Mean-field backward stochastic differential equations in general probability spaces

Applied Mathematics and Computation 263 (2015) 1–11

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Mean-field backward stochastic differential equations in general probability spaces ✩ Wen Lu a, Yong Ren b,∗, Lanying Hu b a b

School of Mathematics and Informational Science, Yantai University, Yantai 264005, China Department of Mathematics, Anhui Normal University, Wuhu 241000, China

a r t i c l e

i n f o

MSC: 60H10 60G40 60H30

a b s t r a c t In this paper, we deal with a class of mean-field backward stochastic differential equations in continuous time with an arbitrary filtered probability space. We prove the existence and uniqueness of a solution for those equations with strengthened Lipschitz assumption. A comparison theorem is also established.

Keywords: Mean-field backward stochastic Differential equation Stieltjes measure Comparison theorem

© 2015 Elsevier Inc. All rights reserved.

1. Introduction The general (nonlinear) backward stochastic differential equations (BSDEs) were firstly introduced by Pardoux and Peng [27] in 1990. Since then, BSDEs have been studied with great interest, and they have gradually become an important mathematical tool in many fields such as, financial mathematics, stochastic games and optimal control, etc, see for example, Peng [28], Hamadène and Lepeltier [18] and El Karoui et al. [17]. McKean–Vlasov stochastic differential equation of the form

dX (t) = b(X (t), μ(t))dt + dW (t), where

b(X (t), μ(t)) =

 

t ∈ [0, T],

X (0) = x,

b(X (t, ω), X (t; ω ))P (dω ) = E[b(ξ , X (t))]|ξ =X (t),

b: RK × RK → R being a (locally) bounded Borel measurable function and μ(t; ·) being the probability distribution of the K-dimensional unknown process X(t), was suggested by Kac [19] as a stochastic toy model for the Vlasov kinetic equation of plasma and the study of which was initiated by Mckean [26]. Since then, many authors made contributions on McKean–Vlasov type SDEs and applications, see for example, Ahmed [1], Ahmed and Ding [2], Borkar and Kumar [4], Chan [7], Crisan and Xiong [15], Kotelenez and Kurtz [20] and the references therein. ✩ The work of Wen Lu is supported partially by the National Natural Science Foundation of China (61273128, 11371029) and a Project of Shandong Province Higher Educational Science and Technology Program (J13LI06). The work of Yong Ren is supported by the National Natural Science Foundation of China (11371029) and the Natural Science Foundation of Anhui Province (1508085JGD10). The work of Lanying Hu is supported by the National Natural Science Foundation of China (11201004). ∗ Corresponding author. Tel.: +86 553 5910643. E-mail addresses: [email protected] (W. Lu), [email protected], [email protected] (Y. Ren), [email protected] (L. Hu).

http://dx.doi.org/10.1016/j.amc.2015.04.014 0096-3003/© 2015 Elsevier Inc. All rights reserved.

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W. Lu et al. / Applied Mathematics and Computation 263 (2015) 1–11

Mathematical mean-field approaches have been used in many fields, not only in physics and chemistry, but also recently in economics, finance and game theory, see for example, Lasry and Lions [21], they have studied mean-field limits for problems in economics and finance, and also for the theory of stochastic differential games. Recently, Buckdahn et al. [5] studied a special mean-field problem in a purely stochastic approach. Furthermore, Buckdahn et al. [6] deepened the investigation of mean-field BSDEs in a rather general setting, they gave the existence and uniqueness of solutions for mean-field BSDEs with Lipschitz condition on coefficients, they also established the comparison principle for these mean-field BSDEs. On the other hand, since the works [5,6] on the mean-field BSDEs, there are some efforts devoted to its generalization. Li and Luo [22] studied reflected BSDEs of mean-field type, they proved the existence and the uniqueness for reflected mean-field BSDEs. Li [23] studied reflected meanfiled BSDEs in a purely probabilistic method, and gave a probabilistic interpretation of the nonlinear and nonlocal PDEs with the obstacles. Xu [29] obtained the existence and uniqueness of solutions for mean-field backward doubly stochastic differential equations with locally monotone coefficient as well as the comparison theorem for these equations. However, most previous contributions to BSDEs and mean-field BSDEs have been obtained in the framework of continuous time diffusion. Recently, there are some works have been done on backward stochastic differential/difference equations with finite states (see, e.g., An et al. [3], Cohen and Elliott [9–11], Lu and Ren [24]) or in general probability spaces (see, e.g., Cohen [8], Cohen et al. [12], Cohen and Elliott [13–14]). Here, we highlight Cohen’s great contribution. More precisely, Cohen and Elliott [14] studied a new kind of BSDEs of the form

