Open Access

Mean-field backward doubly stochastic differential equations and related SPDEs

Boundary Value Problems20122012:114

DOI: 10.1186/1687-2770-2012-114

Received: 14 May 2012

Accepted: 2 October 2012

Published: 17 October 2012

Abstract

Existence and uniqueness result of the solutions to mean-field backward doubly stochastic differential equations (BDSDEs in short) with locally monotone coefficients as well as the comparison theorem for these equations are established. As a preliminary step, the existence and uniqueness result for the solutions of mean-field BDSDEs with globally monotone coefficients is also established. Furthermore, we give the probabilistic representation of the solutions for a class of stochastic partial differential equations by virtue of mean-field BDSDEs, which can be viewed as the stochastic Feynman-Kac formula for SPDEs of mean-field type.

Keywords

mean-field backward doubly stochastic differential equations locally monotone coefficients comparison theorem stochastic partial differential equations

1 Introduction

In this paper, we study a new kind of stochastic partial differential equations (SPDEs):
u ( t , x ) = E [ Φ ( X T 0 , x 0 , x ) ] + t T L u ( s , x ) d s + t T E [ f ( s , X s 0 , x 0 , x , u ( s , X s 0 , x 0 ) , u ( s , x ) , σ ˆ u ( s , X s 0 , x 0 ) , σ ˆ u ( s , x ) ) ] d s + t T E [ g ( s , X s 0 , x 0 , x , u ( s , X s 0 , x 0 ) , u ( s , x ) , σ ˆ u ( s , X s 0 , x 0 ) , σ ˆ u ( s , x ) ) ] d B s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ1_HTML.gif
(1.1)
where σ ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq1_HTML.gif is the transpose of σ ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq2_HTML.gif which is defined by σ ˆ : = E [ σ ( s , X s 0 , x 0 , x ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq3_HTML.gif, and is a second-order differential operator given by ( L u ) i = ( L u i ) 1 i n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq4_HTML.gif with
L : = 1 2 i , j = 1 d a i j 2 x i x j + i = 1 d E [ b i ( t , X t 0 , x 0 , x ) ] x i , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equa_HTML.gif
and
a : = ( a i , j ) = ( E [ σ ( t , X t 0 , x 0 , x ) ] E [ σ ( t , X t 0 , x 0 , x ) ] ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equb_HTML.gif

Here, the function u ( t , x ) : [ 0 , T ] × R d R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq5_HTML.gif is the unknown function, and { B t , 0 t T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq6_HTML.gif is an l-dimensional Brownian motion process defined on a given complete probability space ( Ω , F , P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq7_HTML.gif. X t 0 , x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq8_HTML.gif, a stochastic process starting at x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq9_HTML.gif when t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq10_HTML.gif, is the solution of one class of stochastic differential equations (SDEs), and E denotes expectation with respect to the probability P. In this paper, we call this kind of equations (1.1) McKean-Vlasov SPDEs, because they are analogous to McKean-Vlasov PDEs except the stochastic term d B t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq11_HTML.gif.

McKean-Vlasov PDEs involving models of large stochastic particle systems with mean-field interaction have been studied by stochastic methods in recent years (see [14] and the references therein). Mean-field approaches have applications in many areas such as statistical mechanics and physics, quantum mechanics and quantum chemistry. Recently, Lasry and Lions introduced mean-field approaches for high-dimensional systems of evolution equations corresponding to a large number of ‘agents’ or ‘particles’. They extended the field of such mean-field approaches to problems in economics, finance and game theory (see [5] and the references therein).

As is well known, to give a probabilistic representation (Feynman-Kac formula) of quasilinear parabolic SPDEs, Pardoux and Peng [6] introduced a new class of backward stochastic differential equations (BSDEs) called backward doubly stochastic differential equations which have two different types of stochastic integrals: a standard (forward) stochastic integral d W t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq12_HTML.gif and a backward stochastic integral d B t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq11_HTML.gif. They proved the existence and uniqueness for solutions of BDSDEs under uniformly Lipschitz coefficients. When the coefficients are smooth enough, they also established the connection between BDSDEs and a certain kind of quasilinear SPDEs. BDSDEs have a practical background in finance. The extra noise B can be regarded as some extra inside information in a derivative security market. Since 1990s, BDSDEs have drawn more attention from many authors (cf. [713] and the references therein). Shi, Gu and Liu gave the comparison theorem of BDSDEs and investigated the existence of solutions for BDSDEs with continuous coefficients in [11]. To relax the Lipschitz conditions, Wu and Zhang studied two kinds of BDSDEs under globally (respectively, locally) monotone assumptions and obtained the uniqueness and existence results of the solutions (see [12]).

Mean-field BSDEs are deduced by Buckdahn, Djehiche, Li and Peng [14] when they studied a special mean-field problem with a purely stochastic method. Later, Buckdahn, Li and Peng [15] investigated the properties of these equations in a Markovian framework, obtained the uniqueness of the solutions of mean-field BSDEs as well as the comparison theorem and also gave the viscosity solutions of a class of McKean-Vlasov PDEs in terms of mean-field BSDEs.

In this paper, we study a new type of BDSDEs, that is, the so called mean-field BDSDEs, under the globally (respectively locally) monotone coefficients. We obtain the existence and uniqueness result of the solution by virtue of the technique proposed by Wu and Zhang [12] and the contraction mapping theorem under certain conditions. Also, the comparison principle for mean-field BDSDEs is discussed when the coefficients satisfy some stricter assumptions. A comparison theorem is a useful result in the theory of BSDEs. For instance, it can be used to study viscosity solutions of PDEs. Here, we point out that it is more delicate to prove the comparison theorem for mean-field BDSDEs because of the mean-field term.

