Mean-field backward doubly stochastic differential equations and related SPDEs

Here, the function $u(t,x):[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^n$ is the unknown function, and $\{B_t,0\le t\le T\}$ is an l-dimensional Brownian motion process defined on a given complete probability space $(\mathrm{\Omega },\mathcal{F},P)$. $X_t^{0,x_0}$, a stochastic process starting at $x_0$ when $t=0$, is the solution of one class of stochastic differential equations (SDEs), and E denotes expectation with respect to the probability P. In this paper, we call this kind of equations (1.1) McKean-Vlasov SPDEs, because they are analogous to McKean-Vlasov PDEs except the stochastic term $d{B}_t$.

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 $d{W}_t$ and a backward stochastic integral $d{B}_t$. 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.

In Eq. (1.2), the integral $d{W}_t$ is a forward Itô integral, and the integral $d{B}_t$ denotes a backward Itô integral. $\{W_t,0\le t\le T\}$ and $\{B_t,0\le t\le T\}$ are two mutually independent standard Brownian motion processes with values respectively in ${\mathbb{R}}^d$ and in ${\mathbb{R}}^l$. 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.

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].

with ${\mathcal{F}}_{s,t}^W=\sigma \{W_r;s\le r\le t\}\vee \mathcal{N}$ and ${\mathcal{F}}_{t,s}^B=\sigma \{B_r-B_t;t\le r\le s\}\vee \mathcal{N}$.

Note that $\{{\mathcal{F}}_t,t\in [0,T]\}$ is not an increasing family of σ-fields, so it is not a filtration.

We will also use the following spaces:

Moreover, for all $(y^{\prime },z^{\prime },y,z)$, $f=f({\omega }^{\prime },\omega ,t,y^{\prime },z^{\prime },y,z):\overline{\mathrm{\Omega }}\times [0,T]\times {\mathbb{R}}^n\times {\mathbb{R}}^{n\times d}\times {\mathbb{R}}^n\times {\mathbb{R}}^{n\times d}\to {\mathbb{R}}^n$, $g=g({\omega }^{\prime },\omega ,t,y^{\prime },z^{\prime },y,z):\overline{\mathrm{\Omega }}\times [0,T]\times {\mathbb{R}}^n\times {\mathbb{R}}^{n\times d}\times {\mathbb{R}}^n\times {\mathbb{R}}^{n\times d}\to {\mathbb{R}}^{n\times l}$ are two ${\overline{\mathcal{F}}}_t$-measurable functions which satisfy

Assumption 2.1 (A1) $g(t,0,0,0,0)\in {\mathcal{H}}_{\overline{\mathbb{F}}}^2(0,T;{\mathbb{R}}^{n\times l})$, and there exist $L>0$ and $0<\alpha <\frac{1}{2}$ such that

(A2) for any fixed $({\omega }^{\prime },\omega ,t)$, $f({\omega }^{\prime },\omega ,t,\cdot ,\cdot ,\cdot ,\cdot )$ is continuous;

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

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

Definition 2.1 A pair of ${\mathcal{F}}_t$-measurable processes $\{(Y_t,Z_t);0\le t\le T\}$ is called a solution of mean-field BDSDE (2.1) if $(Y,Z)\in {\mathcal{S}}_{\mathbb{F}}^2([0,T];{\mathbb{R}}^n)\times {\mathcal{H}}_{\mathbb{F}}^2(0,T;{\mathbb{R}}^{n\times d})$ and it satisfies mean-field BDSDE (2.1).

The main result of this section is the following theorem.

Theorem 2.1 For any random variable $\xi \in L^2(\mathrm{\Omega },{\mathcal{F}}_T,P;{\mathbb{R}}^n)$, under Assumption  2.1, mean-field BDSDE (2.1) admits a unique solution $(Y,Z)\in {\mathcal{S}}_{\mathbb{F}}^2([0,T];{\mathbb{R}}^n)\times {\mathcal{H}}_{\mathbb{F}}^2(0,T;{\mathbb{R}}^{n\times d})$.

and f fulfills

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

The parameters $\overline{c}$ and β are specified later.

For any $(y^1,z^1),(y^2,z^2)\in {\mathcal{H}}_{\mathbb{F}}^2(0,T;{\mathbb{R}}^n\times {\mathbb{R}}^{n\times d})$, we set $(Y^1,Z^1)=I[(y^1,z^1)]$, $(Y^2,Z^2)=I[(y^2,z^2)]$, $(\overline{y},\overline{z})=(y^1-y^2,z^1-z^2)$ and $(\overline{Y},\overline{Z})=(Y^1-Y^2,Z^1-Z^2)$. Then applying Itô’s formula to $e^{\beta s}{|{\overline{Y}}_s|}^2$ and by virtue of $Y^1,Y^2\in {\mathcal{S}}_{\mathbb{F}}^2([0,T];{\mathbb{R}}^n)$, we have

(2.3)

From condition (A4) and noting that $E^{\prime }[Y_s({\omega }^{\prime })]=E[Y_s(\omega )]$, for any $M>0$, we get