Yt = ξ +



T

t

f (s, Ys− , Zs )dμs −

 i

t

T

Zsi dMsi , t ∈ [0, T],

(1.1)

using only a separability assumption on Hilbert space L2 (FT ) (which will be defined later), they established the existence and uniqueness as well as a comparison theorem for solutions of these BSDEs. We point out that both the martingale and driver terms in BSDE (1.1) are permitted to jump, this provide a unification of the discrete and continuous time theory of BSDEs. Furthermore, Cohen [8] deepened the investigation on this topic, where he gave a g-expectation representation for filtration consistent nonlinear expectations in general probability space. Moreover, Cohen et al. [12] established a general comparison theorem for BSDEs based on arbitrary martingales and gave its applications to the theory of nonlinear expectations. Motivated by above works, the present paper deal with a class of mean-field BSDEs in general probability space of the form

Yt = ξ +



T

t

 E [f (s, Ys− , Zs , Ys− , Zs )]dμs −

 i

T t

Zsi dMsi , 0 ≤ t ≤ T,

(1.2)

for details of the coefficient f, one can see Subsection 2.3. To the best of our knowledge, so far little is known about this new kind of BSDEs. This type of equation encompasses forms of mean-field BSDEs in Buckdahn et al. [5]. Our aim is to find a pair of adapted processes (Y, Z) in an appropriate space such that (1.2) hold. We also present a comparison theorem for the solutions of these BSDEs. Compared with the comparison theorem of Buckdahn et al. [5], we need not make additional hypotheses on coefficients. Therefore, we extend, in some sense, some existing results. Moreover, it should be pointed out that the approach of this paper is inspired by Cohen and Elliott [9] and Cohen et al. [12]. The paper is organized as follows. In Section 2, we introduce some notations, preliminaries and basic assumptions. Section 3 is devoted to the proof of the existence and uniqueness of the solution to mean-field BSDEs in general probability spaces. In Section 4, we give a comparison theorem for the solutions of mean-field BSDEs.

2. Notations, preliminaries and basic assumptions In this section, we introduce some notations, preliminaries and basic assumptions.

2.1. Martingale representations In the sequel, let T > 0 be a fixed terminal time. Let (, F , P ) be a probability space with a filtration {Ft }, t ∈ [0, T], satisfying the usual conditions of completeness and right-continuity. The standard Euclidean norm on RK is denote by  · . Definition 2.1. For any nondecreasing process of finite variation μ, the measure introduced by μ is a measure over  × [0, T] given by

 A→E

[0,T]

 IA (ω, t)dμ .

Here, A ∈ F ⊗ B([0, T]), and the integral is taken pathwise in the Stieltjes sense. The key result used in the construction of BSDEs is the martingale representation theorem. Under the assumption that the Hilbert space L2 (FT ) is separable, the following result is presented in [16] (see also [25]).

W. Lu et al. / Applied Mathematics and Computation 263 (2015) 1–11

3

Theorem 2.1. Suppose that L2 (FT ) is a separable Hilbert space, with an inner product (X, Y) = E[XY]. Then there exists a finite or countable sequence of square-integrable {Ft }-martingales M1 , M2 , . . . such that every square integrable {Ft }-martingale N has a representation

Nt = N0 +



[0,t]

i

Zui dMui

for some sequence of predictable processes Zi , which satisfies that



E



 i

  M u < +∞. Zui 2 d

[0,t]

i

(2.1)

These martingales are orthogonal (that is, E[MTi MT ] = 0 for all i  j), and the predictable quadratic variation processes Mi satisfy that j

Mi Mi . . . , where  denotes absolute continuity of the induced measures (Definition 2.1). Furthermore, these martingales are unique, in the sense that if Ni is another such sequence, then Ni  Mi , where  denotes equivalence of the induced measures.  i i Corollary 2.1. For any predictable processes Zi satisfying (2.1), the process ∞ i=0 [0,t] Zu dMu is well defined, and is a square-integrable martingale. In the sequel, let RK ×  denote the space of infinite RK valued sequences. Definition 2.2. (Stochastic seminorm  · Mt ) For each i ∈ N, let Mi be a measure on the predictable σ -algebra and Mi have the following Lebesgue decomposition i,2 Mi t = mi,1 t + mt ,

i,2 where mi,1 t is absolutely continuous with respect to (w.r.t.) μ × P and mt is orthogonal to μ × P. As they represent bounded i,1 i,2 measures on the predictable σ -algebra, both mt and mt are nondecreasing predictable processes. We define, for zt  RK ×  ,  

i,1 i K  K zt 2M := i zti 2 ddm (μ×P) , where zt ∈ R is the i th element in zt , considered as a series of values in R . t

Remark 2.1. For any predictable, progressively measurable process Z taking values in RK ×  , in particular for processes satisfying (2.1) in each of their K components, it holds that

 E A



 

Zt 2Mt d

μt ≤ E



  i

A

  Zti 2 d

Mti



⎡ 2 ⎤ 2      i i i i ⎣ Zt dMt ⎦ =E Zt dMt = E 

i

A

i

A

for any predictable A   × [0, T]. 2.2. Some results for Stieltjes integrals In this subsection, for completeness, we recall some results for Stieltjes integrals, one can see Cohen and Elliott [14] for more details. Definition 2.3. (Stieltjes exponentials) For any càdlàg function of finite variation ν : [0, [ → R, we write