We also present the connection between McKean-Vlasov SPDEs and mean-field BDSDEs. In detail, let { X s t , x , t s T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq13_HTML.gif be the solution of
{ d X s t , x = E [ b ( s , ( X s 0 , x 0 ) , X s t , x ) ] d s + E [ σ ( s , ( X s 0 , x 0 ) , X s t , x ) ] d W s , s [ t , T ] , X t t , x = x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equc_HTML.gif
Assume that Eq. (1.1) has a classical solution. Then the couple ( Y s t , x , Z s t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq14_HTML.gif, where Y s t , x = u ( s , X s t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq15_HTML.gif and Z s t , x = E [ σ ( s , ( X s 0 , x 0 ) , X s t , x ) ] D u ( s , X s t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq16_HTML.gif, verifies the following mean-field BDSDE :
Y s t , x = E [ Φ ( ( X T 0 , x 0 ) , X T t , x ) ] + s T E [ f ( r , ( X r 0 , x 0 ) , X r t , x , ( Y r 0 , x 0 ) , Y r t , x , ( Z r 0 , x 0 ) , Z r t , x ) ] d r + s T E [ g ( r , ( X r 0 , x 0 ) , X r t , x , ( Y r 0 , x 0 ) , Y r t , x , ( Z r 0 , x 0 ) , Z r t , x ) ] d B r s T Z r t , x d W r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ2_HTML.gif
(1.2)

In Eq. (1.2), the integral d W t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq12_HTML.gif is a forward Itô integral, and the integral d B t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq11_HTML.gif denotes a backward Itô integral. { W t , 0 t T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq17_HTML.gif and { B t , 0 t T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq6_HTML.gif are two mutually independent standard Brownian motion processes with values respectively in R d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq18_HTML.gif and in R l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq19_HTML.gif. This conclusion gives a probabilistic representation of McKean-Vlasov SPDEs (1.1), which can be regarded as a stochastic Feynman-Kac formula for Mckean-Vlasov SPDEs.

Our paper is organized as follows. In Section 2, we present the existence and uniqueness results about mean-field BDSDEs with globally monotone coefficients. We investigate the properties of mean-field BDSDEs with locally monotone assumptions in Section 3. We first prove the existence and uniqueness of the solutions of mean-field BDSDEs and then derive the comparison theorem when the mean-field BDSDEs are one-dimensional. In Section 4, we introduce the decoupled mean-field forward-backward doubly stochastic differential equation and study the regularity of its solution with respect to x, which is the initial condition of the McKean-Vlasov SDE. Finally, Section 5 is devoted to the formulation of McKean-Vlasov SPDEs and provides the relationship between the solutions of SPDEs and those of mean-field BDSDEs.

2 Mean-field BDSDEs with globally monotone coefficients

In this section, we study mean-field BDSDEs with globally monotone coefficients, which is helpful for the case of locally monotone coefficients. To this end, we firstly introduce some notations and recall some results on mean-field BSDEs obtained by Buckdahn, Li and Peng [15].

Let { W t , 0 t T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq20_HTML.gif and { B t , 0 t T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq6_HTML.gif be two mutually independent standard Brownian motion processes, with values respectively in R d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq18_HTML.gif and R l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq19_HTML.gif, defined over some complete probability space ( Ω , F , P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq7_HTML.gif, where T is a fixed positive number throughout this paper. Moreover, let N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq21_HTML.gif denote the class of P-null sets of . For each 0 s T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq22_HTML.gif, we define
F t F 0 , t W F t , T B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equd_HTML.gif

with F s , t W = σ { W r ; s r t } N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq23_HTML.gif and F t , s B = σ { B r B t ; t r s } N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq24_HTML.gif.

Note that { F t , t [ 0 , T ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq25_HTML.gif is not an increasing family of σ-fields, so it is not a filtration.

We will also use the following spaces:

  • For any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq26_HTML.gif, let H F 2 ( 0 , T ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq27_HTML.gif denote the set of (classes of d P × d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq28_HTML.gif a.e. equal) n-dimensional jointly measurable random processes { ψ t ; t [ 0 , T ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq29_HTML.gif which satisfy: Evidently, H F 2 ( 0 , T ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq30_HTML.gif is a Banach space endowed with the canonical norm ψ = { E 0 T | ψ s | 2 d s } 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq31_HTML.gif.

  1. (i)

    E 0 T | ψ t | 2 d t < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq32_HTML.gif,

     
  2. (ii)

    ψ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq33_HTML.gif is F t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq34_HTML.gif measurable, for a.e. 0 t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq35_HTML.gif.

     
  • We denote similarly by S F 2 ( [ 0 , T ] ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq36_HTML.gif the set of continuous n-dimensional random processes { ψ t ; t [ 0 , T ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq37_HTML.gif which satisfy:

  1. (i)

    E ( sup 0 t T | ψ t | 2 ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq38_HTML.gif,

     
  2. (ii)

    ψ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq33_HTML.gif is F t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq34_HTML.gif measurable, for a.e. 0 t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq35_HTML.gif.

     
  • L 0 ( Ω , F , P ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq39_HTML.gif denotes the space of all R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq40_HTML.gif valued -measurable random variables.

  • For 1 p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq41_HTML.gif, L p ( Ω , F , P ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq42_HTML.gif is the space of all R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq40_HTML.gif valued -measurable random variables such that E [ | ξ | p ] < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq43_HTML.gif.

Let ( Ω ¯ , F ¯ , P ¯ ) = ( Ω × Ω , F F , P P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq44_HTML.gif be the (non-completed) product of ( Ω , F , P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq45_HTML.gif with itself, and we define F ¯ = { F ¯ t = F F t , 0 t T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq46_HTML.gif on this product space. A random variable ξ L 0 ( Ω , F , P ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq47_HTML.gif originally defined on Ω is extended canonically to Ω ¯ : ξ ( ω , ω ) = ξ ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq48_HTML.gif, ( ω , ω ) Ω ¯ = Ω × Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq49_HTML.gif. For any θ L 1 ( Ω ¯ , F ¯ , P ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq50_HTML.gif, the variable θ ( , ω ) : Ω R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq51_HTML.gif belongs to L 1 ( Ω , F , P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq52_HTML.gif, P ( d ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq53_HTML.gif-a.s., whose expectation is denoted by
E [ θ ( , ω ) ] = Ω θ ( ω , ω ) P ( d ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Eque_HTML.gif
Notice that E [ θ ] = E [ θ ( , ω ) ] L 1 ( Ω , F , P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq54_HTML.gif and
E ¯ [ θ ] ( = Ω ¯ θ d P ¯ = Ω E [ θ ( , ω ) ] P ( d ω ) ) = E [ E [ θ ] ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equf_HTML.gif

Moreover, for all ( y , z , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq55_HTML.gif, f = f ( ω , ω , t , y , z , y , z ) : Ω ¯ × [ 0 , T ] × R n × R n × d × R n × R n × d R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq56_HTML.gif, g = g ( ω , ω , t , y , z , y , z ) : Ω ¯ × [ 0 , T ] × R n × R n × d × R n × R n × d R n × l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq57_HTML.gif are two F ¯ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq58_HTML.gif-measurable functions which satisfy

