally, as has been pointed out in Briand et al. (2003). It has been known that it is more difficult to solve BSDEs with only integrable parameters than those with square-integrable parameters. In fact, to our knowledge, BSDEs with only integrable parameters were solved in only few papers such as Peng (1997), ✩ Supported by the National Natural Science Foundation of China (No. 10971220), the FANEDD (No. 200919) and the Fundamental Research Funds for the Central Universities. ∗ Corresponding author. E-mail addresses: [email protected] (S. Fan), [email protected] (D. Liu).
0167-7152/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2010.09.009
S. Fan, D. Liu / Statistics and Probability Letters 80 (2010) 2024–2031
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Briand et al. (2002, 2003). The objective of this paper is to establish an existence and uniqueness result in this direction, where the generator g of the BSDE is α -Hölder (0 < α < 1) continuous in z. Let us close this introduction by giving the notations which we will use in the whole paper. In what follows, let us fix a real number T > 0. Let (Ω , F , P ) be a probability space carrying a standard d-dimensional Brownian motion (Bt )t ≥0 . Let (Ft )t ≥0 be the natural σ -algebra generated by (Bt )t ≥0 and assume FT = F . For every positive integer n, we use | · | to denote the norm of Euclidean space Rn . For each real p > 0, S p denotes the set of real-valued, adapted and continuous processes (Yt )t ∈[0,T ] such that
p
1∧1/p
‖Y ‖S p := E[ sup |Yt | ] t ∈[0,T ]
< +∞.
If p ≥ 1, ‖ · ‖S p is a norm on S p and if p ∈ (0, 1), (X , X ′ ) −→ ‖X − X ′ ‖S p defines a distance on S p . Under this metric, S p is complete. Moreover, let Mp denote the set of (equivalent classes of) (Ft )-progressively measurable, Rd -valued processes {Zt , t ∈ [0, T ]} such that
∫
T 2
‖Z ‖Mp := E
p/2 1∧1/p
|Zt | dt
< +∞.
0
For p ≥ 1, Mp is a Banach space endowed with this norm and for p ∈ (0, 1), Mp is a complete metric space with the resulting distance. We set S = ∪p>1 S p and let us recall that a continuous process (Yt )t ∈[0,T ] belongs to the class (D) if the family {Yτ : τ is stopping time bounded by T } is uniformly integrable. For a process Y in the class (D), we put
‖Y ‖1 = sup{E[Yτ ], τ is stopping time bounded by T }. The space of (Ft )-progressively measurable continuous processes which belong to the class (D) is complete under this norm. As mentioned above, we will deal only with the one-dimensional BSDE which is an equation of type (1), where the terminal condition ξ is FT -measurable and the generator g is (Ft )-progressively measurable for each (y, z ). In this paper, by a solution to the BSDE (1) we mean a pair of (Ft )-adapted processes (y· , z· ) with values in R × Rd such that dP-a.s., t −→ yt is continuous, t −→ zt belongs to L2 (0, T ), t −→ f (t , yt , zt ) belongs to L1 (0, T ) and dP-a.s., the BSDE (1) holds true for each t ∈ [0, T ]. 2. Main result In this section, we will put forward and prove our main result (Theorem 1). Firstly, let us introduce the following assumptions (H1) and (H2):
|g (s, 0, 0)|ds < +∞. (H2) There exist two constants µ, α > 0 such that dP × dt-a.s., (H1) E |ξ | +
T 0
∀ y1 , y2 , z1 , z2 ,
|g (ω, t , y1 , z1 ) − g (ω, t , y2 , z2 )| ≤ µ|y1 − y2 | + µ|z1 − z2 |α .
Remark 1. The condition (H2), one of the main assumptions of this paper, is equivalent to the following conditions (A1) and (A2): (A1) g is Lipschitz continuous in y uniformly with respect to (ω, t , z ), i.e., there exists a constant µ > 0, such that, dP × dt-a.s.,
∀ y1 , y2 , z ,
|g (ω, t , y1 , z ) − g (ω, t , y2 , z )| ≤ µ|y1 − y2 |.
(A2) g is Hölder continuous in z uniformly with respect to (ω, t , y), i.e., there exist two constants µ, α > 0 such that dP × dt-a.s.,
∀ y, z1 , z2 ,
|g (ω, t , y, z1 ) − g (ω, t , y, z2 )| ≤ µ|z1 − z2 |α .
The main result of this paper is as follows. Theorem 1. Let (H1) and (H2) with α ∈ (0, 1) hold. Then, the BSDE with parameters (ξ , T , g ) has a unique solution (y· , z· ) such that y· is of class (D) and z· ∈ Mβ for some β > α . Moreover, (y· , z· ) belongs to S β × Mβ for all β ∈ (0, 1). Remark 2. It is clear that under the conditions of Theorem 1, the generator g is not necessarily uniformly Lipschitz continuous with respect to z. The following Example 1, to our knowledge, is not covered by any existing results.
√
Example 1. Let g (t , y, z ) = sin y + |z | + |Bt |. Then, it follows from Theorem 1 that for each ξ ∈ L1 (Ω , FT , P ), the BSDE with parameters (ξ , T , g ) has a unique solution (y· , z· ) such that y· is of class (D) and z· ∈ Mβ for some β > 1/2.
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Now, let us start the proof of Theorem 1. Proof of Theorem 1. Uniqueness: We first establish the following Proposition 1, and the uniqueness result is a direct consequence of it. Proposition 1. Let g satisfy (H2) with α ∈ (0, 1], and let (y· , z· ) and (y′· , z·′ ) be respectively a solution for the BSDE with parameters (ξ , T , g ) and (ξ ′ , T , g ) such that both y· and y′· are of class (D), and both z· and z·′ belong to Mβ for some β > α . If dP-a.s., ξ ≤ ξ ′ , then for each t ∈ [0, T ], dP-a.s.,
yt ≤ y′t .
Proof of Proposition 1. The main idea of the proof is motivated from Briand and Hu (2008). Let us fix n ∈ N∗ and denote
τn the stopping time ∫ t 2 |zs | + |zs′ |2 ds ≥ n ∧ T . τn = inf t ∈ [0, T ] : 0
Tanaka’s formula leads to the equation, setting yˆ t = yt − y′t , zˆt = zt − zt′ , τn
∫
µτn + eµ(t ∧τn ) yˆ + yˆ τn − t ∧τn ≤ e
eµs 1yˆ s >0 zˆs · dBs , +
τn
∫
t ∧τn
eµs {1yˆ s >0 [g (s, ys , zs ) − g (s, y′s , zs′ )] − µˆy+ s }ds.
t ∧τn
It follows from (H2) that τn
∫
µτn + eµ(t ∧τn ) yˆ + yˆ τn + t ∧τn ≤ e
µeµs 1yˆ s >0 |ˆzs |α ds −
∫
t ∧τn
τn
eµs 1yˆ s >0 zˆs · dBs
(2)
t ∧τn
and thus that eµ(t ∧τn ) yˆ + t ∧τn
[
µT
≤e E
∫
+µ
yˆ + τn
τn
] [ ∫ µT |ˆzs | ds Ft ≤ e E yˆ + τn + µ
T
α
t ∧τn
0
] |ˆzs | ds Ft . α
′ β Firstly, let us show that yˆ + · ∈ S . In fact, since y· and y· belong to the class (D) and zˆ· belongs to M for some β > α , we ′ can send n to ∞ in the previous inequality, in view of ξ ≤ ξ , to get that for each t ∈ [0, T ],
yˆ t ≤ µe +
µ(T −t )
T
[∫ E 0
] |ˆzs | ds Ft , α
and thus Jensen’s inequality, Doob’s inequality and Hölder’s inequality lead to the following estimate + β/α
E[ sup |ˆyt | t ∈[0,T ]
∫
β/2
T 2
] ≤ CE
|ˆzs | ds
< +∞,
0
where C is a constant. That is to say, yˆ + · ∈ S. In the sequel, we prove a simple estimate on the function xα with α ∈ (0, 1], which will play a key role in the following proof: xα ≤ mx +
∀ m ≥ 1, x ∈ R+ ,
1 mα
.
