Mean-field anticipated BSDEs driven by time-changed Lévy noises

The objective of this work is to show a new kind of mean-field anticipated backward stochastic differential equation (in short MF-ABSDE) driven by time-changed Lévy noises. We give two methods to prove the existence and uniqueness of the solution of those equations by the fixed point theorem and the Picard iterative sequence. Finally, we obtain a comparison theorem for the solutions.


Introduction
With the pioneering work of Pardoux and Peng [13], different properties of backward stochastic differential equations (in short BSDEs) in wider areas have attracted many researchers' great interests. Applying these results to finance, biology and physics, such processes appear in many different applications. Many achievements have been made in the research of BSDEs in a more general framework. We note there are several constructions of BSDEs in the literature. See e.g. El Karou et al. [5], Peng [14], Buckdahn et al. [2] and Douissi et al. [4]. In mathematical terms, following for instance Peng and Yang [15], this amounts to solving a fundamental class of BSDE, called anticipated BSDE; that is, where u(t), v(t) ∈ C[0, T] and they satisfy (A1) ∃K > 0, ∀t ∈ [0, T] such that (A2) ∃L > 0, ∀t ∈ [0, T], g(t) ≥ 0 and g(t) ∈ L[0, T + K] such that Clearly we note that the generator f (·) contains many values (Y t , Z t ) of current time and future time in Eq. (1). And the authors dealt with the fact that the anticipated BSDE has a unique result and established a comparison theorem for the solution. Also, they verified by a duality relationship between anticipated BSDEs and stochastic differential delay equations. Using the duality, they solved some stochastic optimal control problems (see e.g. Xu [20], Yu [21], Giulia and Steffen [7], Klimsiak [9], Wang [19], Wang, Shi and Meng [18], Zhang and Yan [22], among others).
Furthermore, Lu and Ren [12] studied anticipated backward stochastic differential equations on Markov chains. Richter [17] where μ is the structure of the mixture of a conditional Brownian measure and a doubly stochastic Poisson measure as follows: They talked about the classical problem of the solutions in depth and showed the connection between the two kinds of equations. Liu [10] continued the study of these equations.
He gave a direct proof using useful a priori estimates of the solution and included some applications of the classical Feynman-Kac formula.
On the other hand, mean-field limits have played a very important role in different fields of physics and chemistry, but have found in recent work also application in economics and game theory (Buckdahn et al. [2]). Specifically, as far as we are aware of, mean-field limits also have been researched in the context of their applications, such as the optimal control problem, the McKean-Vlasov equation and stochastic games. More and more scholars begin to study stochastic systems with mean-field interaction. Buckdahn et al. [2] investigated mean-field SDEs associated McKean-Vlasov forward SPEs and PDEs. Buckdahn et al. [1] studied the optimal control problem for a kind of general mean-field SDEs. Douissi et al. [4] researched a related stochastic optimal control problem of MF-BSDEs when the noises are fractional Brownian.
In our approach here we are concerned with a kind of MF-ABSDE driven by timechanging Lévy noises, it follows that The main objective of this manuscript is the profound study of the anticipated BSDEs (e.g. Liu (2016), [10]). We study the equation in the sense of mean-field limits. In other words, the aim of this manuscript is to discuss some applications of the mean-field anticipated backward stochastic differential equation driven by time-changing Lévy noises. One of the motivations is to study the solution of MF-ABSDE (3). And we give two methods to prove the existence and uniqueness of this solution. Finally, we explore the comparison theorem.
An outline of the paper is as follows: Sect. 2 is devoted to recalling of concepts and auxiliary results. In Sect. 3, we present the existence and uniqueness of the solution for the MF-ABSDE with two methods. In Sect. 4, we obtain the comparison theorem of the solutions.

The framework
The aim of this section is to present some concepts and notation. For more details, one can see Buckdahn, Li and Peng [2], Di Nunno and Sjursen [3], Douissi, Wen and Shi [4] and the references therein. Suppose . and let (¯ ,F ,P) := ( × , F ⊗ F, P ⊗ P) be the product of ( , F, P) with itself. Suppose T > 0, let Consider the following mixture of measures: Let we now define the noises driving (3), represents the density function of the standard normal distribution function.

Definition 2
We define the singed random measure μ on the Borel subsets of X by Clearly, from (C1)-(C5), we get where α 1 ∩ α 2 = φ. So α 1 and α 2 are orthogonal given F . The random measure B and H are related to a specific form of time change for Brownian motion and a pure Lévy process. To be more specific, for convenience, we define B t : Indeed it is not a natural choice of filtration because it includes some anticipating information, the future values of B and H .

An existence and uniqueness result for MF-ABSDE
The aim of this section is to seek out a pair of processes (Y (t), Z(t)) ∈ S 2 G (0, T + K; R) × L 2 G (0, T + K; R) satisfying the mean-field anticipated BSDE (3). According to Liu and Ren [11], Lemma 4, we can easily draw the following conclusion.
Remark 7 In fact, we almost surely get the solution of the mean-field anticipated BSDE (3) by the Picard iterative sequence.