E(νt ) := eνt



(1 + νs )e− νs

0≤s≤t

and call this as the Stieltjes exponentials of ν . Definition 2.4. Let ν be a càdlàg function of finite variation with ν t > −1 for all t. Then, the left-jump-inversion of ν is defined by

ν¯ t := νt −



(νs )2 . 1 + ν s 0≤s≤t

Similarly, if ν t < 1 for all t, the right-jump-inversion is defined by

ν˜t := νt +

 0≤s≤t

(νs )2

1 − ν s

.

Remark 2.2. For ν a function defined above, the left- and right-jump-inversions are finite, (whenever they are defined), and satisfy that E(νt )−1 = E(−ν¯ t ) and E(−νt ) = E(ν˜t )−1 . Definition 2.5. Let u, v be two measures on a σ -algebra A, we write du  dv if, for any A ∈ A, u(A)  v(A).

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W. Lu et al. / Applied Mathematics and Computation 263 (2015) 1–11

The following lemma is Lemma 4.6 of Cohen and Elliott [14]. Lemma 2.1. Let u, ν , w be signed Stieltjes measures on B([0, T]), such that ν t < 1 for all t, and

dut ≥ −ut− dνt + dwt , then

d(ut E(ν˜t )) ≥ (1 − νt )−1 E(ν˜t− )dwt , where ν˜ is the right-jump-inversion of ν . 2.3. Mean-field BSDEs in general spaces-a definition ¯ , F¯ , P¯ ) = ( × , F ⊗ F , P ⊗ P ) be the (non-completed) product of (, F , P ) with itself. We denote the filtration of this Let ( product space by F¯ = {F¯ t = F ⊗ Ft , 0 ≤ t ≤ T }. A random variable ξ ∈ L0 (, F , P; RK ) originally defined on  is extended canon¯ , F¯ , P¯ ) the variable θ ( ·, ω):  → R belongs to L1 (, F , P ), ¯ =  × . For any θ ∈ L1 ( ically to  : ξ  (ω , ω) = ξ (ω ), (ω , ω) ∈  P(dω)-a.s., we denote its expectation by

E [θ (·, ω)] =





θ (ω , ω)P(dω ).

Notice that E [θ ] = E [θ (·, ω)] ∈ L1 (, F , P ), and

  ¯ θ] = E[



θ dP¯ =





 E [θ (·, ω)]P (dω) = E[E [θ ]].

In what follows, we need the following spaces. For any 0  s < S  T, • L2 (FS ): the space of FS -measurable random variables ξ :  → RK such that Eξ 2 < ;  • L2M (s, S): the space of predictable processes Z:  × [s, S] → RK ×  such that E[ i [s,S] Zui 2 d Mi u ] < ∞. 2 (s, S): the space of adapted processes Y:  × [s, S] → RK such that E[sup 2 • SF u∈[s,S] Yu  ] < ∞.

Two elements Z, Z¯ of L2M (s, S) are deemed equivalent if E[

 i ]s,S]

Zui − Z¯ ui 2 d Mi u ] = 0.

Remark 2.3. In the sequel, we always assume that L2 (FT ) is separable. For convenience, we rewrite mean-field BSDEs (1.2) as below:

Yt = ξ +



T

 E [f (s, Ys− , Zs , Ys− , Zs )]dμs −

t

 i

T t

Zsi dMsi , 0 ≤ t ≤ T,

(2.2)

¯ × [0, T] × RK × RK×∞ × RK × RK×∞ → RK is a predictable, progressively measurable function such that where the coefficient f :  (H1) E¯ [0,T] f (ω , ω, t, 0, 0, 0, 0)2 dμt < ∞; (H2) there exists a measurable deterministic function ct > 0 uniformly bounded by some c > 0 such that for all t  [0, T], yt , yt , ξt , ξt ∈ RK , zt , zt , ηt , ηt ∈ RK×∞ , we have

f (ω , ω, t, yt , zt , ξt , ηt ) − f (ω , ω, t, yt , zt , ξt , ηt )2 ≤ ct (yt − yt 2 + ξt − ξt 2 ) +c and ct μt <



zt − zt 2Mt + ηt − ηt 2Mt



dμ × dP − a.s.

1 2.

Now, let us introduce the definition of a solution to mean-field BSDEs (1.2). Definition 2.6. A solution to mean-field BSDE (2.2) is a pair of Ft -adapted stochastic processes (Y, Z) = (Yt , Zt )0  t  T satisfying 2 (0, T ) × L2 (0, T ). mean-field BSDE (2.2) such that (Y, Z ) ∈ SF M Remark 2.4. We emphasize that, due to our notations, the driving coefficient f of (2.2) has to be interpreted as follows

E [f (s, Ys , Zs , Ys , Zs )](ω) = E [f (s, Ys , Zs , Ys (ω), Zs (ω))]  = f (s, Ys (ω ), Zs (ω ), Ys (ω), Zs (ω))P (dω ). 