Assumption 2.1 (A1) g ( t , 0 , 0 , 0 , 0 ) H F ¯ 2 ( 0 , T ; R n × l ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq59_HTML.gif, and there exist L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq60_HTML.gif and 0 < α < 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq61_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equg_HTML.gif

(A2) for any fixed ( ω , ω , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq62_HTML.gif, f ( ω , ω , t , , , , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq63_HTML.gif is continuous;

(A3) there exist a process f ¯ t H F ¯ 2 ( 0 , T ; R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq64_HTML.gif and a constant L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq65_HTML.gif such that
| f ( t , y , z , y , z ) | f ¯ t + L ( | y | + | z | + | y | + | z | ) ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equh_HTML.gif
(A4) there exist constants λ 1 , λ 2 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq66_HTML.gif such that for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq67_HTML.gif, y i , y i R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq68_HTML.gif, z i , z i R n × d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq69_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq70_HTML.gif),
( y 1 y 2 ) ( f ( t , y 1 , z , y 1 , z ) f ( t , y 2 , z , y 2 , z ) ) λ 1 ( y 1 y 2 ) ( y 1 y 2 ) + λ 2 | y 1 y 2 | 2 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equi_HTML.gif
(A5) there exists L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq71_HTML.gif such that, P ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq72_HTML.gif-a.s., for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq73_HTML.gif, y , y R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq74_HTML.gif, z 1 , z 2 , z 1 , z 2 R n × d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq75_HTML.gif,
| f ( t , y , z 1 , y , z 1 ) f ( t , y , z 2 , y , z 2 ) | 2 L ( | z 1 z 2 | 2 + | z 1 z 2 | 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equj_HTML.gif
We now consider the following mean-field BDSDEs with the form:
Y t = ξ + t T E [ f ( s , Y s , Z s , Y s , Z s ) ] d s + t T E [ g ( s , Y s , Z s , Y s , Z s ) ] d B s t T Z s d W s , 0 t T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ3_HTML.gif
(2.1)
Remark 2.1 Due to our notation, the coefficients of (2.1) are interpreted as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equk_HTML.gif
Remark 2.2 If coefficient f meets the following Lipschitz assumption: There exists a constant L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq71_HTML.gif such that, P ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq72_HTML.gif-a.s., for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq67_HTML.gif, y i , y i R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq68_HTML.gif, z i , z i R n × d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq76_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq70_HTML.gif),
| f ( t , y 1 , z 1 , y 1 , z 1 ) f ( t , y 2 , z 2 , y 2 , z 2 ) | 2 L ( | y 1 y 2 | 2 + | z 1 z 2 | 2 + | y 1 y 2 | 2 + | z 1 z 2 | 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equl_HTML.gif

then it must satisfy conditions (A4) and (A5).

Definition 2.1 A pair of F t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq34_HTML.gif-measurable processes { ( Y t , Z t ) ; 0 t T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq77_HTML.gif is called a solution of mean-field BDSDE (2.1) if ( Y , Z ) S F 2 ( [ 0 , T ] ; R n ) × H F 2 ( 0 , T ; R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq78_HTML.gif and it satisfies mean-field BDSDE (2.1).

The main result of this section is the following theorem.

Theorem 2.1 For any random variable ξ L 2 ( Ω , F T , P ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq79_HTML.gif, under Assumption  2.1, mean-field BDSDE (2.1) admits a unique solution ( Y , Z ) S F 2 ( [ 0 , T ] ; R n ) × H F 2 ( 0 , T ; R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq78_HTML.gif.

Proof Step 1: For any ( y , z ) H F 2 ( 0 , T ; R n × R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq80_HTML.gif, BDSDE
Y t = ξ + t T E [ f ( s , y s , z s , Y s , Z s ) ] d s + t T E [ g ( s , y s , z s , Y s , Z s ) ] d B s t T Z s d W s , 0 t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ4_HTML.gif
(2.2)
has a unique solution. In order to get this conclusion, we define
φ ( y , z ) ( s , μ , ν ) : = E [ φ ( s , y s , z s , μ , ν ) ] , for  φ = f , g . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equm_HTML.gif
Then (2.2) can be rewritten as
Y t = ξ + t T f ( y , z ) ( Y s , Z s ) d s + t T g ( y , z ) ( Y s , Z s ) d B s t T Z s d W s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equn_HTML.gif
Due to Assumption 2.1, for all ( μ , ν ) , ( μ 1 , ν 1 ) , ( μ 2 , ν 2 ) R n × R n × d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq81_HTML.gif, g satisfies
| g ( y , z ) ( μ 1 , ν 1 ) g ( y , z ) ( μ 2 , ν 2 ) | 2 L | μ 1 μ 2 | 2 + α | ν 1 ν 2 | 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equo_HTML.gif
and f fulfills
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equp_HTML.gif

According to Theorem 2.2 in [12], BDSDE (2.2) has a unique solution.

Step 2: Now, we introduce a norm on the space H F 2 ( 0 , T ; R n × R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq82_HTML.gif which is equivalent to the canonical norm
( y , z ) β = { E 0 T e β s ( c ¯ | y s | 2 + | z s | 2 ) d s } 1 2 , c ¯ , β > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equq_HTML.gif

The parameters c ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq83_HTML.gif and β are specified later.

From Step 1, we can introduce the mapping ( Y , Z ) = I [ ( y , z ) ] : H F 2 ( 0 , T ; R n × R n × d ) H F 2 ( 0 , T ; R n × R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq84_HTML.gif through the equation
Y t = ξ + t T E [ f ( s , y s , z s , Y s , Z s ) ] d s + t T E [ g ( s , y s , z s , Y s , Z s ) ] d B s t T Z s d W s , 0 t T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equr_HTML.gif
For any ( y 1 , z 1 ) , ( y 2 , z 2 ) H F 2 ( 0 , T ; R n × R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq85_HTML.gif, we set ( Y 1 , Z 1 ) = I [ ( y 1 , z 1 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq86_HTML.gif, ( Y 2 , Z 2 ) = I [ ( y 2 , z 2 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq87_HTML.gif, ( y ¯ , z ¯ ) = ( y 1 y 2 , z 1 z 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq88_HTML.gif and ( Y ¯ , Z ¯ ) = ( Y 1 Y 2 , Z 1 Z 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq89_HTML.gif. Then applying Itô’s formula to e β s | Y ¯ s | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq90_HTML.gif and by virtue of Y 1 , Y 2 S F 2 ( [ 0 , T ] ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq91_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ5_HTML.gif
(2.3)
From condition (A4) and noting that E [ Y s ( ω ) ] = E [ Y s ( ω ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq92_HTML.gif, for any M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq93_HTML.gif, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equs_HTML.gif
Then we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equt_HTML.gif
If we set M = 2 L 1 2 α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq94_HTML.gif, c ¯ = 4 1 + 2 α ( | λ 1 | M + L ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq95_HTML.gif, β = | λ 1 | M + 2 λ 2 + + 3 M + L + 1 2 c ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq96_HTML.gif, then it yields
E [ t T e β s ( c ¯ | Y ¯ s | 2 + | Z ¯ s | 2 ) d s ] 1 + 2 α 2 E [ t T e β s ( c ¯ | y ¯ s | 2 + | z ¯ s | 2 ) d s ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equu_HTML.gif

