Abstract
Existence and uniqueness result of the solutions to meanfield 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 meanfield 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 meanfield BDSDEs, which can be viewed as the stochastic FeynmanKac formula for SPDEs of meanfield type.
Keywords:
meanfield; backward doubly stochastic differential equations; locally monotone coefficients; comparison theorem; stochastic partial differential equations1 Introduction
In this paper, we study a new kind of stochastic partial differential equations (SPDEs):
where
and
Here, the function
McKeanVlasov PDEs involving models of large stochastic particle systems with meanfield interaction have been studied by stochastic methods in recent years (see [14] and the references therein). Meanfield approaches have applications in many areas such as statistical mechanics and physics, quantum mechanics and quantum chemistry. Recently, Lasry and Lions introduced meanfield approaches for highdimensional systems of evolution equations corresponding to a large number of ‘agents’ or ‘particles’. They extended the field of such meanfield approaches to problems in economics, finance and game theory (see [5] and the references therein).
As is well known, to give a probabilistic representation (FeynmanKac 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
Meanfield BSDEs are deduced by Buckdahn, Djehiche, Li and Peng [14] when they studied a special meanfield 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 meanfield BSDEs as well as the comparison theorem and also gave the viscosity solutions of a class of McKeanVlasov PDEs in terms of meanfield BSDEs.
In this paper, we study a new type of BDSDEs, that is, the so called meanfield 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 meanfield 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 meanfield BDSDEs because of the meanfield term.
We also present the connection between McKeanVlasov SPDEs and meanfield BDSDEs.
In detail, let
Assume that Eq. (1.1) has a classical solution. Then the couple
In Eq. (1.2), the integral
Our paper is organized as follows. In Section 2, we present the existence and uniqueness results about meanfield BDSDEs with globally monotone coefficients. We investigate the properties of meanfield BDSDEs with locally monotone assumptions in Section 3. We first prove the existence and uniqueness of the solutions of meanfield BDSDEs and then derive the comparison theorem when the meanfield BDSDEs are onedimensional. In Section 4, we introduce the decoupled meanfield forwardbackward doubly stochastic differential equation and study the regularity of its solution with respect to x, which is the initial condition of the McKeanVlasov SDE. Finally, Section 5 is devoted to the formulation of McKeanVlasov SPDEs and provides the relationship between the solutions of SPDEs and those of meanfield BDSDEs.
2 Meanfield BDSDEs with globally monotone coefficients
In this section, we study meanfield 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 meanfield BSDEs obtained by Buckdahn, Li and Peng [15].
Let
with
Note that
We will also use the following spaces:
• For any
(i)
(ii)
• We denote similarly by
(i)
(ii)
•
• For
Let
Notice that
Moreover, for all
Assumption 2.1 (A1)
(A2) for any fixed
(A3) there exist a process
(A4) there exist constants
(A5) there exists
We now consider the following meanfield BDSDEs with the form:
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
then it must satisfy conditions (A4) and (A5).
Definition 2.1 A pair of
The main result of this section is the following theorem.
Theorem 2.1For any random variable
ProofStep 1: For any
has a unique solution. In order to get this conclusion, we define
Then (2.2) can be rewritten as
Due to Assumption 2.1, for all
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
The parameters
From Step 1, we can introduce the mapping
For any
From condition (A4) and noting that
Then we have
If we set
Consequently, I is a strict contraction on
Suppose that: For some
where
Corollary 2.1Suppose that
where
The proof of the above corollary is similar to that of Theorem 2.1 and is therefore omitted.
3 Meanfield BDSDEs with locally monotone coefficients
In this section, we investigate meanfield BDSDEs with locally monotone coefficients. The results can be regarded as an extension of the results in [12] to the meanfield type.
We assume
(A3′) there exist
(A4′) for any
(A5′)
Remark 3.1 Since
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
(i) for fixed
(ii) ∀m,
(iii) ∀N,
(iv) ∀m,
for anyt,
(v) ∀m,
for anyt,
Proof We define
where
We now present the main result of this section.
Theorem 3.1Let (A1), (A2), (A3′)(A5′) hold. Assume, moreover,
whereθis an arbitrarily fixed constant such that
Proof We now construct an approximate sequence. Let
admits a unique solution
where
and
Hence,
Then it follows from Gronwall’s inequality and the BDG inequality that
where
For any
and
Next, we will conclude that
admits a unique solution
where
We next estimate I, II and III.
For the first term I, based on Hölder’s inequality and Chebyshev’s inequality, we have
where
For the second term II, due to the local monotonicity of
For the last term, we have
Choose
Applying Gronwall’s inequality and the BDG inequality to the above inequality yields
where
as
Next, we show that
Set
and
Since
where
then we have
where
there exists a subsequence of
as
Now, passing to the limit as
Therefore, we come to the conclusion of this theorem. □
Now, we discuss the comparison theorem for meanfield BDSDEs. We only consider onedimensional
meanfield BDSDEs, i.e.,
We consider the following meanfield BDSDEs: (
Theorem 3.2 (Comparison theorem)
Assume meanfield BDSDEs (3.9) and (3.10) satisfy the conditions of Theorem 3.1. Let
(i) One of the two generators is independent of
(ii) One of the two generators is nondecreasing in
Then if
Remark 3.2 The conditions (i) and (ii) of Theorem 3.2 are, in particular, satisfied if they
hold for the same generator
Proof Without loss of generality, we suppose that (i) is satisfied by
By Itô’s formula applied to
Since
Then we have
With the assumption (A1), we obtain
Combining (3.12), (3.13) with (3.11) yields
By Gronwall’s inequality, it follows that
that is,
4 Decoupled meanfield forwardbackward doubly SDEs
In this section, we study the decoupled meanfield forwardbackward doubly stochastic
differential equations. First, we recall some results of Buckdahn, Li and Peng [15] on McKeanVlasov SDEs. Given continuous functions
Assumption 4.1 (i)
(ii) b and σ are Lipschitz in x,
For any
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
Proposition 4.1
for all
Now, let
Assumption 4.2 (i) Φ:
(ii) For all
(iii) f, g and Φ satisfy a linear growth condition, i.e., there exists some
Next, we investigate the solution of the following BDSDE:
Firstly, we study the case
and
For BDSDE (4.3), we give the following proposition.
Proposition 4.2For any
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 Meanfield BDSDEs and McKeanVlasov SPDEs
We now pay attention to investigation of the following system of quasilinear backward
stochastic partial differential equations which are called McKeanVlasov SPDEs: for
any
with
with
where
Note that
In fact, Eq. (5.1) is a new kind of nonlocal SPDE because of the meanfield term.
Here, the functions b, σ, f, g and Φ are supposed to satisfy Assumption 4.1 and Assumption 4.2 respectively, and
Now, we give the main theorem of this section.
Theorem 5.1Suppose that Assumption 4.1 and Assumption 4.2 hold. Let
Then we have
Proof It suffices to show that
According to our notations introduced in Section 2, we know that
Let
The condition that
Then we have
So,
Remark 5.1 Formula (5.2) generalizes the stochastic FeynmanKac formula for SPDEs of the meanfield 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.
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