We conclude this section by recalling a main result of Cohen and Elliott [14]. Lemma 2.2. Assume that (H1) and (H2) hold. For any given ξ ∈ L2 (FT ), the BSDE (1.1) has a unique solution.

W. Lu et al. / Applied Mathematics and Computation 263 (2015) 1–11

5

3. Existence and uniqueness of the solution 3.1. A fundamental result In this subsection, we aim to study the existence and uniqueness of the solution for mean-field BSDE (2.2) under the hypothesis that μT  1. As μ is of finite variation, its discontinuity part μ is bounded. We assume that μ assigns positive measure to any nonempty open interval in [0, T]. In order to get the existence and uniqueness result for BSDE (2.2), we firstly prove the following unique result. Proposition 3.1. Given that ξ ∈ L2 (FT ) and the assumptions (H1) and (H2) hold. Then, the mean-field BSDE (2.2) has at most one solution. Proof. Let (Y·1 , Z·1 ), (Y·2 , Z·2 ) be two solutions of BSDE (2.2), define

Yˆ· := Y·1 − Y·2 , Zˆ· := Z·1 − Z·2 . By application of the differential rule for stochastic integrals, we have

      1 1 2 2 dYˆs 2 = −2Yˆs− E f1 s, Ys− , Zs1 , Ys− , Zs1 − f2 s, Ys− , Zs2 , Ys− , Zs2 dμs   j +2 Zˆsi Zˆs d[Mi , Mj ]s Yˆs− Zˆsi dMsi + i

i,j

    2   1 1 2 2 + E f1 s, Ys− , Zs1 , Ys− , Zs1 − f2 s, Ys− , Zs2 , Ys− , Zs2 (μs )2 . Using the fact that (μs )2 = μs dμs , integrating on any A ∈ B([0, T]) and taking expectation shows that





   1 1 2 2 dEYˆs 2 = −2E Yˆs− E f1 (s, Ys− , Zs1 , Ys− , Zs1 ) − f2 (s, Ys− , Zs2 , Ys− , Zs2 ) dμs A  A    1 1 2 2 +E Zˆsi 2 d Mi s + E μs E f1 (s, Ys− , Zs1 , Ys− , Zs1 ) − f2 (s, Ys− , Zs2 , Ys− , Zs2 )2 dμs . A

A

i

By Young’s inequality, for any measurable functions xs 0 and ws 0, provided x−1 s − μs ≥ 0, we get



A

dEYˆs 2 ≥ −



A

xs EYˆs− 2 dμs + E

 

 i

A

Zˆsi 2 d Mi s

     2 1 1 1 2 2 2 −E − μs E f s, Ys− , Zs− , Ys− , Zs1 − f s, Ys− , Zs− , Ys− , Zs2 dμs A   ≥− xs EYˆs− 2 dμs + E Zˆsi 2 d Mi s x−1 s

A

−E −E where

  A  A



i

A

 2 x−1 s − μs (1 + ws )E fs  dμs

x−1 s − μ s





  1 + w−1 E δ fs 2 dμs , s





1 1 1 1 2 1 fs := f s, Ys− , Zs− , Ys− , Zs1 − f s, Ys− , Zs− , Ys− , Zs2 ,









1 2 1 2 2 2 δ fs := f s, Ys− , Zs− , Ys− , Zs2 − f s, Ys− , Zs− , Ys− , Zs2 .

By (H2), we have



A

dEYˆs 2 ≥ − −

Set

νt := πt := ρ := i t

 [0,t]



[0,t]



[0,t]

 A

xs EYˆs− 2 dμs + E

  A

   2 

ˆi i 1 − 2c x−1 s − μs (1 + ws ) Zs d M s i

A

  x−1 1 + w−1 E¯δ fs 2 dμs . s − μ s s

xs dμ s , 

x−1 s − μ s

  1 + w−1 (1 − νs )−1 dμs , s

   −1 i 1 − 2c x−1 s − μs (1 + ws ) (1 − νs ) d M s .

(3.1)

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W. Lu et al. / Applied Mathematics and Computation 263 (2015) 1–11

With proper choices of xs and ws , without loss of generality, we can assume that the integrands defining ν , π and ρ are uniformly bounded. Thus, the inequality (3.1) becomes



A

dEYˆs 2 ≥ −



A

EYˆs− 2 dνs + E

φ(A) := E

i

A

A

i

Thus, provide ν s < 1 and





x−1 s

[1 − νs ]Zˆsi 2 dρsi −



A

[1 − νs ]E¯δ fs 2 dπs .

