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 is the transpose of which is defined by , and ℒ is a secondorder differential operator given by with
and
Here, the function is the unknown function, and is an ldimensional Brownian motion process defined on a given complete probability space . , a stochastic process starting at when , 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) McKeanVlasov SPDEs, because they are analogous to McKeanVlasov PDEs except the stochastic term .
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 and a backward stochastic integral . They proved the existence and uniqueness for solutions of BDSDEs under uniformly Lipschitz coefficients. When the coefficients are smooth enough, they also established the connection between BDSDEs and a certain kind of quasilinear SPDEs. BDSDEs have a practical background in finance. The extra noise B can be regarded as some extra inside information in a derivative security market. Since 1990s, BDSDEs have drawn more attention from many authors (cf.[713] and the references therein). Shi, Gu and Liu gave the comparison theorem of BDSDEs and investigated the existence of solutions for BDSDEs with continuous coefficients in [11]. To relax the Lipschitz conditions, Wu and Zhang studied two kinds of BDSDEs under globally (respectively, locally) monotone assumptions and obtained the uniqueness and existence results of the solutions (see [12]).
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 be the solution of
Assume that Eq. (1.1) has a classical solution. Then the couple , where and , verifies the following meanfield BDSDE :
In Eq. (1.2), the integral is a forward Itô integral, and the integral denotes a backward Itô integral. and are two mutually independent standard Brownian motion processes with values respectively in and in . This conclusion gives a probabilistic representation of McKeanVlasov SPDEs (1.1), which can be regarded as a stochastic FeynmanKac formula for MckeanVlasov SPDEs.
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 and be two mutually independent standard Brownian motion processes, with values respectively in and , defined over some complete probability space , where T is a fixed positive number throughout this paper. Moreover, let denote the class of Pnull sets of ℱ. For each , we define
Note that is not an increasing family of σfields, so it is not a filtration.
We will also use the following spaces:
• For any , let denote the set of (classes of a.e. equal) ndimensional jointly measurable random processes which satisfy: Evidently, is a Banach space endowed with the canonical norm .
(ii) is measurable, for a.e. .
• We denote similarly by the set of continuous ndimensional random processes which satisfy:
(ii) is measurable, for a.e. .
• denotes the space of all valued ℱmeasurable random variables.
• For , is the space of all valued ℱmeasurable random variables such that .
Let be the (noncompleted) product of with itself, and we define on this product space. A random variable originally defined on Ω is extended canonically to , . For any , the variable belongs to , a.s., whose expectation is denoted by
Moreover, for all , , are two measurable functions which satisfy
Assumption 2.1 (A1) , and there exist and such that
(A2) for any fixed , is continuous;
(A3) there exist a process and a constant such that
(A4) there exist constants such that for all , , (),
(A5) there exists such that, a.s., for all , , ,
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 such that, a.s., for all , , (),
then it must satisfy conditions (A4) and (A5).
Definition 2.1 A pair of measurable processes is called a solution of meanfield BDSDE (2.1) if and it satisfies meanfield BDSDE (2.1).
The main result of this section is the following theorem.
Theorem 2.1For any random variable, under Assumption 2.1, meanfield BDSDE (2.1) admits a unique solution.
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 , g satisfies
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 which is equivalent to the canonical norm
The parameters and β are specified later.
From Step 1, we can introduce the mapping through the equation
For any , we set , , and . Then applying Itô’s formula to and by virtue of , we have
From condition (A4) and noting that , for any , we get
Then we have
If we set , , , then it yields
Consequently, I is a strict contraction on equipped with the norm for . With the contraction mapping theorem, there admits a unique fixed point such that . On the other hand, from Step 1, we know that if , then , which is the unique solution of Eq. (2.1). □
Suppose that: For some satisfying (A2)(A5), the generators , are of the form
where . Then we have the following corollary.
Corollary 2.1Suppose thatis the solution of meanfield BDSDE (2.1) with data, , whereare two arbitrary terminal values. The difference ofandsatisfies the following estimate:
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 and such that ;
(A4′) for any , there exist constants such that, satisfying (), we have
(A5′) , there exists such that, for any , y, , satisfying (), it holds
Remark 3.1 Since , , (A3′) implies that
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 ofsuch that
(i) for fixed, , ω, t, is continuous;
(iv) ∀m, is globally monotone iny; moreover, for anym, Nwith, it holds that
for anyt, , , , zsatisfying ();
(v) ∀m, is globally Lipschitz in, z; moreover, for anym, Nwith, it holds that
for anyt, , y, , satisfying ().
where is a sequence of smooth functions such that , for , and for . Similarly, we define the sequences , , . It should be pointed out that , , and 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,
whereθis an arbitrarily fixed constant such that. Then meanfield BDSDE (2.1) has a unique solution.