Then we have

Consequently, I is a strict contraction on ${\mathcal{H}}_{\mathbb{F}}^2(0,T;{\mathbb{R}}^n\times {\mathbb{R}}^{n\times d})$ equipped with the norm ${\parallel \cdot \parallel }_{\beta }$ for $0<\alpha <\frac{1}{2}$. With the contraction mapping theorem, there admits a unique fixed point $(Y,Z)\in {\mathcal{H}}_{\mathbb{F}}^2(0,T;{\mathbb{R}}^n\times {\mathbb{R}}^{n\times d})$ such that $I(Y,Z)=(Y,Z)$. On the other hand, from Step 1, we know that if $I(Y,Z)=(Y,Z)$, then $(Y,Z)\in {\mathcal{S}}_{\mathbb{F}}^2([0,T];{\mathbb{R}}^n)\times {\mathcal{H}}_{\mathbb{F}}^2(0,T;{\mathbb{R}}^{n\times d})$, which is the unique solution of Eq. (2.1). □

where ${\phi }_i\in {\mathcal{H}}_{\overline{\mathbb{F}}}^2(0,T;{\mathbb{R}}^n)$. Then we have the following corollary.

Corollary 2.1 Suppose that $(Y^i,Z^i)$ is the solution of mean-field BDSDE (2.1) with data $({\xi }^i,f_i,g)$, $i=1,2$, where ${\xi }^1,{\xi }^2\in L^2(\mathrm{\Omega },{\mathcal{F}}_T,P;{\mathbb{R}}^n)$ are two arbitrary terminal values. The difference of $(Y^1,Z^1)$ and $(Y^2,Z^2)$ satisfies the following estimate:

(2.4)

where $\beta =2|{\lambda }_1|+2{\lambda }_2^{+}+\frac{4L}{1-2\alpha }+2+2L$.

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

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

We assume

(A3′) there exist $L>0$ and $0\le \gamma <1$ such that $|f(t,y^{\prime },z^{\prime },y,z)|\le L(1+{|y^{\prime }|}^{\gamma }+{|z^{\prime }|}^{\gamma }+{|y|}^{\gamma }+{|z|}^{\gamma })$;

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

for any t, $y^{\prime }$, y, $z_i^{\prime }$, $z_i$ satisfying $|y^{\prime }|,|y|,|z_i^{\prime }|,|z_i|\le N$ ($i=1,2$).

where ${\varphi }_m:{\mathbb{R}}^n\to {\mathbb{R}}^{+}$ is a sequence of smooth functions such that $0\le {\varphi }_m\le 1$, ${\varphi }_m(x)=1$ for $|x|\le m$, and ${\varphi }_m(x)=0$ for $|x|\ge m+1$. Similarly, we define the sequences ${\phi }_m:{\mathbb{R}}^{n\times d}\to {\mathbb{R}}^{+}$, ${\psi }_m:{\mathbb{R}}^n\to {\mathbb{R}}^{+}$, ${\eta }_m:{\mathbb{R}}^{n\times d}\to {\mathbb{R}}^{+}$. It should be pointed out that ${\varphi }_m$, ${\phi }_m$, ${\psi }_m$ and ${\eta }_m$ 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.

where θ is an arbitrarily fixed constant such that $0<\theta <1-2\alpha$. Then mean-field BDSDE (2.1) has a unique solution $(Y,Z)\in {\mathcal{S}}_{\mathbb{F}}^2([0,T];{\mathbb{R}}^n)\times {\mathcal{H}}_{\mathbb{F}}^2(0,T;{\mathbb{R}}^{n\times d})$.

admits a unique solution $(Y^m,Z^m)\in {\mathcal{S}}_{\mathbb{F}}^2([0,T];{\mathbb{R}}^n)\times {\mathcal{H}}_{\mathbb{F}}^2(0,T;{\mathbb{R}}^{n\times d})$ for each $m\ge N$. Applying Itô’s formula to ${|Y_t^m|}^2$ yields

where

and

Hence,

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

where $C>0$ only depends on T, α, L and is independent of m.

and $\overline{A}:=\mathrm{\Omega }\setminus A$.

admits a unique solution $(Y^k,Z^k)\in {\mathcal{S}}_{\mathbb{F}}^2([0,T];{\mathbb{R}}^n)\times {\mathcal{H}}_{\mathbb{F}}^2(0,T;{\mathbb{R}}^{n\times d})$ for each $k\ge N$. Applying Itô’s formula to ${|Y_t^m-Y_t^k|}^2$, we have

(3.3)

where

We next estimate I, II and III.

where $C>0$ depends on T, L, α and $E[{\int }_0^T{|E^{\prime }[g(s,0,0,0,0)]|}^{2}ds]$.

Choose $M>0$ such that $\theta :=\frac{2L_N}{M}<1-2\alpha$. Then from (3.3)-(3.6), we obtain

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

as $m\to \mathrm{\infty }$.

Next, we show that $(Y,Z)$ 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)