− μs ≥ 0, let’s define a signed Stieltjes measure φ as

(1 − νs )Zˆsi 2 dρsi −



A

(1 − νs )E¯δ fs 2 dπs ,

and we equate it with its distribution function φ t φ ([0, t]). In virtue of Lemma 2.1, we have



A

dEYˆsn+1 2 E(ν˜s ) ≥



A

=E

(1 − νs )−1 E(ν˜s− )dφs



A

i

For A = [t, T], we get

EYˆt 2 E(ν˜t ) + E

[t,t]

i

 ≤



[t,T]

E(ν˜s− )Zˆsi 2 dρsi −

 A

E¯|δ fs |2 E(ν˜s− )dπs .

E(ν˜s− )Zˆsi 2 dρsi

   1 + w−1 2cs x−1 (1 − νs )−1 EYˆs− 2 E(ν˜s− )dμs . s − μ s s

(3.2)

Taking a left-limit in t on both sides of (3.2) gives, by the dominated convergence theorem

EYˆt− 2 E(ν˜t− ) + E

i

 ≤



[t,T]

[t,T]

E(ν˜s− )Zˆsi 2 dρsi

   1 + w−1 2cs x−1 (1 − νs )−1 EYˆs− 2 E(ν˜s− )dμs . s − μ s s

Therefore, by integration and Fubini’s theorem, we have



[0,T]

EYˆt− 2 E(ν˜t− )dμt + E

i



≤



[0,T]



[0,T]







−1 μt 1 − 2c x−1 ˜t− )Zˆti 2 d Mi t t − μt (1 + wt ) (1 − νt ) E(ν

   2μt ct x−1 (1 − νt )−1 EYˆt− 2 E(ν˜t− )dμt . 1 + w−1 t − μ t t

(3.3)

From (H2) and Lemma 5.5 of Cohen and Elliott [14], we know that there exists a fixed ε > 0 such that 2ct μt  1 − ε . Thus, by choosing

x−1 t =

1  2 + + μt , w−1 , t = −1 2 8 − 4 2c(1 + 2 )

we get

   2μt ct x−1 (1 − νt )−1 1 + w−1 t − μ t t    1  2 + = 2μt ct +  μ 1 + t 2 8 − 4 2c(1 + 2 −1 )    2ct  2 +1− 1+ + ≤ μt −1 2 8 − 4 2c(1 + 2 )    1  2 ≤ + + 1 −  1 + 2 8 − 4 (1 + 2 −1 ) = 1−

2 8

<1

(3.4)

and

  1 − 2c x−1 s − μs (1 + ws ) = 1 −

16 − 2 2 > 0. 16 − 2 2 + 4

(3.5)

W. Lu et al. / Applied Mathematics and Computation 263 (2015) 1–11

7

This together with (3.3) leads to Yˆt− = 0, Zˆt = 0, P-a.s. In virtue of (3.2), we have Yˆt = 0, Zˆt = 0, P-a.s. The proof is complete.  Next, we aim to study the existence of solution for the mean-field BSDE (2.2). Lemma 3.1. Assume assumptions (H1)–(H2) hold. Then the following mean-field BSDE

Yt = ξ +



T t

 E [f (s, Ys− , Ys− , Zs )]dμs −



T t

i

Zsi dMsi

(3.6)

has a unique solution. Proof. The uniqueness of the solution is a consequence of Proposition 3.1. For the existence, let Yt0 = 0, t  [0, T], consider the following BSDE:

Ytn+1 = ξ +



T t

   n n E f s, Ys− , Ys− , Zsn+1 dμs − i

T t

i

Zsn+1 dMsi .

(3.7)

According to Theorem 5.1 of Cohen and Elliott [14], we can define recursively (Y·n+1 , Z·n+1 ) be the unique solution of mean-field BSDE (3.7). For t  [0, T], we set

δ Ytn+1 := Ytn+1 − Ytn , δ Ztn+1 := Ztn+1 − Ztn . By (H2), for any measurable functions xs 0 and ws 0, provided x−1 s − μs ≥ 0, we have



A

dEδ Ytn+1 2 ≥ −



A

−E

n+1 2 xt Eδ Ys−  dμ s + E

  A 

x−1 s

 i

i

A

δ Zsn+1 2 d Mi s

    2 n n n n − μs (1 + ws )E f s, Ys− , Ys− , Zsn+1 − f s, Ys− , Ys− , Zsn dμs 

  x−1 1 + w−1 E δ1 fs 2 dμs −E s − μ s s A     

n+1 2 n+1 i 2 1 − c x−1 ≥− xt Eδ Ys−  dμ s + E  d Mi s s − μs (1 + ws ) δ Zs A

i

A

    x−1 1 + w−1 E¯δ1 fs 2 dμs , − s − μ s s

(3.8)

A

n  , Y n , Z n ) − f (s, Y n−1  , Y n−1 , Z n ). where δ1 fs := f (s, Ys− s− s− s− s s In this case, we set

νt := πt := ρti :=



[0,t]



[0,t]



[0,t]

xs dμ s , 

x−1 s − μ s

  1 + w−1 (1 − νs )−1 dμs , s

   −1 i 1 − c x−1 s − μs (1 + ws ) (1 − νs ) d M s .