(3)
In fact, if 0 ≤ x ≤ 1/m, (3) is obvious since xα ≤ 1/mα . And, if 1/m < x < 1, then mx > 1 > xα . Finally, in the case of x ≥ 1, we also have mx ≥ x ≥ xα . It follows from (2) and (3) that for each m ≥ 1, eµ(t ∧τn ) yˆ + t ∧τn
≤
eµτn yˆ + τn
∫
τn
+
µs
µe 1yˆ s >0 m|ˆzs | +
t ∧τn
µT ≤ eµτn yˆ + τn + T e
µ mα
τn
∫
1
∫
mα
[
t ∧τn
eµs 1yˆ s >0 zˆs · dBs
t ∧τn
eµs 1yˆ s >0 zˆs · −
−
τn
ds − mµˆzs
|ˆzs |
Let Pm be the probability on (Ω , F ) which is equivalent to P and defined by: dPm dP
T
∫ := exp mµ 0
zˆs
|ˆzs |
1|ˆzs |̸=0 · dBs −
1 2
m2 µ2
T
∫
1|ˆzs |̸=0 ds . 0
]
1|ˆzs |̸=0 ds + dBs .
(4)
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It should be noted that dPm /dP has moments of all order. By Girsanov’s theorem, under Pm the process Bm (t ) = Bt − t ∧τ t mµˆzs 1|ˆzs |̸=0 ds, t ∈ [0, T ], is an (Ft , Pm )-Brownian motion. Moreover, the process 0 n eµs 1yˆ s >0 zˆs · dBm (s) is an 0 |ˆz | s
0≤t ≤T
(Ft , Pm )-martingale. Let Em [X |Ft ] represent the conditional expectation of the random variable X with respect to Ft under Pm . By taking the conditional expectation with respect to Ft under Pm in (4), we deduce that for each m ≥ 1 and each t ∈ [0, T ], µτn + yˆ τn |Ft ] + eµ(t ∧τn ) yˆ + t ∧τn ≤ Em [e
T eµT µ mα
.
(5)
′ Finally, since yˆ + · belongs to S , we can easily send n to infinity in (5), in view of ξ ≤ ξ , to get that for each t ∈ [0, T ] and each m ≥ 1, µT eµt yˆ + t ≤ Te
µ mα
,
from which, the conclusion of Proposition 1 follows by letting m → ∞. We have completed the proof of the uniqueness part of Theorem 1, let us now turn to the proof of the existence part of Theorem 1. Existence: Let us first introduce the whole idea. Firstly, with the help of Theorems 6.2 and 6.3 in Briand et al. (2003), we construct a sequence of solutions {(yn· , z·n )}∞ n=1 for BSDEs with parameters (ξ , T , gn ), where gn are some uniformly approximate functions of g. Secondly, we prove this sequence of solutions is a Cauchy sequence and hence converges to some process (y· , z· ) in the desired space. Finally, the (y· , z· ) will be shown to be a solution for the BSDE with parameters (ξ , T , g ). First step: It follows from (H2) that dP × dt-a.s., for each n ≥ 1 and each (y, z , u) ∈ R × Rd × Rd , g (ω, t , y, u) + (n + µ)|u − z | ≥ g (ω, t , y, 0) − µ|u|α + µ(|u| − |z |)
≥ g (ω, t , y, 0) − µ|u| − µ + µ(|u| − |z |) ≥ g (ω, t , y, 0) − µ − µ|z |. Thus, for each n ≥ 1 and (y, z ) ∈ R × Rd , dP × dt-a.s., we can define the following (Ft )-progressively measurable function gn (ω, t , y, z ) = inf (g (ω, t , y, u) + (n + µ)|u − z |).
(6)
u∈Rd
With respect to gn , motivated by Jia (2008), we have the following conclusions: (i) For each ε > 0, there exists an Nε ≥ 0 such that for all n > Nε , dP × dt-a.s., |gn (ω, t , y, z ) − g (ω, t , y, z )| ≤ ε holds true for all y, z. (ii) For each n ≥ 1, gn (ω, t , y, z ) satisfies (H1) and (H2). Moreover, gn is also Lipschitz continuous in z uniformly with respect to (ω, t , y), i.e., there exists a constant Cn > 0, such that dP × dt-a.s.,
∀ y, z1 , z2 ,
|gn (ω, t , y, z1 ) − gn (ω, t , y, z2 )| ≤ Cn |z1 − z2 |.
(7)
In fact, for each n ≥ 1 and (y, z ) ∈ R × Rd , let Sn = {u ∈ Rd : (n + µ)|u − z | > µ|u − z | + 2µ} =
u ∈ Rd : |u − z | >
2µ n
,
then Snc
d
= u ∈ R : |u − z | ≤
2µ n
,
which is a compact set in Rd . It is clear that gn (ω, t , y, z ) ≤ g (ω, t , y, z ), on the other hand, for each u ∈ Sn , it follows from (H2) that dP × dt-a.s., g (ω, t , y, u) + (n + µ)|u − z | > g (ω, t , y, z ) − µ|u − z |α + µ|u − z | + 2µ
≥ g (ω, t , y, z ) + µ > g (ω, t , y, z ). Thus, dP × dt-a.s., for each (y, z ), gn (ω, t , y, z ) = inf (g (ω, t , y, u) + (n + µ)|u − z |).
(8)
u∈Snc
It then follows from (8) and (H2) that dP × dt-a.s., for each (y, z ), gn (ω, t , y, z ) ≥ inf (g (ω, t , y, z ) − µ|u − z |α ) = g (ω, t , y, z ) − µ u∈Snc
2µ n
α
.