Proof We now construct an approximate sequence. Let be associated to f by Lemma 3.1. Then for each m, is globally monotone in y and globally Lipschitz in z. By Theorem 2.1, the following meanfield BDSDE
admits a unique solution for each . Applying Itô’s formula to yields
where
and
Hence,
Then it follows from Gronwall’s inequality and the BDG inequality that
where only depends on T, α, L and is independent of m.
Next, we will conclude that is a Cauchy sequence in . Actually, since meanfield BDSDE
admits a unique solution for each . Applying Itô’s formula to , we have
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 depends on T, L, α and .
For the second term II, due to the local monotonicity of in y and the local Lipschitz condition of in z, we obtain that for , the following holds:
For the last term, we have
Choose such that . Then from (3.3)(3.6), we obtain
Applying Gronwall’s inequality and the BDG inequality to the above inequality yields
where is independent of m, k. Now passing to the limit successively on m, k and N, we see that is a Cauchy (hence convergent) sequence in ; denote the limit by , which satisfies
Next, we show that is the solution of meanfield BDSDE (2.1). To this end, we only need to prove that the following conclusion holds along a subsequence:
Set
Since
where
then we have
there exists a subsequence of , still denoted by , such that a.e., a.s. It then follows from the continuity of f in y and the dominated convergence theorem that
Now, passing to the limit as and in (3.8) successively, it follows that (3.7) holds. Then letting in (3.2) yields
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. Letandbe the solutions of meanfield BDSDEs (3.9) and (3.10), respectively. Moreover, for the two generators ofand, we suppose:
(i) One of the two generators is independent of.
(ii) One of the two generators is nondecreasing in.
Then if, a.s., , a.s., there also holds that, a.s. .
Remark 3.2 The conditions (i) and (ii) of Theorem 3.2 are, in particular, satisfied if they hold for the same generator (), 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 and (ii) by . For notational simplicity, we set , , then
By Itô’s formula applied to and noting that , it easily follows that
Since a.s. and is nondecreasing in , we have
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
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 and which are supposed to satisfy the following conditions:
Assumption 4.1 (i) and are measurable continuous processes and there exists some constant such that
(ii) b and σ are Lipschitz in x, , i.e., there is some constant such that
For any , we consider the following SDE parameterized by the initial condition :
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 has a continuous version with the following wellknown standard estimates.
Proposition 4.1, there existssuch that, for alland,
Now, let , and be realvalued functions and satisfy the following conditions.
Assumption 4.2 (i) Φ: is an measurable random variable, and are two measurable processes such that , are measurable, for all .
(ii) For all , , , , there exist constants , and such that
(iii) f, g and Φ satisfy a linear growth condition, i.e., there exists some such that, a.s., for all
Next, we investigate the solution of the following BDSDE:
Firstly, we study the case . From Theorem 2.1, we know that there exists a unique solution to the meanfield BDSDE (4.3). Once we have , Eq. (4.3) becomes a classical BDSDE with coefficients
and . Then due to Theorem 2.2 in [12], we obtain that there exists a unique solution to Eq. (4.3).
For BDSDE (4.3), we give the following proposition.
Proposition 4.2For anyand, there exists a constantsuch that
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
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 is the solution of the meanfield SDE (4.1) with .
Now, we give the main theorem of this section.
Theorem 5.1Suppose that Assumption 4.1 and Assumption 4.2 hold. Letbe ameasurable random field such thatsatisfies Eq. (5.1) and for each, a.s. Moreover, we assume thatfor a.s. .
Then we have, whereis the unique solution of the meanfield BDSDEs (4.3) and
Proof It suffices to show that solves the meanfield BDSDE (4.3). To simplify the notation, we define
According to our notations introduced in Section 2, we know that
Let and . For each , applying Itô’s formula to and noticing that u satisfies Eq. (5.1), we get
The condition that and the continuity of f and g are adopted in the last equation.
Then we have
So, , solves the meanfield BDSDE (4.3). The proof is now complete. □
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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