Now, we define a signed Stieltjes measure φ as

φ(A) := E



i

A

i

(1 − νs )δ Zsn+1 2 dρsi −

 A

(1 − νs )E¯δ1 fs 2 dπs ,

which we equate with its distribution function φ t φ (]0, t]). Following the proof of Proposition 3.1, we can deduce that {Y·n+1 } and hence {Z·n+1 } are Cauchy sequences with proper norms 2 (0, T ) and L2 (0, T ). Denote their limits by Y and Z, respectively, one can easily to check that (Y, Z ) ∈ S2 (0, T ) × L2 (0, T ) is the in SF F M M unique solution to mean-field BSDE (3.6). The proof is complete.  We state the main result of this section. Theorem 3.1. Assume the assumptions (H1)–(H2) hold. Then, for any given terminal conditions ξ ∈ L2 (FT ), the mean-field BSDE (2.2) has a unique solution. Proof. The uniqueness is proved by Proposition 3.1. Let’s begin to prove the existence. Let Zt0 = 0, t  [0, T], in virtue of Lemma 3.1, we can define recursively the pair of processes (Y·n+1 , Z·n+1 ) be the unique solution of the following mean-field BSDE

Ytn+1 = ξ +



T t

   n+1  n+1 E f s, Ys− , Zsn  , Ys− , Zsn+1 dμs − i

t

T

i

Zsn+1 dMsi .

(3.9)

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W. Lu et al. / Applied Mathematics and Computation 263 (2015) 1–11

With the same discussions as above, for any measurable functions xs 0 and ws 0, provided x−1 s − μs ≥ 0, we have



A

dEδ Ysn+1 2 ≥ −



A

−E

n+1 2 xs Eδ Ys−  dμ s + E

  A



i

A

i

δ Zsn+1 2 d Mi s

    n+1  n n+1 n+1 x−1 s − μs E f s, Ys− , Zs , Ys− , Zs

 2  n n , Zsn−1  , Ys− , Zsn dμs − f s, Ys−      n+1 2 x−1 1 + w−1 E δ2 fs 2 dμs ≥ − xs Eδ Ys−  dμ s − E s − μ s s A A  

    i n+1 2 +E δ Zsn+1 2 d Mi s − E c x−1 dμ s s − μs (1 + wt ) δ Zs A

i

Ms

A

    n 2 c x−1 −E s − μs (1 + ws ) δ Zs dμs Ms  A     n+1 2 −1 ≥ − xs Eδ Ys−  dμs − xs − μs 1 + w−1 E¯δ2 fs 2 dμs s A A    

n+1 i 2 1 − c x−1 +E d Mi s s − μs (1 + ws ) δ Zs A

i

    n 2 c x−1 − s − μs (1 + ws ) E δ Zs dμs ,

(3.10)

Ms

A

n+1  n+1 n  , Z n−1  , Y n , Z n ). where δ2 fs := f (s, Ys− , Zsn−1  , Ys− , Zsn ) − f (s, Ys− s s− s We denote



νt :=

xs dμ s ,

[0,t]



λt :=

  −1 c x−1 s − μs (1 + ws )(1 − νs ) dμs ,

[0,t]



πt :=

   x−1 1 + w−1 (1 − νs )−1 dμs , s − μ s s

[0,t]



ρ := i t

   −1 i 1 − c x−1 s − μs (1 + ws ) (1 − νs ) d M s

[0,t]

and a signed Stieltjes measure φ as

φ(A) := E



A

i

 −

A

i

δ Zsn+1 2 (1 − νs )dρsi

E¯δ2 fs 2 (1 − νs )dπs −

 A

Eδ Zsn 2MS (1 − νs )dλs ,

we still equate it with its distribution function φt := φ([0, t]). From (3.10), provided ν s < 1 and x−1 s − μs ≥ 0, in virtue of Lemma 2.1, for A = [t, T], we get

Eδ Ytn+1 2 E(ν˜t ) + E

[t,T]

i

 ≤



[t,T]

E(ν˜s )δ Zsn+1

E¯δ2 fs 2 E(ν˜s )dπs +

 [t,T]

i 2

 dρsi

Eδ Zsn 2Ms E(ν˜s )dλs .