Consequently, I is a strict contraction on H F 2 ( 0 , T ; R n × R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq97_HTML.gif equipped with the norm β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq98_HTML.gif for 0 < α < 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq99_HTML.gif. With the contraction mapping theorem, there admits a unique fixed point ( Y , Z ) H F 2 ( 0 , T ; R n × R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq100_HTML.gif such that I ( Y , Z ) = ( Y , Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq101_HTML.gif. On the other hand, from Step 1, we know that if I ( Y , Z ) = ( Y , Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq102_HTML.gif, then ( Y , Z ) S F 2 ( [ 0 , T ] ; R n ) × H F 2 ( 0 , T ; R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq103_HTML.gif, which is the unique solution of Eq. (2.1). □

Suppose that: For some f : Ω ¯ × [ 0 , T ] × R n × R n × d × R n × R n × d R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq104_HTML.gif satisfying (A2)-(A5), the generators f i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq105_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq70_HTML.gif are of the form
f i ( s , ( Y s i ) , ( Z s i ) , Y s i , Z s i ) = f ( s , ( Y s i ) , ( Z s i ) , Y s i , Z s i ) + φ i ( s ) , d s d P ¯ -a.e. , i = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equv_HTML.gif

where φ i H F ¯ 2 ( 0 , T ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq106_HTML.gif. Then we have the following corollary.

Corollary 2.1 Suppose that ( Y i , Z i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq107_HTML.gif is the solution of mean-field BDSDE (2.1) with data ( ξ i , f i , g ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq108_HTML.gif, i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq70_HTML.gif, where ξ 1 , ξ 2 L 2 ( Ω , F T , P ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq109_HTML.gif are two arbitrary terminal values. The difference of ( Y 1 , Z 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq110_HTML.gif and ( Y 2 , Z 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq111_HTML.gif satisfies the following estimate:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ6_HTML.gif
(2.4)

where β = 2 | λ 1 | + 2 λ 2 + + 4 L 1 2 α + 2 + 2 L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq112_HTML.gif.

The proof of the above corollary is similar to that of Theorem 2.1 and is therefore omitted.

3 Mean-field BDSDEs with locally monotone coefficients

In this section, we investigate mean-field BDSDEs with locally monotone coefficients. The results can be regarded as an extension of the results in [12] to the mean-field type.

We assume

(A3′) there exist L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq65_HTML.gif and 0 γ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq113_HTML.gif such that | f ( t , y , z , y , z ) | L ( 1 + | y | γ + | z | γ + | y | γ + | z | γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq114_HTML.gif;

(A4′) for any N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq115_HTML.gif, there exist constants λ N , λ ¯ N R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq116_HTML.gif such that, y i , y i , z , z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq117_HTML.gif satisfying | y i | , | y i | , | z | , | z | N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq118_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq70_HTML.gif), we have
( y 1 y 2 ) ( f ( t , y 1 , z , y 1 , z ) f ( t , y 2 , z , y 2 , z ) ) λ N ( y 1 y 2 ) ( y 1 y 2 ) + λ ¯ N | y 1 y 2 | 2 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equw_HTML.gif
(A5′) N N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq119_HTML.gif, there exists L N > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq120_HTML.gif such that, for any y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq121_HTML.gif, y, z i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq122_HTML.gif, z i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq123_HTML.gif satisfying | y | , | y | , | z i | , | z i | N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq124_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq70_HTML.gif), it holds
| f ( t , y , z 1 , y , z 1 ) f ( t , y , z 2 , y , z 2 ) | 2 L N ( | z 1 z 2 | 2 + | z 1 z 2 | 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equx_HTML.gif
Remark 3.1 Since | x | γ 1 + | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq125_HTML.gif, γ [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq126_HTML.gif, (A3′) implies that
| f ( t , y , z , y , z ) | L ( 5 + | y | + | z | + | y | + | z | ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equy_HTML.gif

We need the following lemma, which plays an important role in the proof of the main result.

Lemma 3.1 Under (A2), (A3′)-(A5′) there exists a sequence of { f m } m = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq127_HTML.gif such that
  1. (i)

    for fixed m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq128_HTML.gif, ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq129_HTML.gif, ω, t, f m ( ω , ω , t , , , , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq130_HTML.gif is continuous;

     
  2. (ii)

    m, | f m ( t , y , z , y , z ) | | f ( t , y , z , y , z ) | L ( 1 + | y | γ + | z | γ + | y | γ + | z | γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq131_HTML.gif;

     
  3. (iii)

    N, ρ N ( f m f ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq132_HTML.gif as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq133_HTML.gif, where ρ m 2 ( f ) : = E [ 0 T sup | y | , | z | , | y | , | z | m | f ( t , y , z , y , z ) | 2 d t ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq134_HTML.gif;

     
  4. (iv)
    m, f m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq135_HTML.gif is globally monotone in y; moreover, for any m, N with m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq136_HTML.gif, it holds that
    ( y 1 y 2 ) ( f m ( t , y 1 , z , y 1 , z ) f m ( t , y 2 , z , y 2 , z ) ) λ N ( y 1 y 2 ) ( y 1 y 2 ) + λ ¯ N | y 1 y 2 | 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equz_HTML.gif
     
for any t, y i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq137_HTML.gif, y i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq138_HTML.gif, z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq139_HTML.gif, z satisfying | y i | , | y i | , | z | , | z | N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq140_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq70_HTML.gif);
  1. (v)
    m, f m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq135_HTML.gif is globally Lipschitz in z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq139_HTML.gif, z; moreover, for any m, N with m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq136_HTML.gif, it holds that
    | f m ( t , y , z 1 , y , z 1 ) f m ( t , y , z 2 , y , z 2 ) | 2 L N ( | z 1 z 2 | 2 + | z 1 z 2 | 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equaa_HTML.gif
     

for any t, y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq121_HTML.gif, y, z i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq122_HTML.gif, z i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq123_HTML.gif satisfying | y | , | y | , | z i | , | z i | N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq141_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq70_HTML.gif).

