General Mean-Field BDSDEs with Stochastic Linear Growth and Discontinuous Generator

: In this paper, we consider the general mean-field backward doubly stochastic differential equations (mean-field BDSDEs) whose generator f can be discontinuous in y . We prove the existence theorem of solutions under stochastic linear growth conditions and also obtain the related comparison theorem. Naturally, we present those results under the linear growth condition, which is a special case of the stochastic condition. Finally, a financial claim sale problem is discussed, which demonstrates the application of the general mean-field BDSDEs in finance.


Introduction
It is well known that backward stochastic differential equations (BSDEs) can be regarded as a class of stochastic differential equations (SDEs) with a given terminal condition (not an initial condition).In 1990, Pardoux and Peng [1] published a famous article and studied nonlinear BSDEs for the first time, In the past 30 years, research on nonlinear BSDEs has developed rapidly.Many scholars have discovered that this theory has important applications in many fields, such as mathematical finance, stochastic control, partial differential equations (PDEs), and so on.Afterward, Pardoux and Peng [2] proposed backward doubly stochastic differential equations (BDSDEs), which contain two random integrals in opposite directions, leading to two opposite information flows, and thus have more complex measurability.Then.Shi, Gu, and Liu [3] proved the comparison theorem of BDSDEs.Recently, Owo [4][5][6] generalized these results under a series of stochastic conditions, including the existence and uniqueness theorem of solution for BDSDEs with stochastic Lipschitz generator, the existence theorem of solutions under stochastic linear growth and continuous or discontinuous conditions, and he also proved the associated comparison theorems.Inspired by this literature, in this paper, we study a new class of BDSDEs called general mean-field BDSDEs to obtain the corresponding results, and the equation's form is as follows: where the coefficients of BDSDEs depend not only on the solution processes but also on the law of the solution processes, which acts as the mean-field term.
Mean-field theory is also a hot research topic that has infiltrated various fields, such as statistical mechanics, physics, economics, finance, and so on.In 2007, Lasry and Lions [7] formally proposed the concept of mean-field games, which studied the problem of stochastic differential games with N particles and the limit behavior of random moving particles when N goes to infinity.Inspired by this idea, Buckdahn, Djehiche, Li, Peng [8] and Buckdahn, Li, Peng [9] used purely random methods to investigate a special class of mean-field problems, and proposed a new type of BSDEs, called mean-field BSDEs.Since then, more and more scholars have devoted their energies to the study of mean-field problems (see [10,11], etc.).Li, Liang and Zhang [12] studied mean-field BSDEs under continuous conditions and proposed a technical lemma by which the existence of solutions was obtained.Wang, Zhao and Shi [13] extended this result to discontinuous conditions.In recent years, Li and Xing [14] combined the results of BDSDEs and mean-field theory, and investigated the existence of a solution for general mean-field BDSDEs with continuous coefficients.Furthermore, Shi, Wang and Zhao [15] obtained the related results of the general mean-field BDSDEs under stochastic linear growth and continuous conditions.
It is worth emphasizing that the theory of mean-field is new, and there are still many conclusions to explore.On the one hand, the ordinary continuous condition or linear growth condition cannot be satisfied in many applications, which the example in Section 4 can reflect: Consider a financial claim with a contingent ξ and there is an investor who has additional information not detected in the financial market and wants to sell the claim.Moreover, suppose that the interest rate is applied only to portfolios whose value remains above a nominal value at any time.This problem is equivalent to solving the following mean-field BDSDE: Since f (t, p, y, z) = θ(t)e − βA(t) 2 E[y] + r(t)yI {y>1} + γ(t)z is not continuous in y, we cannot apply the existence result in [15].Therefore, we relax the restriction on the generator f (t, p, y, z) that f is discontinuous in y, continuous in p and z, and we solve the above problem, shown in Section 4. On the other hand, mean field theory is a useful tool when we study problems related to large numbers of particles.Because when the number of particles N tends to infinity, it is impractical to deal with the behavior of each particle, but through the mean-field term, we only need to pay attention to the limited behavior of randomly moving particles when N tends to infinity.In conclusion, it is meaningful to study the general mean-field BDSDE (3) with discontinuous and stochastic linear growth coefficients, which can solve some problems in physics, finance and so on.
Our paper is organized as follows: In Section 2, we give some preliminary results of general mean-field BDSDEs which are needed in what follows, and we also list some existing results related to our paper.Section 3 is devoted to giving the main results, including the existence theorem of solutions and the related comparison theorem under stochastic linear growth and discontinuous conditions.Then, we naturally introduce the existence theorem of solutions under linear growth conditions, which is a special case of stochastic conditions, and we also propose the associated comparison theorem.In Section 4, we study the application of the general mean-field BDSDEs to the financial claim sales problem.Finally, we conclude in Section 5.