(3.11)

With the same procedure as above, we can deduce that



[0,T]

n+1 2 Eδ Ys−  E(ν˜s )dμs



+E

i

[0,T]



≤E  +

i

i

[0,T]



[0,T]









−1 i μs E(ν˜s )δ Zsn+1 2 1 − c x−1 s − μs (1 + ws ) (1 − νs ) d M s

2 



−1 i μs E(ν˜s )E δ Zsn c x−1 s − μs (1 + ws )(1 − νs ) d M s i

   n+1 2 1 + w−1 E(ν˜s )E δ Ys− (1 − νs )−1 dμs . 2μs cs x−1 s − μ s s

(3.12)

W. Lu et al. / Applied Mathematics and Computation 263 (2015) 1–11

9

Thus, choosing

1 + μ s , 2c(1 + 2 −1 )

x−1 s =

w−1 s =

 2

+

2

8 − 4

and remembering that for there exists a fixed ε > 0 such that 2cs μs < 1 − ε , we have

8 − 2 1 < , 2 16 − 2 2 + 4 8 − 2 1 −1 1 − c(xs − μs )(1 + ws ) = 1 − > . 2 16 − 2 2 + 4 c(x−1 s − μs )(1 + ws ) =

(3.13)

From (3.4) and (3.12), we get

2



8

[0,T]

 n+1 2 Eδ Ys−  E(ν˜s )dμs + 1 −

 8 − 2 ≤ E 16 − 2 2 + 4

i



i

  8 − 2 i μs E(ν˜s )δ Zsn+1 2 (1 − νs )−1 d Mi s E 16 − 2 2 + 4 [0,T]

μs E(ν˜s )Eδ Zsn 2 (1 − νs )−1 d Mi s . i

[0,T]

Thus, {Zn } is a Cauchy sequence with proper norm in L2M (0, T ). As an immediate consequence, in virtue of (3.11) and Lipschitz 2 (0, T ) . Denote their limits by Z and Y, assumption of f, we deduce that, with proper norm, {Yn } is also a Cauchy sequence in SF respectively. Then (Y, Z) solves BSDE (2.2) uniquely. We complete the proof.  3.2. The general result In this subsection, we concern with the extension of the above existence and uniqueness result to allow μ to be any Stieltjes measure, by relaxing the condition that μT < 1. We make the following assumption. (H2’) There exists a measurable deterministic function ct > 0 uniformly bounded by some c > 0 such that for all t  [0, T], yt , yt , ξt , ξt ∈ RK , zt , zt , ηt , ηt ∈ RK×∞ , we have f (ω , ω, t, yt , zt , ξt , ηt ) − f (ω , ω, t, yt , zt , ξt , ηt )2

≤ ct









yt − yt 2 + ξt − ξt 2 + c zt − zt 2Mt + ηt − ηt 2Mt , dμ × dP − a.s.

and ct (μt )2 <

1 2.

Theorem 3.2. Let μ be any deterministic Stieltjes measure assigning positive measure to every open interval. Assume assumptions (H1) and (H2’) hold. Then, for any given ξ ∈ L2 (FT ), the mean-field BSDE (2.2) has a unique solution. Proof. By Lemma 5.5 of Cohen and Elliott [14], as (μt )2 is a nondecreasing càdlàg function of finite variation, and ct (μt )2 < for all t, there exists an 0 < ε < 1 such that ct (μt )2 ≤ 12 (1 − ) Let’s define

νt =



[0,t]

2(1 +  −1 )c dμ =:  + 2(1 +  −1 )cμs s



[0,t]

1 2

λ−1 dμs .

Then, ν  μ and ν t = λ−1 μt < 1. Therefore, by Lemma 6.1 of Cohen and Elliott [14], there exist an η > 0 and a finite sequence {t0 = 0 < t1 <  < tN = T} such that ν (]tn , tn + 1 ])  1 − η for all n. With those notations, the mean-field BSDE (2.2) can be written as

Yt = ξ +



T t

 E [λs f (s, Ys− , Zs , Ys− , Zs )]dνs −

 i

T t

Zsi dMsi .

(3.14)

In this case, one can easily to check that the coefficient satisfying the following Lipschitz assumption λs f (ω, s, ys , zs , ξs , ηs ) − λs f (ω, s, ys , zs , ξs , ηs )2

≤ c¯s









ys − ys 2 + ξs − ξs 2 + c¯ zs − zs 2Ms + ηs − ηs 2Ms , dμ × dP − a.s.,

with c¯ = sups {λ2s c} ≤ [ 2(1+ −1 )c + μT ]2 c < ∞ and c¯s = λ2s cs . We assume that, without loss of generality, c 1. Noting that ν s < 1, we then have

c¯s νs =

 + 2(1 +  −1 )cμ 2 2(1 +  −1 )c

≤ (1 +  −1 )

s

cs νs

 2 cs + (1 + )cs (μs )2 4(1 +  −1 )2 c2

10

W. Lu et al. / Applied Mathematics and Computation 263 (2015) 1–11

2

1 + (1 + )(1 − ) 4 2 2 1 ≤ − 2 4 1 < . 2 ≤

Defining the measures

νtn =





[0,t∧tn+1 ]







η η + 1− I dν s . νtn νtn s>tn

We can easily check that, on [0, tn + 1 ], ν n is a measure considered in our Theorem 3.1 and ν n agrees with ν for all subsets of [tn , tn+1 ]. Then, we consider the sequence of mean-field BSDEs