Proof We define f m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq135_HTML.gif by
f m ( t , y , z , y , z ) = f ( t , y , z , y , z ) ϕ m ( y ) φ m ( z ) ψ m ( y ) η m ( z ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equab_HTML.gif

where ϕ m : R n R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq142_HTML.gif is a sequence of smooth functions such that 0 ϕ m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq143_HTML.gif, ϕ m ( x ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq144_HTML.gif for | x | m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq145_HTML.gif, and ϕ m ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq146_HTML.gif for | x | m + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq147_HTML.gif. Similarly, we define the sequences φ m : R n × d R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq148_HTML.gif, ψ m : R n R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq149_HTML.gif, η m : R n × d R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq150_HTML.gif. It should be pointed out that ϕ m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq151_HTML.gif, φ m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq152_HTML.gif, ψ m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq153_HTML.gif and η m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq154_HTML.gif are continuously differentiable with bounded derivatives for each m. The conclusion of this lemma can be easily obtained by arguments similar to those of Lemma 3.3 in [12]. □

We now present the main result of this section.

Theorem 3.1 Let (A1), (A2), (A3′)-(A5′) hold. Assume, moreover,
1 + exp ( 2 L + 2 | λ N | + 2 λ ¯ N + + 2 L N θ 1 + 2 ) N 2 ( 1 γ ) 0 , as N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ7_HTML.gif
(3.1)

where θ is an arbitrarily fixed constant such that 0 < θ < 1 2 α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq155_HTML.gif. Then mean-field BDSDE (2.1) has a unique solution ( Y , Z ) S F 2 ( [ 0 , T ] ; R n ) × H F 2 ( 0 , T ; R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq156_HTML.gif.

Proof We now construct an approximate sequence. Let f m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq135_HTML.gif be associated to f by Lemma 3.1. Then for each m, f m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq135_HTML.gif is globally monotone in y and globally Lipschitz in z. By Theorem 2.1, the following mean-field BDSDE
Y t m = ξ + t T E [ f m ( s , ( Y s m ) , ( Z s m ) , Y s m , Z s m ) ] d s + t T E [ g ( s , ( Y s m ) , ( Z s m ) , Y s m , Z s m ) ] d B s t T Z s m d W s , 0 t T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ8_HTML.gif
(3.2)
admits a unique solution ( Y m , Z m ) S F 2 ( [ 0 , T ] ; R n ) × H F 2 ( 0 , T ; R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq157_HTML.gif for each m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq136_HTML.gif. Applying Itô’s formula to | Y t m | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq158_HTML.gif yields
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equac_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equad_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equae_HTML.gif
Hence,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equaf_HTML.gif
Then it follows from Gronwall’s inequality and the B-D-G inequality that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equag_HTML.gif

where C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq159_HTML.gif only depends on T, α, L and is independent of m.

For any m , k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq160_HTML.gif, set
A : = { ( ω , ω , s ) : | ( Y s m ) | + | ( Z s m ) | + | ( Y s k ) | + | ( Z s k ) | + | Y s m | + | Z s m | + | Y s k | + | Z s k | N } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equah_HTML.gif

and A ¯ : = Ω A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq161_HTML.gif.

Next, we will conclude that ( Y m , Z m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq162_HTML.gif is a Cauchy sequence in S F 2 ( [ 0 , T ] ; R n ) × H F 2 ( 0 , T ; R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq163_HTML.gif. Actually, since mean-field BDSDE
Y t k = ξ + t T E [ f k ( s , ( Y s k ) , ( Z s k ) , Y s k , Z s k ) ] d s + t T E [ g ( s , ( Y s k ) , ( Z s k ) , Y s k , Z s k ) ] d B s t T Z s k d W s , 0 t T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equai_HTML.gif
admits a unique solution ( Y k , Z k ) S F 2 ( [ 0 , T ] ; R n ) × H F 2 ( 0 , T ; R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq164_HTML.gif for each k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq165_HTML.gif. Applying Itô’s formula to | Y t m Y t k | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq166_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ9_HTML.gif
(3.3)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equaj_HTML.gif

We next estimate I, II and III.

For the first term I, based on Hölder’s inequality and Chebyshev’s inequality, we have
I E t T | Y s m Y s k | 2 d s + E [ t T E [ | f m ( s , ( Y s m ) , ( Z s m ) , Y s m , Z s m ) f k ( s , ( Y s k ) , ( Z s k ) , Y s k , Z s k ) | 2 ] I A d s ] E t T | Y s m Y s k | 2 d s + C N 2 ( 1 γ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ10_HTML.gif
(3.4)

where C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq159_HTML.gif depends on T, L, α and E [ 0 T | E [ g ( s , 0 , 0 , 0 , 0 ) ] | 2 d s ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq167_HTML.gif.

For the second term II, due to the local monotonicity of f m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq135_HTML.gif in y and the local Lipschitz condition of f m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq135_HTML.gif in z, we obtain that for M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq168_HTML.gif, the following holds:
II = 2 E [ t T ( Y s m Y s k ) E [ f m ( s , ( Y s m ) , ( Z s m ) , Y s m , Z s m ) f m ( s , ( Y s k ) , ( Z s m ) , Y s k , Z s m ) ] I A ¯ d s ] + 2 E [ t T ( Y s m Y s k ) E [ f m ( s , ( Y s k ) , ( Z s m ) , Y s k , Z s m ) f m ( s , ( Y s k ) , ( Z s k ) , Y s k , Z s k ) ] I A ¯ d s ] 2 E t T [ λ N ( Y s m Y s k ) E ( Y s m Y s k ) + λ ¯ N | Y s m Y s k | 2 ] d s + M E t T | Y s m Y s k | 2 d s + L N M E t T [ E [ | ( Z s m ) ( Z s k ) | 2 ] + | Z s m Z s k | 2 ] d s ( 2 | λ N | + 2 λ ¯ N + + M ) E t T | Y s m Y s k | 2 d s + 2 L N M E t T | Z s m Z s k | 2 d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ11_HTML.gif
(3.5)
For the last term, we have
III E t T | Y s m Y s k | 2 d s + ρ N 2 ( f m f k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ12_HTML.gif
(3.6)
Choose M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq93_HTML.gif such that θ : = 2 L N M < 1 2 α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq169_HTML.gif. Then from (3.3)-(3.6), we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equak_HTML.gif
Applying Gronwall’s inequality and the B-D-G inequality to the above inequality yields
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equal_HTML.gif
where c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq170_HTML.gif is independent of m, k. Now passing to the limit successively on m, k and N, we see that ( Y m , Z m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq162_HTML.gif is a Cauchy (hence convergent) sequence in S F 2 ( [ 0 , T ] ; R n ) × H F 2 ( 0 , T ; R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq171_HTML.gif; denote the limit by ( Y , Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq172_HTML.gif, which satisfies
E [ sup 0 t T | Y t m Y t | 2 ] + E [ 0 T | Z s m Z s | 2 d s ] 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equam_HTML.gif

as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq173_HTML.gif.