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As a result, dP × dt-a.s., for each (y, z ),
0 ≤ g (ω, t , y, z ) − gn (ω, t , y, z ) ≤ µ
2µ
α
n
,
(9)
from which (i) follows. In the sequel, we prove (ii). Firstly, it follows from (9) that dP × dt-a.s.,
|gn (ω, t , 0, 0)| ≤ |g (ω, t , 0, 0)| + µ
2µ
α
n
,
so (H1) holds true for gn . Secondly, it follows from (6) that gn (ω, t , y, z ) = inf (g (ω, t , y, z − v) + (n + µ)|v|). v∈Rd
(10)
Thus, in view of (10), the following basic inequality
inf f (x) − inf g (x) ≤ sup |f (x) − g (x)| x∈D x∈D x∈D
(11)
and (H2), we get, dP × dt-a.s., for each (y1 , y2 , z1 , z2 ) ∈ R × R × Rd × Rd ,
|gn (ω, t , y1 , z1 ) − gn (ω, t , y2 , z2 )| ≤ sup |g (ω, t , y1 , z1 − v) − g (ω, t , y2 , z2 − v)| v∈Rd
≤ µ|y1 − y2 | + µ|z1 − z2 |α . That is to say, (H2) is also true for each gn . Finally, it follows from (6) and (11) that, dP × dt-a.s., for each (y, z1 , z2 ) ∈ R × Rd × Rd ,
|gn (ω, t , y, z1 ) − gn (ω, t , y, z2 )| ≤ sup |(n + µ)|u − z1 | − (n + µ)|u − z2 | | u∈Rd
≤ (n + µ)|z1 − z2 |, which means that (7) holds true for each gn . The statement (ii) is also proved. Thus, in view of (i) and (ii), it follows from Theorems 6.2 and 6.3 in Briand et al. (2003) that for each n ≥ 1, the BSDE with parameters (ξ , T , gn ) has a unique solution (yn· , z·n ) such that yn· is of class (D) and z·n ∈ Mβ for some β > α . Moreover, (yn· , z·n ) belongs to S β × Mβ for all 0 < β < 1. Second step: For each n ≥ 1 and p ≥ 1, let yˆ n· ,p = yn· +p − yn· , zˆ·n,p = z·n+p − z·n , then n ,p
yˆ t
T
∫ =
gˆ n,p (s, yˆ ns ,p , zˆsn,p )ds −
t
T
∫
zˆsn,p · dBs ,
t ∈ [0, T ],
(12)
t
where for each (y, z ) ∈ R × Rd , gˆ n,p (s, y, z ) = gn+p (s, y + yns , z + zsn ) − gn (s, yns , zsn ). The (i) in the first step shows that for each ε > 0, there is an Nε ≥ 0 such that for all n > Nε and p ≥ 1, dP × dt-a.s.,
∀ (y, z ) ∈ R × Rd ,
|gn+p (ω, t , y + yns , z + zsn ) − gn (ω, t , y + yns , z + zsn )| ≤ ε.
Thus, thanks to (H2) for gn , one gets that for all n > Nε and p ≥ 1, dP × dt-a.s., sgn(y)ˆg n,p (s, y, z ) = sgn(y)(gn+p (ω, t , y + yns , z + zsn ) − gn (ω, t , y + yns , z + zsn ))
+ sgn(y)(gn (ω, t , y + yns , z + zsn ) − gn (ω, t , yns , zsn )) ≤ µ|y| + µ1y̸=0 |z |α + ε
(13)
holds true for each (y, z ) ∈ R × Rd . Furthermore, applying (3) one get that for all n > Nε and p ≥ 1, dP × dt-a.s., for each m ≥ 1 and (y, z ) ∈ R × Rd , sgn(y)ˆg n,p (s, y, z ) ≤ µ|y| + µ1y̸=0 m|z | +
≤ µ|y| + µm|z | +
µ mα
µ mα
+ε
+ ε.
Now, let us fix k ∈ N∗ and denote τk the stopping time
∫ t τk = inf t ∈ [0, T ] : |ˆzsn,p |2 ds ≥ k ∧ T . 0
(14)
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Applying Tanaka’s formula to the BSDE (12) leads to the equation, n ,p eµ(t ∧τk ) |ˆyt ∧τk | ≤ eµτk |ˆynτk,p | −
τk
∫
eµs sgn(ˆyns ,p )ˆzsn,p · dBs +
τk
∫
eµs [sgn(ˆyns ,p )ˆg n,p (s, yˆ ns ,p , zˆsn,p ) − µ|ˆyns ,p |]ds.
(15)
t ∧τk
t ∧τk
Combining (15) and (13) we can deduce that for all n > Nε , p ≥ 1 and t ∈ [0, T ], e
µ(t ∧τk )
n ,p yt ∧τk
|ˆ
|≤e
µτk
τk
∫
n,p
|ˆyτk | −
µs
yns ,p
e sgn(ˆ
zsn,p
)ˆ
τk
∫ · dBs +
t ∧τk
t ∧τk
[eµs µ1yˆ ns ,p ̸=0 |ˆzsn,p |α + eµs ε]ds,
and thus that e
µ(t ∧τk )
n ,p yt ∧τk
|ˆ
τk
] |ˆ | Ft + T eµT ε | ≤ e E |ˆyτk | + µ t ∧τk ] [ ∫ T |ˆzsn,p |α ds Ft + T eµT ε. ≤ eµT E |ˆynτk,p | + µ [
µT
∫
n ,p
zsn,p α ds
0
n,p
Then, since yˆ · belongs to the class (D) and zˆ·n,p belongs to Mβ for some β > α , we can send k to ∞ in the previous inequality, n ,p in view of |ˆyT | = 0, to get that for each t ∈ [0, T ], T
[∫
|ˆynt ,p | ≤ µeµ(T −t ) E
0
] |ˆzsn,p |α ds Ft + T eµ(T −t ) ε,
and thus Jensen’s inequality, Doob’s inequality and Hölder’s inequality show that
[ E
n,p yt β/α
sup |ˆ
t ∈[0,T ]
∫
]
|
T
≤ c1 E
zsn,p 2 ds
|ˆ
β/2
|
+ c2 < +∞,
0
where c1 and c2 are two constants. Hence, |ˆyn· ,p | ∈ S . In the sequel, combining (15) and (14) we can deduce that for all n > Nε , p ≥ 1, m ≥ 1 and t ∈ [0, T ], n ,p eµ(t ∧τk ) |ˆyt ∧τk | ≤ eµτk |ˆynτk,p | −
≤e
µτk
τk
∫
eµs sgn(ˆyns ,p )ˆzsn,p · dBs +
|ˆyτk | + T e
µT
τk
eµs 1yˆ n,p ̸=0 µm|ˆzsn,p | + eµs s
t ∧τk
t ∧τk
n ,p
∫
µ mα
+ε −
∫
τk
µs
yns ,p
e sgn(ˆ
zsn,p
)ˆ
t ∧τk
µ mα
+ε
ds
[ ] n,p n ,p sgn(ˆys )µmzˆs · − 1|ˆz n,p |̸=0 ds + dBs . (16) s |ˆzsn,p |
Let Pm,n,p be the probability on (Ω , F ) which is equivalent to P and defined by: T
∫ := exp µm
dPm,n,p dP
n ,p
n,p
sgn(ˆys )ˆzs
0
n,p zs
|ˆ
1|ˆz n,p |̸=0 · dBs − s
|
1 2
µ2 m 2
T
∫
1yˆ n,p ̸=0 1|ˆz n,p |̸=0 ds . 0
s
s
It should be noted that dPm,n,p /dP has moments of all order. By Girsanov’s theorem, under Pm,n,p the process Bm,n,p (t ) = Bt −
t
n,p
n,p
sgn(ˆys
)µmzˆs n,p |ˆzs |
0
t ∧τn
∫
1|ˆz n,p |̸=0 ds, t ∈ [0, T ], is an (Ft , Pm,n,p )-Brownian motion. Moreover, the process s
eµs sgn(ˆyns ,p )ˆzsn,p · dBm,n,p (s)
0
0≤t ≤T
is an (Ft , Pm,n,p )-martingale. Let Em,n,p [X |Ft ] represent the conditional expectation of the random variable X with respect to Ft under Pm,n,p . By taking the conditional expectation with respect to Ft under Pm,n,p in (16), we deduce that for all n > Nε , p ≥ 1, m ≥ 1 and t ∈ [0, T ], n ,p
eµ(t ∧τk ) |ˆyt ∧τk | ≤ Em,n,p [eµτk |ˆynτk,p | |Ft ] + T eµT
µ mα
+ε .
(17) n ,p
Since |ˆyn· ,p | ∈ S , we can easily send k to infinity in (17), in view of |ˆyT | = 0, to get that for all n > Nε , p ≥ 1, m ≥ 1 and t ∈ [0, T ], n,p
eµt |ˆyt | ≤ T eµT
µ mα
+ε .
