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 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 3.1 Under (A2), (A3′)-(A5′) there exists a sequence of such that
-
(i)
for fixed , , ω, t, is continuous;
-
(ii)
∀m, ;
-
(iii)
∀N, as , where ;
-
(iv)
∀m, is globally monotone in y; moreover, for any m, N with , it holds that
for any t, , , , z satisfying ();
-
(v)
∀m, is globally Lipschitz in , z; moreover, for any m, N with , it holds that
for any t, , y, , satisfying ().
Proof We define by
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 3.1 Let (A1), (A2), (A3′)-(A5′) hold. Assume, moreover,
(3.1)
where θ is an arbitrarily fixed constant such that . Then mean-field 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 mean-field BDSDE
(3.2)
admits a unique solution for each . Applying Itô’s formula to yields
where
and
Hence,
Then it follows from Gronwall’s inequality and the B-D-G inequality that
where only depends on T, α, L and is independent of m.
For any , set
and .
Next, we will conclude that is a Cauchy sequence in . Actually, since mean-field 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
(3.4)
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:
(3.5)
For the last term, we have
(3.6)
Choose such that . Then from (3.3)-(3.6), we obtain
Applying Gronwall’s inequality and the B-D-G 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
as .
Next, we show that 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:
Set
and .
Since
where
then we have
where is independent of m. As
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
as .
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 mean-field BDSDEs. We only consider one-dimensional mean-field BDSDEs, i.e., .
We consider the following mean-field BDSDEs: ()
Theorem 3.2 (Comparison theorem)
Assume mean-field BDSDEs (3.9) and (3.10) satisfy the conditions of Theorem 3.1. Let and be the solutions of mean-field BDSDEs (3.9) and (3.10), respectively. Moreover, for the two generators of and , 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
(3.12)
With the assumption (A1), we obtain
Combining (3.12), (3.13) with (3.11) yields
By Gronwall’s inequality, it follows that
that is, , P-a.s., . □