Next, we show that ( Y , Z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq172_HTML.gif is the solution of mean-field BDSDE (2.1). To this end, we only need to prove that the following conclusion holds along a subsequence:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ13_HTML.gif
(3.7)
Set
A m : = { ( ω , ω , s ) : | ( Y s m ) | + | ( Z s m ) | + | Y s | + | Z s | + | Y s m | + | Z s m | + | Y s | + | Z s | N } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equan_HTML.gif

and A ¯ m : = Ω A m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq174_HTML.gif.

Since
f m ( s , ( Y s m ) , ( Z s m ) , Y s m , Z s m ) f ( s , Y s , Z s , Y s , Z s ) = I 1 ( m , s ) + I 2 ( m , s ) + I 3 ( m , s ) + I 4 ( m , s ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equao_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equap_HTML.gif
then we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ14_HTML.gif
(3.8)
where C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq159_HTML.gif is independent of m. As
E [ sup 0 t T | Y t m Y t | 2 ] + E [ 0 T | Z s m Z s | 2 d s ] 0 and sup m N E [ sup 0 t T | Y t m | 2 ] < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equaq_HTML.gif
there exists a subsequence of Y m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq175_HTML.gif, still denoted by Y m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq175_HTML.gif, such that Y t m Y t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq176_HTML.gif a.e., a.s. It then follows from the continuity of f in y and the dominated convergence theorem that
E [ t T E [ | f ( s , ( Y s m ) , Z s , Y s m , Z s ) f ( s , Y s , Z s , Y s , Z s ) | 2 ] d s ] 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equar_HTML.gif

as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq173_HTML.gif.

Now, passing to the limit as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq173_HTML.gif and N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq177_HTML.gif in (3.8) successively, it follows that (3.7) holds. Then letting m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq173_HTML.gif in (3.2) yields
Y t = ξ + t T E [ f ( s , Y s , Z s , Y s , Z s ) ] d s + t T E [ g ( s , Y s , Z s , Y s , Z s ) ] d B s t T Z s d W s , 0 t T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equas_HTML.gif

Therefore, we come to the conclusion of this theorem. □

Now, we discuss the comparison theorem for mean-field BDSDEs. We only consider one-dimensional mean-field BDSDEs, i.e., n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq178_HTML.gif.

We consider the following mean-field BDSDEs: ( 0 t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq179_HTML.gif)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ15_HTML.gif
(3.9)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ16_HTML.gif
(3.10)

Theorem 3.2 (Comparison theorem)

Assume mean-field BDSDEs (3.9) and (3.10) satisfy the conditions of Theorem  3.1. Let ( Y 1 , Z 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq110_HTML.gif and ( Y 2 , Z 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq111_HTML.gif be the solutions of mean-field BDSDEs (3.9) and (3.10), respectively. Moreover, for the two generators of f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq180_HTML.gif and f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq181_HTML.gif, we suppose:
  1. (i)

    One of the two generators is independent of z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq139_HTML.gif.

     
  2. (ii)

    One of the two generators is nondecreasing in y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq121_HTML.gif.

     

Then if ξ 1 ξ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq182_HTML.gif, a.s., f 1 ( t , y , z , y , z ) f 2 ( t , y , z , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq183_HTML.gif, a.s., there also holds that Y t 1 Y t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq184_HTML.gif, a.s. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq185_HTML.gif.

Remark 3.2 The conditions (i) and (ii) of Theorem 3.2 are, in particular, satisfied if they hold for the same generator f j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq186_HTML.gif ( j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq187_HTML.gif), but also if (i) is satisfied by one generator and (ii) by the other one.

Proof Without loss of generality, we suppose that (i) is satisfied by f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq180_HTML.gif and (ii) by f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq181_HTML.gif. For notational simplicity, we set ξ ¯ : = ξ 1 ξ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq188_HTML.gif, ( Y ¯ , Z ¯ ) : = ( Y 1 Y 2 , Z 1 Z 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq189_HTML.gif, then
Y ¯ t = ξ ¯ + t T E [ f 1 ( s , ( Y s 1 ) , Y s 1 , Z s 1 ) f 2 ( s , ( Y s 2 ) , ( Z s 2 ) , Y s 2 , Z s 2 ) ] d s + t T E [ g ( s , ( Y s 1 ) , ( Z s 1 ) , Y s 1 , Z s 1 ) g ( s , ( Y s 2 ) , ( Z s 2 ) , Y s 2 , Z s 2 ) ] d B s t T Z ¯ s d W s , 0 t T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equat_HTML.gif
By Itô’s formula applied to | Y ¯ t + | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq190_HTML.gif and noting that ξ 1 ξ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq182_HTML.gif, it easily follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ17_HTML.gif
(3.11)
Since f 1 ( t , y , z , y , z ) f 2 ( t , y , z , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq183_HTML.gif a.s. and f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq181_HTML.gif is nondecreasing in y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq121_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equau_HTML.gif
Then we have
Δ : = 2 E [ Y ¯ s + ( E [ f 1 ( s , ( Y s 1 ) , Y s 1 , Z s 1 ) f 2 ( s , ( Y s 2 ) , ( Z s 2 ) , Y s 2 , Z s 2 ) ] ) ] ( 2 λ ¯ N + + L N 1 2 α + 2 | λ N | ) E [ | Y ¯ s + | 2 ] + ( 1 2 α ) E [ I { Y ¯ s > 0 } | Z ¯ s | 2 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ18_HTML.gif
(3.12)
With the assumption (A1), we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ19_HTML.gif
(3.13)
Combining (3.12), (3.13) with (3.11) yields
E [ | Y ¯ t + | 2 ] ( 2 λ ¯ N + + L N 1 2 α + 2 | λ N | + 2 L ) t T E [ ( Y ¯ s + ) 2 ] d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equav_HTML.gif
By Gronwall’s inequality, it follows that
E [ | Y ¯ t + | 2 ] = 0 , t [ 0 , T ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equaw_HTML.gif

that is, Y t 1 Y t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq184_HTML.gif, P-a.s., t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq185_HTML.gif. □