Then, by letting m → ∞ in the above inequality we know that for each ε > 0, there exists an Nε ≥ 0 such that for all n > Nε and p ≥ 1, n +p
sup |yt
t ∈[0,T ]
− ynt | ≤ T eµT ε.
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S. Fan, D. Liu / Statistics and Probability Letters 80 (2010) 2024–2031
Thus, since for each n ≥ 1, yn· belongs to the class (D) and the space S β for each β ∈ (0, 1), there exists a process y· which belongs also to the class (D) and the space S β for each β ∈ (0, 1) such that limn→∞ ‖ynt − yt ‖1 = 0 and
[
∀ β ∈ (0, 1),
lim E
n→∞
sup |ynt − yt |β
]
t ∈[0,T ]
= 0.
(18)
Moreover, applying Itô’s formula to the BSDE (12) leads to the equation, τk
∫
|ˆy0n,p |2 +
|ˆzsn,p |2 ds = |ˆynτk,p |2 + 2
0
τk
∫
τk
∫
yˆ ns ,p gˆ n,p (s, yˆ ns ,p , zˆsn,p ) ds − 2
yˆ ns ,p zˆsn,p · dBs .
0
0
On the other hand, it follows from (14) that for each ε > 0, there exists an Nε ≥ 0 such that, for all n > Nε , p ≥ 1 and m ≥ 1,
|ˆzsn,p |2
2yˆ ns ,p gˆ n,p (s, yˆ ns ,p , zˆsn,p ) ≤ (2µ + 2µ2 m2 + 1)|ˆyns ,p |2 +
2
+
µ mα
+ε
2
.
Hence, τk
∫
∫ τ k n,p n,p |ˆ | ds + 2T yˆ s zˆs · dBs | ds ≤ 2|ˆyτk | + 2(2µ + 2µ m + 1) + ε + 4 α m 0 ∫ τ 0 µ 2 k n ,p n ,p 2 2 n ,p 2 ˆ ˆ y z ≤ 2(2µT + 2µ m T + T + 1) sup |ˆys | + 2T + 4 · dB + ε s . s s α
zsn,p 2
|ˆ 0
n,p 2
2
τk
∫
2
µ
yns ,p 2
2
m
s∈[0,T ]
0
Thus, since yˆ n· ,p ∈ S β for each β ∈ (0, 1), we have τk
∫ E
β/2
zsn,p 2
|ˆ
| ds
β
≤ cβ (4µT + 4µ m T + 2T + 2) E 2
2
2
0
+ cβ (2T )
β 2
µ mα
+ε
β
[
yns ,p β
sup |ˆ
s∈[0,T ]
∫ + 2 cβ E
τk
β
]
|
yns ,p zsn,p
ˆ
ˆ
0
β/2 · dBs ,
where cβ is a constant depending only on β . Furthermore, it follows from BDG’s inequality that
∫ 2 cβ E
τk
β
ysn,p zsn,p
ˆ
ˆ
0
∫ β/2 · dBs ≤ dβ E ≤
d2β 2
τk
|ˆ
zsn,p 2 ds
| |ˆ
|
0
[ E
yns ,p 2
yns ,p β
sup |ˆ
s∈[0,T ]
|
]
1
β/4 ∫
τk
+ E 2
zsn,p 2
|ˆ
β/2
| ds
,
0
where dβ is a constant depending on cβ and β . Thus, combining the above two inequality one knows that for each ε > 0, there exists an Nε ≥ 0 such that for all n > Nε , p ≥ 1, m ≥ 1 and β ∈ (0, 1), τk
∫ E
β/2
zsn,p 2
|ˆ
| ds
0
β
≤ [2cβ (4µT + 4µ2 m2 T + 2T + 2) 2 + d2β ]E
[
sup |ˆyns ,p |β
s∈[0,T ]
]
β
+ 2cβ (2T ) 2
µ mα
+ε
β
.
Letting k → ∞, then n → ∞, and finally m → ∞ in the above inequality, Fatou’s lemma and the equality (18) imply that for each ε > 0, p ≥ 1 and β ∈ (0, 1),
∫
T
lim sup E
zsn,p 2 ds
|ˆ
n→∞
β/2
|
β
≤ 2cβ (2T ) 2 ε β ,
0
which means that, in view of the fact that z·n belongs to Mβ for each β ∈ (0, 1) and n ≥ 1, there exists a process z· which belongs also to Mβ such that
∫ lim E
n→∞
T
zsn
|
2
− zs | ds
β/2
= 0.
(19)
0
Third step: In view of the (i) in the first step and the continuity of the generator g with respect to (y, z ), we pass to the limit in ucp for the BSDE with parameters (ξ , T , gn ), thanks to (18) and (19), to see that (y· , z· ) solves the BSDE with parameters (ξ , T , g ), y· is of class (D), and (y· , z· ) belongs to S β × Mβ for all 0 < β < 1. This completes the proof of Theorem 1.
S. Fan, D. Liu / Statistics and Probability Letters 80 (2010) 2024–2031
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Acknowledgement The authors would like to thank the referee for his/her careful reading. With the referee’s help, a mistake in this manuscript has been found and corrected. References Briand, Ph., Delyon, B., Hu, Y., 2002. BSDEs with integrable parameters. Preprint 02-20, IRMAR, Universite Rennes. Briand, Ph., Delyon, B., Hu, Y., Pardoux, E., Stoica, L., 2003. Lp solutions of backward stochastic differential equations. Stochastic Processes and their Applications 108, 109–129. Briand, Ph., Hu, Y., 2008. Quadratic BSDEs with convex generators and unbounded terminal conditions. Probability Theory and Related Fields 141, 543–567. Jia, G., 2008. A uniqueness theorem for the solution of backward stochastic differential equations. Comptes Rendus de l’Academie des Sciences, Serie I 346, 439–444. Kobylanski, M., 2000. Backward stochastic differential equations and partial equations with quadratic growth. Annals of Probability 28, 259–276. Lepeltier, J.P., San Martín, J., 1997. Backward stochastic differential equations with continuous coefficient. Statistics and Probability Letters 32, 425–430. Mao, X., 1995. Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients. Stochastic Processes and their Applications 58, 281–292. Pardoux, E., Peng, S., 1990. Adapted solution of a backward stochastic differential equation. Systems & Control Letters 14, 55–61. Peng, S., 1997. Backward SDE and related g-expectation. In: El Karoui, N., Mazliak, L. (Eds.), Backward Stochastic Differential Equations. In: Pitman Research Notes Mathematical Series, vol. 364. Longman, Harlow, pp. 141–159.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
A class of BSDE with integrable parameters✩ ShengJun Fan ∗ , DeQun Liu College of Sciences, China University of Mining & Technology, Xuzhou, Jiangsu 221116, PR China
article
abstract
info
Article history: Received 22 March 2010 Received in revised form 13 September 2010 Accepted 13 September 2010 Available online 19 September 2010
In this paper, we establish an existence and uniqueness result for solutions to onedimensional backward stochastic differential equations (BSDEs) with only integrable parameters, where the generator g is α -Hölder (0 < α < 1) continuous in z. © 2010 Elsevier B.V. All rights reserved.
