Open Access Research

Mean-field backward doubly stochastic differential equations and related SPDEs

Ruimin Xu

Author affiliations

School of Mathematics, Shandong University, Jinan, 250100, China

School of Mathematics, Shandong Polytechnic University, Jinan, 250353, China

Citation and License

Boundary Value Problems 2012, 2012:114  doi:10.1186/1687-2770-2012-114


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/114


Received:14 May 2012
Accepted:2 October 2012
Published:17 October 2012

© 2012 Xu; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M2">View MathML</a> is the transpose of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M3">View MathML</a> which is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M4">View MathML</a>, and ℒ is a second-order differential operator given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M5">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M6">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M7">View MathML</a>

Here, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M8">View MathML</a> is the unknown function, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M9">View MathML</a> is an l-dimensional Brownian motion process defined on a given complete probability space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M10">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M11">View MathML</a>, a stochastic process starting at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M12">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M13">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M14">View MathML</a>.

McKean-Vlasov PDEs involving models of large stochastic particle systems with mean-field interaction have been studied by stochastic methods in recent years (see [1-4] 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M15">View MathML</a> and a backward stochastic integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M14">View MathML</a>. 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.[7-13] 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M17">View MathML</a> be the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M18">View MathML</a>

Assume that Eq. (1.1) has a classical solution. Then the couple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M19">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M21">View MathML</a>, verifies the following mean-field BDSDE :

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M22">View MathML</a>

(1.2)

In Eq. (1.2), the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M15">View MathML</a> is a forward Itô integral, and the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M14">View MathML</a> denotes a backward Itô integral. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M9">View MathML</a> are two mutually independent standard Brownian motion processes with values respectively in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M27">View MathML</a> and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M28">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M9">View MathML</a> be two mutually independent standard Brownian motion processes, with values respectively in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M28">View MathML</a>, defined over some complete probability space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M10">View MathML</a>, where T is a fixed positive number throughout this paper. Moreover, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M34">View MathML</a> denote the class of P-null sets of ℱ. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M35">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M36">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M38">View MathML</a>.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M39">View MathML</a> is not an increasing family of σ-fields, so it is not a filtration.

We will also use the following spaces:

• For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M40">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M41">View MathML</a> denote the set of (classes of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M42">View MathML</a> a.e. equal) n-dimensional jointly measurable random processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M43">View MathML</a> which satisfy: Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M48">View MathML</a> is a Banach space endowed with the canonical norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M49">View MathML</a>.

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M44">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M45">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M46">View MathML</a> measurable, for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M47">View MathML</a>.

• We denote similarly by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M50">View MathML</a> the set of continuous n-dimensional random processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M51">View MathML</a> which satisfy:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M52">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M45">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M46">View MathML</a> measurable, for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M47">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M56">View MathML</a> denotes the space of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M57">View MathML</a> valued ℱ-measurable random variables.

• For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M59">View MathML</a> is the space of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M57">View MathML</a> valued ℱ-measurable random variables such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M61">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M62">View MathML</a> be the (non-completed) product of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M63">View MathML</a> with itself, and we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M64">View MathML</a> on this product space. A random variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M65">View MathML</a> originally defined on Ω is extended canonically to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M67">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M68">View MathML</a>, the variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M69">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M71">View MathML</a>-a.s., whose expectation is denoted by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M72">View MathML</a>

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M73">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M74">View MathML</a>

Moreover, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M77">View MathML</a> are two <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M78">View MathML</a>-measurable functions which satisfy

Assumption 2.1 (A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M79">View MathML</a>, and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M81">View MathML</a> such that

(A2) for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M84">View MathML</a> is continuous;

(A3) there exist a process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M85">View MathML</a> and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M86">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M87">View MathML</a>

(A4) there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M88">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M91">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M93">View MathML</a>

(A5) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M94">View MathML</a> such that, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M95">View MathML</a>-a.s., for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M98">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M99">View MathML</a>

We now consider the following mean-field BDSDEs with the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M100">View MathML</a>

(2.1)

Remark 2.1 Due to our notation, the coefficients of (2.1) are interpreted as follows:

Remark 2.2 If coefficient f meets the following Lipschitz assumption: There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M94">View MathML</a> such that, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M95">View MathML</a>-a.s., for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M106">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M108">View MathML</a>

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

Definition 2.1 A pair of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M46">View MathML</a>-measurable processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M110">View MathML</a> is called a solution of mean-field BDSDE (2.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M111">View MathML</a> and it satisfies mean-field BDSDE (2.1).

The main result of this section is the following theorem.