4 Decoupled mean-field forward-backward doubly SDEs

In this section, we study the decoupled mean-field forward-backward doubly stochastic differential equations. First, we recall some results of Buckdahn, Li and Peng [15] on McKean-Vlasov SDEs. Given continuous functions b : Ω ¯ × [ 0 , T ] × R d × R d R d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq191_HTML.gif and σ : Ω ¯ × [ 0 , T ] × R d × R d R d × d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq192_HTML.gif which are supposed to satisfy the following conditions:

Assumption 4.1 (i) b ( t , 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq193_HTML.gif and σ ( t , 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq194_HTML.gif are F ¯ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq195_HTML.gif-measurable continuous processes and there exists some constant L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq65_HTML.gif such that
| b ( t , x , x ) | + | σ ( t , x , x ) | L ( 1 + | x | ) , a.s., for all  0 t T , x , x R d ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equax_HTML.gif
  1. (ii)
    b and σ are Lipschitz in x, x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq196_HTML.gif, i.e., there is some constant L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq65_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equay_HTML.gif
     
For any x 0 R d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq197_HTML.gif, we consider the following SDE parameterized by the initial condition ( t , ζ ) [ 0 , T ] × L 2 ( Ω , F t , P ; R d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq198_HTML.gif:
{ d X s t , ζ = E [ b ( s , ( X s 0 , x 0 ) , X s t , ζ ) ] d s + E [ σ ( s , ( X s 0 , x 0 ) , X s t , ζ ) ] d W s , s [ t , T ] , X t t , ζ = ζ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ20_HTML.gif
(4.1)

From the result about Eq. (5.1) in [15], we know that under Assumption 4.1, SDE (4.1) has a unique strong solution, and we can obtain that X T t , ζ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq199_HTML.gif has a continuous version with the following well-known standard estimates.

Proposition 4.1 p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq200_HTML.gif, there exists C p R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq201_HTML.gif such that, for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq67_HTML.gif and ζ , ζ L p ( Ω , F t , P ; R d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq202_HTML.gif,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ21_HTML.gif
(4.2)

for all δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq203_HTML.gif with t + δ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq204_HTML.gif.

Now, let f ( t , x , x , y , y , z , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq205_HTML.gif, g ( t , x , x , y , y , z , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq206_HTML.gif and Φ ( x , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq207_HTML.gif be real-valued functions and satisfy the following conditions.

Assumption 4.2 (i) Φ: Ω ¯ × R d × R d R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq208_HTML.gif is an F ¯ T B ( R d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq209_HTML.gif-measurable random variable, f : Ω ¯ × [ 0 , T ] × R d × R d × R n × R n × R n × d × R n × d R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq210_HTML.gif and g : Ω ¯ × [ 0 , T ] × R d × R d × R n × R n × R n × d × R n × d R n × l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq211_HTML.gif are two measurable processes such that f ( t , x , x , y , y , z , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq205_HTML.gif, g ( t , x , x , y , y , z , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq212_HTML.gif are F ¯ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq213_HTML.gif-measurable, for all ( x , x , y , y , z , z ) R d × R d × R n × R n × R n × d × R n × d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq214_HTML.gif.
  1. (ii)
    For all 0 t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq215_HTML.gif, x 1 , x 1 , x 2 , x 2 R d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq216_HTML.gif, y 1 , y 1 , y 2 , y 2 R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq217_HTML.gif, z 1 , z 1 , z 2 , z 2 R n × d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq218_HTML.gif, there exist constants L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq65_HTML.gif, λ 1 , λ 2 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq219_HTML.gif and 0 < α < 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq61_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equaz_HTML.gif
     
  2. (iii)
    f, g and Φ satisfy a linear growth condition, i.e., there exists some L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq220_HTML.gif such that, a.s., for all x , x R d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq221_HTML.gif
    | Φ ( x , x ) | + | f ( t , x , x , 0 , 0 , 0 , 0 ) | + | g ( t , x , x , 0 , 0 , 0 , 0 ) | L ( 1 + | x | + | x | ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equba_HTML.gif
     
Next, we investigate the solution of the following BDSDE:
{ d Y s t , ζ = E [ f ( s , ( X s 0 , x 0 ) , X s t , ζ , ( Y s 0 , x 0 ) , Y s t , ζ , ( Z s 0 , x 0 ) , Z s t , ζ ) ] d s d Y s t , ζ = + E [ g ( s , ( X s 0 , x 0 ) , X s t , ζ , ( Y s 0 , x 0 ) , Y s t , ζ , ( Z s 0 , x 0 ) , Z s t , ζ ) ] d B s Z s t , ζ d W s , Y T t , ζ = E [ Φ ( ( X T 0 , x 0 ) , X T t , ζ ) ] , s [ t , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ22_HTML.gif
(4.3)
Firstly, we study the case ( t , ζ ) = ( 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq222_HTML.gif. From Theorem 2.1, we know that there exists a unique solution ( Y 0 , x 0 , Z 0 , x 0 ) S F 2 ( 0 , T ; R n ) × H F 2 ( 0 , T ; R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq223_HTML.gif to the mean-field BDSDE (4.3). Once we have ( Y 0 , x 0 , Z 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq224_HTML.gif, Eq. (4.3) becomes a classical BDSDE with coefficients
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equbb_HTML.gif

and Φ ˜ ( ω , X T t , ζ ( ω ) ) = E [ Φ ( , ω , ( X T 0 , x 0 ) , X T t , ζ ) ] L 2 ( Ω , F T , P ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq225_HTML.gif. Then due to Theorem 2.2 in [12], we obtain that there exists a unique solution ( Y t , ζ , Z t , ζ ) S F 2 ( [ 0 , T ] ; R n ) × H F 2 ( 0 , T ; R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq226_HTML.gif to Eq. (4.3).

For BDSDE (4.3), we give the following proposition.