MSC: primary 60H10 Keywords: Backward stochastic differential equation Existence and uniqueness Integrable parameters Hölder continuous
1. Introduction In this paper, 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 ,
t ∈ [0, T ],
(1)
t
where ξ is a random variable called the terminal condition, the random function g (ω, t , y, z ) : Ω × [0, T ] × R × Rd → R is progressively measurable for each (y, z ), called the generator of the BSDE (1), and B is a d-dimensional Brownian motion. The solution (y· , z· ) is a pair of adapted processes. The triple (ξ , T , g ) is called the parameters of the BSDE (1). Such equations, in the nonlinear case, were firstly introduced in Pardoux and Peng (1990), who established an existence and uniqueness result for solutions to multidimensional BSDEs with square integrable parameters under the Lipschitz assumption of the generator g. Since then, much attention has been paid towards relaxing the Lipschitz hypothesis on g, for instance, some details can be found in Mao (1995), Lepeltier and San Martín (1997), Kobylanski (2000), Jia (2008) and Briand and Hu (2008) etc., most of which dealt with BSDEs with square-integrable parameters. Peng (1997) introduced the notion of g-martingales by solutions to BSDEs, which can be viewed, in some sense, as nonlinear martingales. Since the classical theory of martingales is carried in the integrable space, the question of solving a BSDE with only integrable parameters comes up naturally, as has been pointed out in Briand et al. (2003). It has been known that it is more difficult to solve BSDEs with only integrable parameters than those with square-integrable parameters. In fact, to our knowledge, BSDEs with only integrable parameters were solved in only few papers such as Peng (1997), ✩ Supported by the National Natural Science Foundation of China (No. 10971220), the FANEDD (No. 200919) and the Fundamental Research Funds for the Central Universities. ∗ Corresponding author. E-mail addresses: [email protected] (S. Fan), [email protected] (D. Liu).
0167-7152/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2010.09.009
S. Fan, D. Liu / Statistics and Probability Letters 80 (2010) 2024–2031
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Briand et al. (2002, 2003). The objective of this paper is to establish an existence and uniqueness result in this direction, where the generator g of the BSDE is α -Hölder (0 < α < 1) continuous in z. Let us close this introduction by giving the notations which we will use in the whole paper. In what follows, let us fix a real number T > 0. Let (Ω , F , P ) be a probability space carrying a standard d-dimensional Brownian motion (Bt )t ≥0 . Let (Ft )t ≥0 be the natural σ -algebra generated by (Bt )t ≥0 and assume FT = F . For every positive integer n, we use | · | to denote the norm of Euclidean space Rn . For each real p > 0, S p denotes the set of real-valued, adapted and continuous processes (Yt )t ∈[0,T ] such that
p
1∧1/p
‖Y ‖S p := E[ sup |Yt | ] t ∈[0,T ]
< +∞.
If p ≥ 1, ‖ · ‖S p is a norm on S p and if p ∈ (0, 1), (X , X ′ ) −→ ‖X − X ′ ‖S p defines a distance on S p . Under this metric, S p is complete. Moreover, let Mp denote the set of (equivalent classes of) (Ft )-progressively measurable, Rd -valued processes {Zt , t ∈ [0, T ]} such that
∫
T 2
‖Z ‖Mp := E
p/2 1∧1/p
|Zt | dt
< +∞.
0
For p ≥ 1, Mp is a Banach space endowed with this norm and for p ∈ (0, 1), Mp is a complete metric space with the resulting distance. We set S = ∪p>1 S p and let us recall that a continuous process (Yt )t ∈[0,T ] belongs to the class (D) if the family {Yτ : τ is stopping time bounded by T } is uniformly integrable. For a process Y in the class (D), we put
‖Y ‖1 = sup{E[Yτ ], τ is stopping time bounded by T }. The space of (Ft )-progressively measurable continuous processes which belong to the class (D) is complete under this norm. As mentioned above, we will deal only with the one-dimensional BSDE which is an equation of type (1), where the terminal condition ξ is FT -measurable and the generator g is (Ft )-progressively measurable for each (y, z ). In this paper, by a solution to the BSDE (1) we mean a pair of (Ft )-adapted processes (y· , z· ) with values in R × Rd such that dP-a.s., t −→ yt is continuous, t −→ zt belongs to L2 (0, T ), t −→ f (t , yt , zt ) belongs to L1 (0, T ) and dP-a.s., the BSDE (1) holds true for each t ∈ [0, T ]. 2. Main result In this section, we will put forward and prove our main result (Theorem 1). Firstly, let us introduce the following assumptions (H1) and (H2):
|g (s, 0, 0)|ds < +∞. (H2) There exist two constants µ, α > 0 such that dP × dt-a.s., (H1) E |ξ | +
T 0
∀ y1 , y2 , z1 , z2 ,
|g (ω, t , y1 , z1 ) − g (ω, t , y2 , z2 )| ≤ µ|y1 − y2 | + µ|z1 − z2 |α .
Remark 1. The condition (H2), one of the main assumptions of this paper, is equivalent to the following conditions (A1) and (A2): (A1) g is Lipschitz continuous in y uniformly with respect to (ω, t , z ), i.e., there exists a constant µ > 0, such that, dP × dt-a.s.,
∀ y1 , y2 , z ,
|g (ω, t , y1 , z ) − g (ω, t , y2 , z )| ≤ µ|y1 − y2 |.
(A2) g is Hölder continuous in z uniformly with respect to (ω, t , y), i.e., there exist two constants µ, α > 0 such that dP × dt-a.s.,
∀ y, z1 , z2 ,
|g (ω, t , y, z1 ) − g (ω, t , y, z2 )| ≤ µ|z1 − z2 |α .
The main result of this paper is as follows. Theorem 1. Let (H1) and (H2) with α ∈ (0, 1) hold. Then, the BSDE with parameters (ξ , T , g ) has a unique solution (y· , z· ) such that y· is of class (D) and z· ∈ Mβ for some β > α . Moreover, (y· , z· ) belongs to S β × Mβ for all β ∈ (0, 1). Remark 2. It is clear that under the conditions of Theorem 1, the generator g is not necessarily uniformly Lipschitz continuous with respect to z. The following Example 1, to our knowledge, is not covered by any existing results.
√
Example 1. Let g (t , y, z ) = sin y + |z | + |Bt |. Then, it follows from Theorem 1 that for each ξ ∈ L1 (Ω , FT , P ), the BSDE with parameters (ξ , T , g ) has a unique solution (y· , z· ) such that y· is of class (D) and z· ∈ Mβ for some β > 1/2.
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S. Fan, D. Liu / Statistics and Probability Letters 80 (2010) 2024–2031
Now, let us start the proof of Theorem 1. Proof of Theorem 1. Uniqueness: We first establish the following Proposition 1, and the uniqueness result is a direct consequence of it. Proposition 1. Let g satisfy (H2) with α ∈ (0, 1], and let (y· , z· ) and (y′· , z·′ ) be respectively a solution for the BSDE with parameters (ξ , T , g ) and (ξ ′ , T , g ) such that both y· and y′· are of class (D), and both z· and z·′ belong to Mβ for some β > α . If dP-a.s., ξ ≤ ξ ′ , then for each t ∈ [0, T ], dP-a.s.,
yt ≤ y′t .