Ytn = Ytn+1 + n+1

E











n n λs f s, Ys− , Zsn , Ys− , Zsn dνsn −

[t,tn+1 ]

 i

i

[t,tn+1 ]

Zsn dMsi

(3.15)

with YTN = ξ . For each n, the existence and uniqueness of solution for BSDE (3.15) is guaranteed by Theorem 3.1. Following the same line as in Cohen and Elliott [14], we can conclude the proof by induction.  4. Comparison theorem In this part, we establish a comparison theorem of the solutions for mean-field BSDEs in general probability spaces. Let (Y·1 , Z·1 ) and (Y·2 , Z·2 ) be respectively the solutions for the following two mean-field BSDEs

Yti = ξ i +



T

t

   E fi s, Ysi , Zsi , Ysi , Zsi dμs − i

t

T

Zsi dMsi , t ∈ [0, T],

with i = 1, 2. Theorem 4.1. Assume that f1 , f2 satisfy (A1) and (A2), ξ 1 , ξ 2 ∈ L2 (, FT , P ). Moreover, we suppose that (i) ξ 1 ξ 2 , P-a.s., ¯ (ii) for any t  [0, T], f1 (ω , ω, t, Yt2 , Zt2 , Yt2 , Zt2 ) ≥ f2 (ω , ω, t, Yt2 , Zt2 , Yt2 , Zt2 ), P-a.s., (iii) for each j, there exists a measure P˜j equivalent to P such that the j-th component of X, as defined fort  [0, T] by

e∗j Xt : = − +



    e∗j E f1 ω , ω, s, Ys2 , Zs1 , Ys2 , Zs1 − f1 ω , ω, s, Ys2 , Zs2 , Ys2 , Zs2 dμs

t 0



t

0

i

  e∗j Zs1i − Zs2i dMsi

is a P˜j supermartingale on [0, T], (iv) if, for all t  [0, T]

e∗i Yt1 − EP˜i



≥ e∗i Yt2 − EP˜i

T t

e∗i E [f1 (ω , ω, s, Ys1 , Zs1 , Ys1 , Zs1 )]dμs |Ft



T t



e∗i E [f1 (ω , ω, s, Ys2 , Zs1 , Ys2 , Zs1 )]dμs |Ft



for all i, then it holds that Yt1 ≥ Yt2 for all t  [0, T]. It is true that Y1 Y2 on [0, T], P-a.s. Proof. The proof is based on the ideas in Cohen et al. [12]. We omit the ω , ω and s for clarity. By assumption (i), for t  [0, T], we have



Yt1 − Yt2 −

T t

     E f1 Ys1 , Zs1 , Ys1 , Zs1 − f2 Ys2 , Zs2 , Ys2 , Zs2 dμs + i

Rearranging it gives that

 Yt1 − Yt2 −  ≥

T t T t

    E f1 Ys1 , Zs1 , Ys1 , Zs1 − f1 Ys2 , Zs1 , Ys2 , Zs1 dμs

    E f1 Ys2 , Zs2 , Ys2 , Zs2 − f2 Ys2 , Zs2 , Ys2 , Zs2 dμs

t

T



 Zs1i − Zs2i dMsi ≥ 0.

W. Lu et al. / Applied Mathematics and Computation 263 (2015) 1–11

 + −

T t

11

    E f1 Ys2 , Zs1 , Ys2 , Zs1 − f1 Ys2 , Zs2 , Ys2 , Zs2 dμs



T



t

i

 Zs1i − Zs2i dMsi .

(4.1)

By assumption (ii), we have



T t

    E f1 Ys2 , Zs2 , Ys2 , Zs2 − f2 Ys2 , Zs2 , Ys2 , Zs2 dμs ≥ 0.

Moreover, by assumption (iii), we know that the process

 e∗j X˜ t : = e∗j Xt − EP˜j∗ XT |Ft 

    T e∗j E f1 Ys2 , Zs1 , Ys2 , Zs1 − f2 Ys2 , Zs2 , Ys2 , Zs2 dμs = EP˜∗ j

−

t

 i

T t

 e∗j Zs1i − Zs2i dMsi |Ft

 (4.2)

is a P˜j∗ supermartingale, on [0, T] with e∗j X˜ T = 0 P˜j∗ -a.s. Therefore, e∗j X˜ t ≥ 0. Then, taking a P˜∗ conditional expectation on both sides of (4.1), together with (4.2), gives that j

e∗j Yt1 − e∗j Yt2 − EP˜∗ j



T t



    e∗j E f1 Ys1 , Zs1 , Ys1 , Zs1 − f1 Ys2 , Zs1 , Ys2 , Zs1 dμs |Ft ≥ 0.

By assumption (iv), we get immediately that Yt1 ≥ Yt2 . The proof is complete.  References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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