Proposition 4.2 For any t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq67_HTML.gif and ζ , ζ L 2 ( Ω , F t , P ; R d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq227_HTML.gif, there exists a constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq228_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ23_HTML.gif
(4.4)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ24_HTML.gif
(4.5)

Proof Combining classical BDSDE estimates (see the proof of Theorem 2.1 in Pardoux and Peng [6]) with the techniques presented in Theorem 3.1, we can get the proof easily. □

5 Mean-field BDSDEs and McKean-Vlasov SPDEs

We now pay attention to investigation of the following system of quasilinear backward stochastic partial differential equations which are called McKean-Vlasov SPDEs: for any ( t , x ) [ 0 , T ] × R d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq229_HTML.gif,
u ( t , x ) = E [ Φ ( X T 0 , x 0 , x ) ] + t T L u ( s , x ) d s + t T E [ f ( s , X s 0 , x 0 , x , u ( s , X s 0 , x 0 ) , u ( s , x ) , σ ˆ u ( s , X s 0 , x 0 ) , σ ˆ u ( s , x ) ) ] d s + t T E [ g ( s , X s 0 , x 0 , x , u ( s , X s 0 , x 0 ) , u ( s , x ) , σ ˆ u ( s , X s 0 , x 0 ) , σ ˆ u ( s , x ) ) ] d B s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ25_HTML.gif
(5.1)
with σ ˆ : = E [ σ ( s , X s 0 , x 0 , x ) ] = E [ σ ( s , ( X s 0 , x 0 ) , x ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq230_HTML.gif, and u : R + × R d R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq231_HTML.gif,
L u = ( L u 1 L u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equbc_HTML.gif
with
L : = 1 2 i , j = 1 d a i j 2 x i x j + i = 1 d E [ b i ( t , X t 0 , x 0 , x ) ] x i , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equbd_HTML.gif
where
a : = ( a i , j ) = ( E [ σ ( t , X t 0 , x 0 , x ) ] E [ σ ( t , X t 0 , x 0 , x ) ] ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Eqube_HTML.gif
Note that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equbf_HTML.gif

In fact, Eq. (5.1) is a new kind of nonlocal SPDE because of the mean-field term. Here, the functions b, σ, f, g and Φ are supposed to satisfy Assumption 4.1 and Assumption 4.2 respectively, and X 0 , x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq232_HTML.gif is the solution of the mean-field SDE (4.1) with ( t , ξ ) = ( 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq233_HTML.gif.

Now, we give the main theorem of this section.

Theorem 5.1 Suppose that Assumption  4.1 and Assumption  4.2 hold. Let { u ( t , x ) ; 0 t T , x R d } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq234_HTML.gif be a F t , T B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq235_HTML.gif-measurable random field such that u ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq236_HTML.gif satisfies Eq. (5.1) and for each ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq237_HTML.gif, u C 0 , 2 ( [ 0 , T ] × R d ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq238_HTML.gif a.s. Moreover, we assume that f , g C ( [ 0 , T ] × R d × R d × R n × R n × R n × d × R n × d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq239_HTML.gif for a.s. ω ¯ Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq240_HTML.gif.

Then we have u ( t , x ) = Y t t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq241_HTML.gif, where { ( Y s t , x , Z s t , x ) ; t s T } t 0 , x R d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq242_HTML.gif is the unique solution of the mean-field BDSDEs (4.3) and
Y s t , x = u ( s , X s t , x ) , Z s t , x = E [ σ ( s , ( X s 0 , x 0 ) , X s t , x ) ] u ( s , X s t , x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equ26_HTML.gif
(5.2)
Proof It suffices to show that { u ( s , X s t , x ) , E [ σ ( s , ( X s 0 , x 0 ) , X s t , x ) ] u ( s , X s t , x ) ; 0 t s } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq243_HTML.gif solves the mean-field BDSDE (4.3). To simplify the notation, we define
φ ˆ ( s , X s 0 , x 0 , x ) E [ φ ( s , X s 0 , x 0 , x , u ( s , X s 0 , x 0 ) , u ( s , x ) , σ ˆ u ( s , X s 0 , x 0 ) , σ ˆ u ( s , x ) ) ] , for  φ = f , g . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equbg_HTML.gif
According to our notations introduced in Section 2, we know that
φ ˆ ( s , X s 0 , x 0 , x ) = E [ φ ( ω , s , X s 0 , x 0 ( ω ) , x , u ( s , X s 0 , x 0 ( ω ) ) , u ( s , x ) , σ ˆ u ( s , X s 0 , x 0 ( ω ) ) , σ ˆ u ( s , x ) ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equbh_HTML.gif
Let t = t 0 < t 1 < t 2 < < t n = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq244_HTML.gif and λ : = max { t i + 1 t i } 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq245_HTML.gif. For each t i [ t , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq246_HTML.gif, applying Itô’s formula to u ( t i , X t i t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq247_HTML.gif and noticing that u satisfies Eq. (5.1), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equbi_HTML.gif

The condition that u C 0 , 2 ( [ 0 , T ] × R d ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq248_HTML.gif and the continuity of f and g are adopted in the last equation.

Then we have
u ( t , x ) u ( T , X T t , x ) = i = 0 n 1 [ u ( t i , X t i t , x ) u ( t i + 1 , X t i + 1 t , x ) ] = i = 0 n 1 t i t i + 1 f ˆ ( s , X s 0 , x 0 , X s t , x ) d s + i = 0 n 1 t i t i + 1 g ˆ ( s , X s 0 , x 0 , X s t , x ) d B s i = 0 n 1 t i t i + 1 E [ σ ( s , ( X s 0 , x 0 ) , X s t , x ) ] u ( s , X s t , x ) d W s = t T f ˆ ( s , X s 0 , x 0 , X s t , x ) d s + t T g ˆ ( s , X s 0 , x 0 , X s t , x ) d B s t T E [ σ ( s , ( X s 0 , x 0 ) , X s t , x ) ] u ( s , X s t , x ) d W s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_Equbj_HTML.gif

So, Y s t , x : = u ( s , X s t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq249_HTML.gif, Z s t , x : = E [ σ ( s , ( X s 0 , x 0 ) , X s t , x ) ] u ( s , X s t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-114/MediaObjects/13661_2012_Article_242_IEq250_HTML.gif solves the mean-field BDSDE (4.3). The proof is now complete. □

Remark 5.1 Formula (5.2) generalizes the stochastic Feynman-Kac formula for SPDEs of the mean-field type.

Declarations

Acknowledgements

The author would like to thank the editor and anonymous referees for their constructive and insightful comments on improving the quality of this revision, and to thank professor Zhen Wu for many helpful suggestions.

Authors’ Affiliations

(1)
School of Mathematics, Shandong University
(2)
School of Mathematics, Shandong Polytechnic University

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