Proof of Proposition 1. The main idea of the proof is motivated from Briand and Hu (2008). Let us fix n ∈ N∗ and denote
τn the stopping time ∫ t 2 |zs | + |zs′ |2 ds ≥ n ∧ T . τn = inf t ∈ [0, T ] : 0
Tanaka’s formula leads to the equation, setting yˆ t = yt − y′t , zˆt = zt − zt′ , τn
∫
µτn + eµ(t ∧τn ) yˆ + yˆ τn − t ∧τn ≤ e
eµs 1yˆ s >0 zˆs · dBs , +
τn
∫
t ∧τn
eµs {1yˆ s >0 [g (s, ys , zs ) − g (s, y′s , zs′ )] − µˆy+ s }ds.
t ∧τn
It follows from (H2) that τn
∫
µτn + eµ(t ∧τn ) yˆ + yˆ τn + t ∧τn ≤ e
µeµs 1yˆ s >0 |ˆzs |α ds −
∫
t ∧τn
τn
eµs 1yˆ s >0 zˆs · dBs
(2)
t ∧τn
and thus that eµ(t ∧τn ) yˆ + t ∧τn
[
µT
≤e E
∫
+µ
yˆ + τn
τn
] [ ∫ µT |ˆzs | ds Ft ≤ e E yˆ + τn + µ
T
α
t ∧τn
0
] |ˆzs | ds Ft . α
′ β Firstly, let us show that yˆ + · ∈ S . In fact, since y· and y· belong to the class (D) and zˆ· belongs to M for some β > α , we ′ can send n to ∞ in the previous inequality, in view of ξ ≤ ξ , to get that for each t ∈ [0, T ],
yˆ t ≤ µe +
µ(T −t )
T
[∫ E 0
] |ˆzs | ds Ft , α
and thus Jensen’s inequality, Doob’s inequality and Hölder’s inequality lead to the following estimate + β/α
E[ sup |ˆyt | t ∈[0,T ]
∫
β/2
T 2
] ≤ CE
|ˆzs | ds
< +∞,
0
where C is a constant. That is to say, yˆ + · ∈ S. In the sequel, we prove a simple estimate on the function xα with α ∈ (0, 1], which will play a key role in the following proof: xα ≤ mx +
∀ m ≥ 1, x ∈ R+ ,
1 mα
.
(3)
In fact, if 0 ≤ x ≤ 1/m, (3) is obvious since xα ≤ 1/mα . And, if 1/m < x < 1, then mx > 1 > xα . Finally, in the case of x ≥ 1, we also have mx ≥ x ≥ xα . It follows from (2) and (3) that for each m ≥ 1, eµ(t ∧τn ) yˆ + t ∧τn
≤
eµτn yˆ + τn
∫
τn
+
µs
µe 1yˆ s >0 m|ˆzs | +
t ∧τn
µT ≤ eµτn yˆ + τn + T e
µ mα
τn
∫
1
∫
mα
[
t ∧τn
eµs 1yˆ s >0 zˆs · dBs
t ∧τn
eµs 1yˆ s >0 zˆs · −
−
τn
ds − mµˆzs
|ˆzs |
Let Pm be the probability on (Ω , F ) which is equivalent to P and defined by: dPm dP
T
∫ := exp mµ 0
zˆs
|ˆzs |
1|ˆzs |̸=0 · dBs −
1 2
m2 µ2
T
∫
1|ˆzs |̸=0 ds . 0
]
1|ˆzs |̸=0 ds + dBs .
(4)
S. Fan, D. Liu / Statistics and Probability Letters 80 (2010) 2024–2031
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It should be noted that dPm /dP has moments of all order. By Girsanov’s theorem, under Pm the process Bm (t ) = Bt − t ∧τ t mµˆzs 1|ˆzs |̸=0 ds, t ∈ [0, T ], is an (Ft , Pm )-Brownian motion. Moreover, the process 0 n eµs 1yˆ s >0 zˆs · dBm (s) is an 0 |ˆz | s
0≤t ≤T
(Ft , Pm )-martingale. Let Em [X |Ft ] represent the conditional expectation of the random variable X with respect to Ft under Pm . By taking the conditional expectation with respect to Ft under Pm in (4), we deduce that for each m ≥ 1 and each t ∈ [0, T ], µτn + yˆ τn |Ft ] + eµ(t ∧τn ) yˆ + t ∧τn ≤ Em [e
T eµT µ mα
.
(5)
′ Finally, since yˆ + · belongs to S , we can easily send n to infinity in (5), in view of ξ ≤ ξ , to get that for each t ∈ [0, T ] and each m ≥ 1, µT eµt yˆ + t ≤ Te
µ mα
,
from which, the conclusion of Proposition 1 follows by letting m → ∞. We have completed the proof of the uniqueness part of Theorem 1, let us now turn to the proof of the existence part of Theorem 1. Existence: Let us first introduce the whole idea. Firstly, with the help of Theorems 6.2 and 6.3 in Briand et al. (2003), we construct a sequence of solutions {(yn· , z·n )}∞ n=1 for BSDEs with parameters (ξ , T , gn ), where gn are some uniformly approximate functions of g. Secondly, we prove this sequence of solutions is a Cauchy sequence and hence converges to some process (y· , z· ) in the desired space. Finally, the (y· , z· ) will be shown to be a solution for the BSDE with parameters (ξ , T , g ). First step: It follows from (H2) that dP × dt-a.s., for each n ≥ 1 and each (y, z , u) ∈ R × Rd × Rd , g (ω, t , y, u) + (n + µ)|u − z | ≥ g (ω, t , y, 0) − µ|u|α + µ(|u| − |z |)
≥ g (ω, t , y, 0) − µ|u| − µ + µ(|u| − |z |) ≥ g (ω, t , y, 0) − µ − µ|z |. Thus, for each n ≥ 1 and (y, z ) ∈ R × Rd , dP × dt-a.s., we can define the following (Ft )-progressively measurable function gn (ω, t , y, z ) = inf (g (ω, t , y, u) + (n + µ)|u − z |).
(6)
u∈Rd
With respect to gn , motivated by Jia (2008), we have the following conclusions: (i) For each ε > 0, there exists an Nε ≥ 0 such that for all n > Nε , dP × dt-a.s., |gn (ω, t , y, z ) − g (ω, t , y, z )| ≤ ε holds true for all y, z. (ii) For each n ≥ 1, gn (ω, t , y, z ) satisfies (H1) and (H2). Moreover, gn is also Lipschitz continuous in z uniformly with respect to (ω, t , y), i.e., there exists a constant Cn > 0, such that dP × dt-a.s.,
∀ y, z1 , z2 ,
|gn (ω, t , y, z1 ) − gn (ω, t , y, z2 )| ≤ Cn |z1 − z2 |.
(7)
In fact, for each n ≥ 1 and (y, z ) ∈ R × Rd , let Sn = {u ∈ Rd : (n + µ)|u − z | > µ|u − z | + 2µ} =
u ∈ Rd : |u − z | >
2µ n
,
then Snc
d
= u ∈ R : |u − z | ≤
2µ n
,
which is a compact set in Rd . It is clear that gn (ω, t , y, z ) ≤ g (ω, t , y, z ), on the other hand, for each u ∈ Sn , it follows from (H2) that dP × dt-a.s., g (ω, t , y, u) + (n + µ)|u − z | > g (ω, t , y, z ) − µ|u − z |α + µ|u − z | + 2µ
≥ g (ω, t , y, z ) + µ > g (ω, t , y, z ). Thus, dP × dt-a.s., for each (y, z ), gn (ω, t , y, z ) = inf (g (ω, t , y, u) + (n + µ)|u − z |).
(8)
u∈Snc
It then follows from (8) and (H2) that dP × dt-a.s., for each (y, z ), gn (ω, t , y, z ) ≥ inf (g (ω, t , y, z ) − µ|u − z |α ) = g (ω, t , y, z ) − µ u∈Snc
2µ n
α
.