Theorem 2.1For any random variable<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M112">View MathML</a>, under Assumption 2.1, mean-field BDSDE (2.1) admits a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M111">View MathML</a>.

ProofStep 1: For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M114">View MathML</a>, BDSDE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M115">View MathML</a>

(2.2)

has a unique solution. In order to get this conclusion, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M116">View MathML</a>

Then (2.2) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M117">View MathML</a>

Due to Assumption 2.1, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M118">View MathML</a>, g satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M119">View MathML</a>

and f fulfills

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

Step 2: Now, we introduce a norm on the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M121">View MathML</a> which is equivalent to the canonical norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M122">View MathML</a>

The parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M123">View MathML</a> and β are specified later.

From Step 1, we can introduce the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M124">View MathML</a> through the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M125">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M126">View MathML</a>, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M130">View MathML</a>. Then applying Itô’s formula to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M131">View MathML</a> and by virtue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M132">View MathML</a>, we have

(2.3)

From condition (A4) and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M134">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M135">View MathML</a>, we get

Then we have

If we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M140">View MathML</a>, then it yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M141">View MathML</a>

Consequently, I is a strict contraction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M142">View MathML</a> equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M143">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M144">View MathML</a>. With the contraction mapping theorem, there admits a unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M145">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M146">View MathML</a>. On the other hand, from Step 1, we know that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M147">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M148">View MathML</a>, which is the unique solution of Eq. (2.1). □

Suppose that: For some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M149">View MathML</a> satisfying (A2)-(A5), the generators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92">View MathML</a> are of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M152">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M153">View MathML</a>. Then we have the following corollary.

Corollary 2.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M154">View MathML</a>is the solution of mean-field BDSDE (2.1) with data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M157">View MathML</a>are two arbitrary terminal values. The difference of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M158">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M159">View MathML</a>satisfies the following estimate:

(2.4)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M161">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M163">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M164">View MathML</a>;

(A4′) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M165">View MathML</a>, there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M166">View MathML</a> such that, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M167">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M168">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M170">View MathML</a>

(A5′) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M171">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M172">View MathML</a> such that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M173">View MathML</a>, y, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M175">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M176">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92">View MathML</a>), it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M178">View MathML</a>

Remark 3.1 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M180">View MathML</a>, (A3′) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M181">View MathML</a>

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

Lemma 3.1Under (A2), (A3′)-(A5′) there exists a sequence of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M182">View MathML</a>such that

(i) for fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M184">View MathML</a>, ω, t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M185">View MathML</a>is continuous;

(ii) ∀m, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M186">View MathML</a>;

(iii) ∀N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M187">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M188">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M189">View MathML</a>;

(iv) ∀m, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190">View MathML</a>is globally monotone iny; moreover, for anym, Nwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M191">View MathML</a>, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M192">View MathML</a>

for anyt, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M195">View MathML</a>, zsatisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M196">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92">View MathML</a>);

(v) ∀m, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190">View MathML</a>is globally Lipschitz in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M195">View MathML</a>, z; moreover, for anym, Nwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M191">View MathML</a>, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M201">View MathML</a>

for anyt, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M173">View MathML</a>, y, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M175">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M205">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M92">View MathML</a>).

Proof We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M208">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M209">View MathML</a> is a sequence of smooth functions such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M211">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M212">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M213">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M214">View MathML</a>. Similarly, we define the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M217">View MathML</a>. It should be pointed out that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M220">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M221">View MathML</a> 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 3.1Let (A1), (A2), (A3′)-(A5′) hold. Assume, moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M222">View MathML</a>

(3.1)

whereθis an arbitrarily fixed constant such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M223">View MathML</a>. Then mean-field BDSDE (2.1) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M224">View MathML</a>.

Proof We now construct an approximate sequence. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190">View MathML</a> be associated to f by Lemma 3.1. Then for each m, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190">View MathML</a> is globally monotone in y and globally Lipschitz in z. By Theorem 2.1, the following mean-field BDSDE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M227">View MathML</a>

(3.2)

admits a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M228">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M191">View MathML</a>. Applying Itô’s formula to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M230">View MathML</a> yields

where

and

Hence,

Then it follows from Gronwall’s inequality and the B-D-G inequality that

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M236">View MathML</a> only depends on T, α, L and is independent of m.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M237">View MathML</a>, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M238">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M239">View MathML</a>.

Next, we will conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M240">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M241">View MathML</a>. Actually, since mean-field BDSDE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M242">View MathML</a>

admits a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M243">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M244">View MathML</a>. Applying Itô’s formula to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M245">View MathML</a>, we have

(3.3)

where

We next estimate I, II and III.