Preliminaries
Now, we begin with introducing some necessary notations and concepts.Let (Ω, F , P) be a complete probability space, that is, all subsets of zero probability sets belong to F , and let T > 0 be an arbitrarily fixed time horizon throughout this paper.Let {W t ; 0 ≤ t ≤ T} and {B t ; 0 ≤ t ≤ T} be two mutually independent standard Brownian Motions with values respectively in R d and R ℓ , defined on (Ω, F , P).Let N denote the class of P-null sets of F , and P 2 (R k ) denotes the set of the probability measures p over (R k , B(R k )) with a finite second moment, that is, R k |x| 2 p(dx) < ∞.Here, B(R k ) denotes the Borel σ-field over R k , and the probability space (Ω, F , P) needs to be rich, so we assume that there is a sub-σ-field F 0 , N ⊂ F 0 ⊂ F , such that (i) Brownian motion (B, W) is independent of F 0 ; (ii) F 0 is 'rich enough', that is, for every p ∈ P 2 (R k ) there is a random variable ξ ∈ L 2 (Ω, F 0 , P; R k ) such that P ξ = p.
Besides, {a(t)} t∈[0,T] is a jointly measurable process with positive values and squareintegrable in [0, T], and we define an increasing process {A(t)} t∈[0,T] by setting A(t) = t 0 a 2 (s)ds.Every β that appears throughout this paper must satisfy β > 0 and be big enough.Here, are the following spaces: . Now, let us consider the following general mean-field BDSDEs: for all t ∈ [0, T], given ξ ∈ L 2,β (Ω, F T , P; R k ), Without loss of generality, in this paper, we consider the case of k = 1.Before discussing the main results of this paper, we will introduce some previous results of general mean-field BDSDEs under some stochastic conditions.Let, coefficients f : jointly measurable and satisfy the following assumptions: There exists a non-negative F W t -measurable process {ν(t)} t∈[0,T] and a constant α, 0 < α < 1, such that for all y 1 , (A3) For all t ∈ [0, T], there exists a positive process {a(t)} t∈[0,T] , which satisfies a(t) 2 = θ(t) 2 + µ(t) + γ(t) 2 + ν(t) ≥ ι > 0 and the definitions of θ(t), µ(t), γ(t) and ν(t) are same as those in assumptions (A1) and (A2).
For the detailed proofs of Lemmas 1-4 , readers can refer to [15].