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S. Fan, D. Liu / Statistics and Probability Letters 80 (2010) 2024–2031
As a result, dP × dt-a.s., for each (y, z ),
0 ≤ g (ω, t , y, z ) − gn (ω, t , y, z ) ≤ µ
2µ
α
n
,
(9)
from which (i) follows. In the sequel, we prove (ii). Firstly, it follows from (9) that dP × dt-a.s.,
|gn (ω, t , 0, 0)| ≤ |g (ω, t , 0, 0)| + µ
2µ
α
n
,
so (H1) holds true for gn . Secondly, it follows from (6) that gn (ω, t , y, z ) = inf (g (ω, t , y, z − v) + (n + µ)|v|). v∈Rd
(10)
Thus, in view of (10), the following basic inequality
inf f (x) − inf g (x) ≤ sup |f (x) − g (x)| x∈D x∈D x∈D
(11)
and (H2), we get, dP × dt-a.s., for each (y1 , y2 , z1 , z2 ) ∈ R × R × Rd × Rd ,
|gn (ω, t , y1 , z1 ) − gn (ω, t , y2 , z2 )| ≤ sup |g (ω, t , y1 , z1 − v) − g (ω, t , y2 , z2 − v)| v∈Rd
≤ µ|y1 − y2 | + µ|z1 − z2 |α . That is to say, (H2) is also true for each gn . Finally, it follows from (6) and (11) that, dP × dt-a.s., for each (y, z1 , z2 ) ∈ R × Rd × Rd ,
|gn (ω, t , y, z1 ) − gn (ω, t , y, z2 )| ≤ sup |(n + µ)|u − z1 | − (n + µ)|u − z2 | | u∈Rd
≤ (n + µ)|z1 − z2 |, which means that (7) holds true for each gn . The statement (ii) is also proved. Thus, in view of (i) and (ii), it follows from Theorems 6.2 and 6.3 in Briand et al. (2003) that for each n ≥ 1, the BSDE with parameters (ξ , T , gn ) has a unique solution (yn· , z·n ) such that yn· is of class (D) and z·n ∈ Mβ for some β > α . Moreover, (yn· , z·n ) belongs to S β × Mβ for all 0 < β < 1. Second step: For each n ≥ 1 and p ≥ 1, let yˆ n· ,p = yn· +p − yn· , zˆ·n,p = z·n+p − z·n , then n ,p
yˆ t
T
∫ =
gˆ n,p (s, yˆ ns ,p , zˆsn,p )ds −
t
T
∫
zˆsn,p · dBs ,
t ∈ [0, T ],
(12)
t
where for each (y, z ) ∈ R × Rd , gˆ n,p (s, y, z ) = gn+p (s, y + yns , z + zsn ) − gn (s, yns , zsn ). The (i) in the first step shows that for each ε > 0, there is an Nε ≥ 0 such that for all n > Nε and p ≥ 1, dP × dt-a.s.,
∀ (y, z ) ∈ R × Rd ,
|gn+p (ω, t , y + yns , z + zsn ) − gn (ω, t , y + yns , z + zsn )| ≤ ε.
Thus, thanks to (H2) for gn , one gets that for all n > Nε and p ≥ 1, dP × dt-a.s., sgn(y)ˆg n,p (s, y, z ) = sgn(y)(gn+p (ω, t , y + yns , z + zsn ) − gn (ω, t , y + yns , z + zsn ))
+ sgn(y)(gn (ω, t , y + yns , z + zsn ) − gn (ω, t , yns , zsn )) ≤ µ|y| + µ1y̸=0 |z |α + ε
(13)
holds true for each (y, z ) ∈ R × Rd . Furthermore, applying (3) one get that for all n > Nε and p ≥ 1, dP × dt-a.s., for each m ≥ 1 and (y, z ) ∈ R × Rd , sgn(y)ˆg n,p (s, y, z ) ≤ µ|y| + µ1y̸=0 m|z | +
≤ µ|y| + µm|z | +
µ mα
µ mα
+ε
+ ε.
Now, let us fix k ∈ N∗ and denote τk the stopping time
∫ t τk = inf t ∈ [0, T ] : |ˆzsn,p |2 ds ≥ k ∧ T . 0
(14)
S. Fan, D. Liu / Statistics and Probability Letters 80 (2010) 2024–2031
2029
Applying Tanaka’s formula to the BSDE (12) leads to the equation, n ,p eµ(t ∧τk ) |ˆyt ∧τk | ≤ eµτk |ˆynτk,p | −
τk
∫
eµs sgn(ˆyns ,p )ˆzsn,p · dBs +
τk
∫
eµs [sgn(ˆyns ,p )ˆg n,p (s, yˆ ns ,p , zˆsn,p ) − µ|ˆyns ,p |]ds.
(15)
t ∧τk
t ∧τk
Combining (15) and (13) we can deduce that for all n > Nε , p ≥ 1 and t ∈ [0, T ], e
µ(t ∧τk )
n ,p yt ∧τk
|ˆ
|≤e
µτk
τk
∫
n,p
|ˆyτk | −
µs
yns ,p
e sgn(ˆ
zsn,p
)ˆ
τk
∫ · dBs +
t ∧τk
t ∧τk
[eµs µ1yˆ ns ,p ̸=0 |ˆzsn,p |α + eµs ε]ds,
and thus that e
µ(t ∧τk )
n ,p yt ∧τk
|ˆ
τk
] |ˆ | Ft + T eµT ε | ≤ e E |ˆyτk | + µ t ∧τk ] [ ∫ T |ˆzsn,p |α ds Ft + T eµT ε. ≤ eµT E |ˆynτk,p | + µ [
µT
∫
n ,p
zsn,p α ds
0
n,p
Then, since yˆ · belongs to the class (D) and zˆ·n,p belongs to Mβ for some β > α , we can send k to ∞ in the previous inequality, n ,p in view of |ˆyT | = 0, to get that for each t ∈ [0, T ], T
[∫
|ˆynt ,p | ≤ µeµ(T −t ) E
0
] |ˆzsn,p |α ds Ft + T eµ(T −t ) ε,
and thus Jensen’s inequality, Doob’s inequality and Hölder’s inequality show that
[ E
n,p yt β/α
sup |ˆ
t ∈[0,T ]
∫
]
|
T
≤ c1 E
zsn,p 2 ds
|ˆ
β/2
|
+ c2 < +∞,
0
where c1 and c2 are two constants. Hence, |ˆyn· ,p | ∈ S . In the sequel, combining (15) and (14) we can deduce that for all n > Nε , p ≥ 1, m ≥ 1 and t ∈ [0, T ], n ,p eµ(t ∧τk ) |ˆyt ∧τk | ≤ eµτk |ˆynτk,p | −
≤e
µτk
τk
∫
eµs sgn(ˆyns ,p )ˆzsn,p · dBs +
|ˆyτk | + T e
µT
τk
eµs 1yˆ n,p ̸=0 µm|ˆzsn,p | + eµs s
t ∧τk
t ∧τk
n ,p
∫
µ mα
+ε −
∫
τk
µs
yns ,p
e sgn(ˆ
zsn,p
)ˆ
t ∧τk
µ mα
+ε
ds
[ ] n,p n ,p sgn(ˆys )µmzˆs · − 1|ˆz n,p |̸=0 ds + dBs . (16) s |ˆzsn,p |
Let Pm,n,p be the probability on (Ω , F ) which is equivalent to P and defined by: T
∫ := exp µm
dPm,n,p dP
n ,p
n,p
sgn(ˆys )ˆzs
0
n,p zs
|ˆ
1|ˆz n,p |̸=0 · dBs − s
|
1 2
µ2 m 2
T
∫
1yˆ n,p ̸=0 1|ˆz n,p |̸=0 ds . 0
s
s
It should be noted that dPm,n,p /dP has moments of all order. By Girsanov’s theorem, under Pm,n,p the process Bm,n,p (t ) = Bt −
t
n,p
n,p
sgn(ˆys
)µmzˆs n,p |ˆzs |
0
t ∧τn
∫
1|ˆz n,p |̸=0 ds, t ∈ [0, T ], is an (Ft , Pm,n,p )-Brownian motion. Moreover, the process s
eµs sgn(ˆyns ,p )ˆzsn,p · dBm,n,p (s)
0
0≤t ≤T
is an (Ft , Pm,n,p )-martingale. Let Em,n,p [X |Ft ] represent the conditional expectation of the random variable X with respect to Ft under Pm,n,p . By taking the conditional expectation with respect to Ft under Pm,n,p in (16), we deduce that for all n > Nε , p ≥ 1, m ≥ 1 and t ∈ [0, T ], n ,p
eµ(t ∧τk ) |ˆyt ∧τk | ≤ Em,n,p [eµτk |ˆynτk,p | |Ft ] + T eµT
µ mα
+ε .