For the first term I, based on Hölder’s inequality and Chebyshev’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M248">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M236">View MathML</a> depends on T, L, α and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M250">View MathML</a>.

For the second term II, due to the local monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190">View MathML</a> in y and the local Lipschitz condition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M190">View MathML</a> in z, we obtain that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M253">View MathML</a>, the following holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M254">View MathML</a>

(3.5)

For the last term, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M255">View MathML</a>

(3.6)

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M135">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M257">View MathML</a>. Then from (3.3)-(3.6), we obtain

Applying Gronwall’s inequality and the B-D-G inequality to the above inequality yields

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M260">View MathML</a> is independent of m, k. Now passing to the limit successively on m, k and N, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M240">View MathML</a> is a Cauchy (hence convergent) sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M262">View MathML</a>; denote the limit by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M263">View MathML</a>, which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M264">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M265">View MathML</a>.

Next, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M263">View MathML</a> 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:

(3.7)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M268">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M269">View MathML</a>.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M270">View MathML</a>

where

then we have

(3.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M236">View MathML</a> is independent of m. As

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M274">View MathML</a>

there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M275">View MathML</a>, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M275">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M277">View MathML</a> a.e., a.s. It then follows from the continuity of f in y and the dominated convergence theorem that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M278">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M265">View MathML</a>.

Now, passing to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M281">View MathML</a> in (3.8) successively, it follows that (3.7) holds. Then letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M265">View MathML</a> in (3.2) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M283">View MathML</a>

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., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M284">View MathML</a>.

We consider the following mean-field BDSDEs: (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M285">View MathML</a>)

(3.9)

(3.10)

Theorem 3.2 (Comparison theorem)

Assume mean-field BDSDEs (3.9) and (3.10) satisfy the conditions of Theorem 3.1. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M158">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M159">View MathML</a>be the solutions of mean-field BDSDEs (3.9) and (3.10), respectively. Moreover, for the two generators of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M290">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M291">View MathML</a>, we suppose:

(i) One of the two generators is independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M195">View MathML</a>.

(ii) One of the two generators is nondecreasing in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M173">View MathML</a>.

Then if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M294">View MathML</a>, a.s., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M295">View MathML</a>, a.s., there also holds that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M296">View MathML</a>, a.s. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M297">View MathML</a>.

Remark 3.2 The conditions (i) and (ii) of Theorem 3.2 are, in particular, satisfied if they hold for the same generator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M298">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M299">View MathML</a>), 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M290">View MathML</a> and (ii) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M291">View MathML</a>. For notational simplicity, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M302">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M303">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M304">View MathML</a>

By Itô’s formula applied to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M305">View MathML</a> and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M294">View MathML</a>, it easily follows that

(3.11)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M295">View MathML</a> a.s. and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M291">View MathML</a> is nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M173">View MathML</a>, we have

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M312">View MathML</a>

(3.12)

With the assumption (A1), we obtain

(3.13)

Combining (3.12), (3.13) with (3.11) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M314">View MathML</a>

By Gronwall’s inequality, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M315">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M296">View MathML</a>, P-a.s., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M297">View MathML</a>. □

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M318">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M319">View MathML</a> which are supposed to satisfy the following conditions:

Assumption 4.1 (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M320">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M321">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M322">View MathML</a>-measurable continuous processes and there exists some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M86">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M324">View MathML</a>

(ii) b and σ are Lipschitz in x, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M325">View MathML</a>, i.e., there is some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M86">View MathML</a> such that

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M328">View MathML</a>, we consider the following SDE parameterized by the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M329">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M330">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M331">View MathML</a> has a continuous version with the following well-known standard estimates.

Proposition 4.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M332">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M333">View MathML</a>such that, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M335">View MathML</a>,

(4.2)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M337">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M338">View MathML</a>.

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M339">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M340">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M341">View MathML</a> be real-valued functions and satisfy the following conditions.

Assumption 4.2 (i) Φ: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M342">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M343">View MathML</a>-measurable random variable, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M344">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M345">View MathML</a> are two measurable processes such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M339">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M347">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M348">View MathML</a>-measurable, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M349">View MathML</a>.

(ii) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M350">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M351">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M352">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M353">View MathML</a>, there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M355">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M81">View MathML</a> such that

(iii) f, g and Φ satisfy a linear growth condition, i.e., there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M358">View MathML</a> such that, a.s., for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M359">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M360">View MathML</a>

Next, we investigate the solution of the following BDSDE:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M361">View MathML</a>

(4.3)

Firstly, we study the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M362">View MathML</a>. From Theorem 2.1, we know that there exists a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M363">View MathML</a> to the mean-field BDSDE (4.3). Once we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M364">View MathML</a>, Eq. (4.3) becomes a classical BDSDE with coefficients

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M366">View MathML</a>. Then due to Theorem 2.2 in [12], we obtain that there exists a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M367">View MathML</a> to Eq. (4.3).