General Mean-Field BDSDEs with Stochastic Linear Growth and Discontinuous Generator
In this section, we focus on general mean-field BDSDEs (5) with stochastic linear growth and a discontinuous generator.We need to add some assumptions for the generator f as follows: (B1) For a.e.(t, ω) ∈ [0, T], f (t, p, y, z) is left-continuous in y, continuous in p and z, especially, with a continuity modulus ρ : R + → R + for p: for all p 1 , Here, ρ is supposed to be non-decreasing, such that ρ(0 + ) = 0; (B2) There exists a continuous function which is non-decreasing with respect to p, and there exist three non-negative and for all
|z|, we note that ( 7) has at least one solution.For each n, because of Lemma 5, the following general mean-field BDSDEs are as follows: for all t ∈ [0, T], has a unique adapted solution, and the solution {(U n , V n )} ∞ n=1 of Equation ( 8) converge to the minimal solution (U, V) of Equation (7).
Before proving the existence of solutions of (5), we first construct a sequence of general mean-field BDSDEs as follows: By Lemma 3, there exists at least one solution of (11).Here, we only consider the minimal solution, denoted as (Y n , Z n ).By Lemma 1 we know that (10) and ( 12) have unique solutions and are denoted as (Y 0 , Z 0 ) and (Y 0 , Z 0 ), respectively.
Proof.Before the proof of this proposition, we first define thus, we know G(s, 0, 0) = 0.
(i) The conclusion can be proved by the induction method.First, prove that Y 1 t ≥ Y 0 t .From the Equations ( 10), ( 11) and ( 13) we have is the solution of Equation ( 10), so  11) and (13), we can obtain . From (B2), we know ψ n s ≥ 0. Similarly, we can also obtain that Y n+1 t ≥ Y n t , for all t ∈ [0, T], P-a.s.
Since (Y n , Z n ) converges to (Y, Z), we can obtain Y s ≤ Y s , P-a.s., for all s ∈ [0, T], which proves that (Y, Z) is the minimal solution of (5).
Remark 2. Similar to the proof of Theorem 1, we can obtain another existence result that Equation ( 5) has a maximal solution.Replace (B1) with (B1) ′ : (B1) ′ : For a.e.(s, ω) ∈ [0, T], f (s, p, y, z) is right-continuous in y, and continuous in p and z, especially with a continuity modulus ρ : R + → R + for p: for all p 1 , Here ρ is supposed to be non-decreasing and such that ρ(0 + ) = 0.
Consider the Equation ( 12) and the following equation: For all n ≥ 1, there exists at least one solution to the general mean-field BDSDEs (15), and here we give the sequence of maximal solutions denoted by {(Y n , Z n )} ∞ n=1 , which will limit to the maximal solution of Equation (5).

Comparison Theorem
The comparison theorem is also an important result in the theory of general mean-field BDSDEs; therefore, we will prove the comparison theorem to the case where the generator f is discontinuous.

A Special Case: General Mean-Field BDSDEs with Linear Growth and Discontinuous Generator
Next, we will discuss the general mean-field BDSDEs under non-stochastic conditions, which is a special case of that under the above stochastic conditions.Let β = 0, and for all t ∈ [0, T], let processes θ(t), µ(t), γ(t), ϕ(t), ν(t) equal to the constant A, then the results under stochastic conditions will degenerate into some classical results, which are shown in Theorems 3 and 4.
Here, we give an example to show the rationality of those mentioned assumptions.
Proof.The proof of Theorems 3 and 4 is similar to that of Theorems 1 and 2, so it is omitted here.

Application in Finance: Selling a Financial Claim
Considering a financial claim with a contingent ξ and there an investor who wants to sell the claim and hedge it.Suppose that the investor has additional information not detected in the financial market, and his decision is also affected by the distribution of all investors' decisions in the market.Moreover, suppose that the interest rate is applied only to portfolios whose value remains above a nominal value at any time.This problem is equivalent to solving the following mean-field BDSDE: ξ ∈ L where the mean-field term E[y] reflects that the investor relies on the distribution of all investors' decisions in the market to make a decision, µ(t) is the interest rate, γ(t) is the risk premium vector and c(t) is the volatility caused by the systemic risks.

Conclusions
This paper studies a class of general mean-field BDSDEs whose generator f depends not only on the solution processes but also on their distribution.
We present the main result in Section 3, that is, the existence of the solutions for the general mean-field BDSDEs and the comparison theorem under discontinuous and stochastic linear growth conditions.
It is worth emphasizing that the general mean-field BDSDEs with discontinuous generators can help to deal with some financial problems, for example, we discuss a financial claim sale problem in Section 4, which can be solved by a class of general meanfield BDSDE.