(17) n ,p
Since |ˆyn· ,p | ∈ S , we can easily send k to infinity in (17), in view of |ˆyT | = 0, to get that for all n > Nε , p ≥ 1, m ≥ 1 and t ∈ [0, T ], n,p
eµt |ˆyt | ≤ T eµT
µ mα
+ε .
Then, by letting m → ∞ in the above inequality we know that for each ε > 0, there exists an Nε ≥ 0 such that for all n > Nε and p ≥ 1, n +p
sup |yt
t ∈[0,T ]
− ynt | ≤ T eµT ε.
2030
S. Fan, D. Liu / Statistics and Probability Letters 80 (2010) 2024–2031
Thus, since for each n ≥ 1, yn· belongs to the class (D) and the space S β for each β ∈ (0, 1), there exists a process y· which belongs also to the class (D) and the space S β for each β ∈ (0, 1) such that limn→∞ ‖ynt − yt ‖1 = 0 and
[
∀ β ∈ (0, 1),
lim E
n→∞
sup |ynt − yt |β
]
t ∈[0,T ]
= 0.
(18)
Moreover, applying Itô’s formula to the BSDE (12) leads to the equation, τk
∫
|ˆy0n,p |2 +
|ˆzsn,p |2 ds = |ˆynτk,p |2 + 2
0
τk
∫
τk
∫
yˆ ns ,p gˆ n,p (s, yˆ ns ,p , zˆsn,p ) ds − 2
yˆ ns ,p zˆsn,p · dBs .
0
0
On the other hand, it follows from (14) that for each ε > 0, there exists an Nε ≥ 0 such that, for all n > Nε , p ≥ 1 and m ≥ 1,
|ˆzsn,p |2
2yˆ ns ,p gˆ n,p (s, yˆ ns ,p , zˆsn,p ) ≤ (2µ + 2µ2 m2 + 1)|ˆyns ,p |2 +
2
+
µ mα
+ε
2
.
Hence, τk
∫
∫ τ k n,p n,p |ˆ | ds + 2T yˆ s zˆs · dBs | ds ≤ 2|ˆyτk | + 2(2µ + 2µ m + 1) + ε + 4 α m 0 ∫ τ 0 µ 2 k n ,p n ,p 2 2 n ,p 2 ˆ ˆ y z ≤ 2(2µT + 2µ m T + T + 1) sup |ˆys | + 2T + 4 · dB + ε s . s s α
zsn,p 2
|ˆ 0
n,p 2
2
τk
∫
2
µ
yns ,p 2
2
m
s∈[0,T ]
0
Thus, since yˆ n· ,p ∈ S β for each β ∈ (0, 1), we have τk
∫ E
β/2
zsn,p 2
|ˆ
| ds
β
≤ cβ (4µT + 4µ m T + 2T + 2) E 2
2
2
0
+ cβ (2T )
β 2
µ mα
+ε
β
[
yns ,p β
sup |ˆ
s∈[0,T ]
∫ + 2 cβ E
τk
β
]
|
yns ,p zsn,p
ˆ
ˆ
0
β/2 · dBs ,
where cβ is a constant depending only on β . Furthermore, it follows from BDG’s inequality that
∫ 2 cβ E
τk
β
ysn,p zsn,p
ˆ
ˆ
0
∫ β/2 · dBs ≤ dβ E ≤
d2β 2
τk
|ˆ
zsn,p 2 ds
| |ˆ
|
0
[ E
yns ,p 2
yns ,p β
sup |ˆ
s∈[0,T ]
|
]
1
β/4 ∫
τk
+ E 2
zsn,p 2
|ˆ
β/2
| ds
,
0
where dβ is a constant depending on cβ and β . Thus, combining the above two inequality one knows that for each ε > 0, there exists an Nε ≥ 0 such that for all n > Nε , p ≥ 1, m ≥ 1 and β ∈ (0, 1), τk
∫ E
β/2
zsn,p 2
|ˆ
| ds
0
β
≤ [2cβ (4µT + 4µ2 m2 T + 2T + 2) 2 + d2β ]E
[
sup |ˆyns ,p |β
s∈[0,T ]
]
β
+ 2cβ (2T ) 2
µ mα
+ε
β
.
Letting k → ∞, then n → ∞, and finally m → ∞ in the above inequality, Fatou’s lemma and the equality (18) imply that for each ε > 0, p ≥ 1 and β ∈ (0, 1),
∫
T
lim sup E
zsn,p 2 ds
|ˆ
n→∞
β/2
|
β
≤ 2cβ (2T ) 2 ε β ,
0
which means that, in view of the fact that z·n belongs to Mβ for each β ∈ (0, 1) and n ≥ 1, there exists a process z· which belongs also to Mβ such that
∫ lim E
n→∞
T
zsn
|
2
− zs | ds
β/2
= 0.
(19)
0
Third step: In view of the (i) in the first step and the continuity of the generator g with respect to (y, z ), we pass to the limit in ucp for the BSDE with parameters (ξ , T , gn ), thanks to (18) and (19), to see that (y· , z· ) solves the BSDE with parameters (ξ , T , g ), y· is of class (D), and (y· , z· ) belongs to S β × Mβ for all 0 < β < 1. This completes the proof of Theorem 1.
S. Fan, D. Liu / Statistics and Probability Letters 80 (2010) 2024–2031
2031
Acknowledgement The authors would like to thank the referee for his/her careful reading. With the referee’s help, a mistake in this manuscript has been found and corrected. References Briand, Ph., Delyon, B., Hu, Y., 2002. BSDEs with integrable parameters. Preprint 02-20, IRMAR, Universite Rennes. Briand, Ph., Delyon, B., Hu, Y., Pardoux, E., Stoica, L., 2003. Lp solutions of backward stochastic differential equations. Stochastic Processes and their Applications 108, 109–129. Briand, Ph., Hu, Y., 2008. Quadratic BSDEs with convex generators and unbounded terminal conditions. Probability Theory and Related Fields 141, 543–567. Jia, G., 2008. A uniqueness theorem for the solution of backward stochastic differential equations. Comptes Rendus de l’Academie des Sciences, Serie I 346, 439–444. Kobylanski, M., 2000. Backward stochastic differential equations and partial equations with quadratic growth. Annals of Probability 28, 259–276. Lepeltier, J.P., San Martín, J., 1997. Backward stochastic differential equations with continuous coefficient. Statistics and Probability Letters 32, 425–430. Mao, X., 1995. Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients. Stochastic Processes and their Applications 58, 281–292. Pardoux, E., Peng, S., 1990. Adapted solution of a backward stochastic differential equation. Systems & Control Letters 14, 55–61. Peng, S., 1997. Backward SDE and related g-expectation. In: El Karoui, N., Mazliak, L. (Eds.), Backward Stochastic Differential Equations. In: Pitman Research Notes Mathematical Series, vol. 364. Longman, Harlow, pp. 141–159.