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

Proposition 4.2For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M369">View MathML</a>, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M370">View MathML</a>such that

(4.4)

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M373">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M374">View MathML</a>

(5.1)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M375">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M376">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M377">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M378">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M379">View MathML</a>

Note that

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M381">View MathML</a> is the solution of the mean-field SDE (4.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M382">View MathML</a>.

Now, we give the main theorem of this section.

Theorem 5.1Suppose that Assumption 4.1 and Assumption 4.2 hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M383">View MathML</a>be a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M384">View MathML</a>-measurable random field such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M385">View MathML</a>satisfies Eq. (5.1) and for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M386">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M387">View MathML</a>a.s. Moreover, we assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M388">View MathML</a>for a.s. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M389">View MathML</a>.

Then we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M390">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M391">View MathML</a>is the unique solution of the mean-field BDSDEs (4.3) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M392">View MathML</a>

(5.2)

Proof It suffices to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M393">View MathML</a> solves the mean-field BDSDE (4.3). To simplify the notation, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M394">View MathML</a>

According to our notations introduced in Section 2, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M395">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M396">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M397">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M398">View MathML</a>, applying Itô’s formula to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M399">View MathML</a> and noticing that u satisfies Eq. (5.1), we get

The condition that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M401">View MathML</a> and the continuity of f and g are adopted in the last equation.

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M402">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M403">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/114/mathml/M404">View MathML</a> 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.

Competing interests

The author declares that he has no competing interests.

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.

References

  1. Bossy, M: Some stochastic particle methods for nonlinear parabolic PDEs. ESAIM Proc.. 15, 18–57 (2005)

  2. Méléard, S: Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. In: Talay D, Tubaro L (eds.) Probabilistic Models for Nonlinear PDE’s, pp. 42–95. Springer, Berlin (1996).

  3. Bossy, M, Talay, D: A stochastic particle method for the McKean-Vlasov and the Burgers equation. Math. Comput.. 217(66), 157–192 (1997)

  4. Talay, D, Vaillant, O: A stochastic particle method with random weights for the computation of statistical solutions of McKean-Vlasov equations. Ann. Appl. Probab.. 13(1), 140–180 (2003)

  5. Lasry, JM, Lions, PL: Mean field games. Jpn. J. Math.. 2, 229–260 (2007). Publisher Full Text OpenURL

  6. Pardoux, E, Peng, SG: Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Relat. Fields. 98, 209–227 (1994). Publisher Full Text OpenURL

  7. Bally, V, Matoussi, A: Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theor. Probab.. 14(1), 125–164 (2001). Publisher Full Text OpenURL

  8. Boufoussi, B, Casteren, J, Mrhardy, N: Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions. Bernoulli. 13(2), 423–446 (2007). Publisher Full Text OpenURL

  9. Han, YC, Peng, SG, Wu, Z: Maximum principle for backward doubly stochastic control systems with applications. SIAM J. Control Optim.. 48(7), 4224–4241 (2010). Publisher Full Text OpenURL

  10. N’zi, M, Owo, J: Backward doubly stochastic differential equations with discontinuous coefficients. Stat. Probab. Lett.. 79, 920–926 (2009). Publisher Full Text OpenURL

  11. Shi, YF, Gu, YL, Liu, K: Comparison theorems of backward doubly stochastic differential equations and applications. Stoch. Anal. Appl.. 23, 97–110 (2005). Publisher Full Text OpenURL

  12. Wu, Z, Zhang, F: BDSDEs with locally monotone coefficients and Sobolev solutions for SPDEs. J. Differ. Equ.. 251, 759–784 (2011). Publisher Full Text OpenURL

  13. Zhang, Q, Zhao, HZ: Stationary solutions of SPDEs and infinite horizon BDSDEs. J. Funct. Anal.. 252, 171–219 (2007). PubMed Abstract | Publisher Full Text OpenURL

  14. Buckdahn, R, Djehiche, B, Li, J, Peng, SG: Mean-field backward stochastic differential equations: a limit approach. Ann. Probab.. 37(4), 1524–1565 (2009). Publisher Full Text OpenURL

  15. Buckdahn, R, Li, J, Peng, SG: Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Process. Appl.. 119(10), 3133–3154 (2009). Publisher Full Text OpenURL