Anticipated BSDEs driven by fractional Brownian motion with time-delayed generator

This paper discusses a new type of anticipated backward stochastic differential equation with a time-delayed generator (DABSDEs, for short) driven by fractional Brownian motion, also known as fractional BSDEs, with Hurst parameter $H\in(1/2,1)$, which extends the results of the anticipated backward stochastic differential equation to the case of the drive is fractional Brownian motion instead of a standard Brownian motion and in which the generator considers not only the present and future times but also the past time. By using the fixed point theorem, we will demonstrate the existence and uniqueness of the solutions to these equations. Moreover, we shall establish a comparison theorem for the solutions.


Introduction
Since Pardoux and Peng [1] first proposed a general form of non-linear backward stochastic differential equations (BSDEs) in 1990, the theoretical research of BSDEs has developed rapidly.In our research, we are looking at the case where there exists a pair of adapted processes (Y•, Z•) that satisfy the following type of BSDE where ξ is the terminal value, f is the generator related to the present time, and Bs is a standard Brownian process.After the already mentioned celebrated work of Pardoux and Peng, the interest in BSDEs has increased, mainly due to the connection of these tools with stochastic control and PDEs, a connection that will be stated clearly soon, for example, various BSDEs models and the uniqueness and existence of the solutions to these models (Bahlali et al. [2]; Abdelhadiet al. [3] Zhang et al. [4]), the numerical solution of BSDEs (Ma et al. [5]; Gobet et al. [6]; Zhao et al. [7]), the relationship between BSDEs and partial differential equations (PDEs) (Ren and Xia [8]; Pardoux and Rȃşcanu [9]), and the numerous applications of BSDEs in various areas including optimal control, finance, biology, and physics (for examples, refer to [11,12,10]).
With the further development of the BSDEs theory, an increasing number of models are being studied.Peng and Yang [13]   The two deterministic R + -valued continuous functions δ(s), ζ(s) defined on [0, T ] satisfy (i) t ≤ t + δ(t) ≤ T + K, t ≤ t + ζ(t) ≤ T + K, and (ii) f (s)ds; the authors also demonstrated the existence and uniqueness of the solution to the above equations.Feng [14] investigated the uniqueness and existence of the solution of an anticipated BSDE with a reflecting boundary.Zhang et al. [15] obtained some results of mean-filed anticipated BSDEs with a time-delayed generator.Wang and Cui [16] also proposed a new type of differential equation called anticipated backward doubly stochastic differential equation; the authors solved certain stochastic control problems by utilizing the duality between anticipated BSDEs and stochastic differential delay equations.
On the other hand, Delong and Imkeller [17] addressed BSDEs with time-delayed generators as follows: where f is a generator that depends on the past value of a solution and 0 ≤ u(s) ≤ T , 0 ≤ v(s) ≤ T .As a generalization of Delong and Imkeller [17] or Peng and Yang [13], He et al. [18] investigated a type of delay and anticipated BSDEs.Under partial information, Zhuang [19] studied non-zero and differential games for the anticipated forward-backward stochastic differential delay equation, which can be used to resolve a problem involving the management of time-delayed pension funds with nonlinear expectations.
On the other hand, first introduced by Kolmogorove [20] in 1940, the fractional Brownian motion (fBm, for short) B H t with Hurst parameter H ∈ (1/2, 1) is a centered Gaussian process with good properties such as self-similarity and long-wall correlation, making it reasonable and efficient to use fBm as a random noise term in stochastic models in the fields of communication engineering, finance and economics.It is therefore important to study the existence uniqueness and stability of solutions of BSDEs driven by fBm.
In 2009, Hu and Peng [21] first proposed BSDEs driven by fractional Brownian motion, that is, the fractional BSDE, has the general form of: Henceforth, the amount of study being done on fractional BSDEs is progressively growing; for example, Borkowska [22] studied generalized BSDEs driven by fractional Brownian motion.Douissi et al. [23] showed a new kind of mean-field anticipated BSDE driven by fractional Brownian motion.Besides, Wen and Shi [24] focused on anticipated BSDEs driven by fractional Brownian motion, while Wen [25] discussed fractional BSDEs with delayed generator.However, under the condition of BSDEs driven by fractional Brownian motion, the case where the generator considers not only the current time and the future time but also the past time has not been studied yet.Therefore, our study will focus on studying the BSDEs of this case to enrich the theory of BSDEs.This study might then encourage researchers to investigate stochastic optimal control problems more realistically.
Based on the motivations discussed above, an essential and meaningful question is, if we construct the anticipated BSDEs with a time-delayed generator driven by fBm with Hurst parameter H ∈ (1/2, 1), how can we prove the existence and uniqueness of its solution?In addition, what about the relative comparison theorem?In this work we are interested in the following anticipated backward stochastic differential equation with a time-delayed generator (DABSDEs for short) driven by fBm The rest of the framework for this study is organized as follows.Section 2 introduces some basic information on the new BSDEs model that we are proposing, which is the fractional DABSDEs.In Section 3, by using the fixed point theorem, we demonstrate the existence and uniqueness of the solutions for this type of BSDE.Then, the comparison theorem of the solutions for this kind of model is obtained in Section 4.

Preliminaries
Let us start with some definitions of the the problem at hand, fractional Brownian motion, assumptions, and some basic results of propositions that will be used throughout the paper.The readers are advised to study papers like Decreusefond and Üstünel [26], Duncan et al. [27] and Hu [28], etc. for a more in-depth discussion.

Preliminaries on the Fractional Brownian Motion
Let B H = {B H t , t ≥ 0} be a fractional Brownian motion with Hurst parameter H ∈ (0, 1), which defined on a complete probability space(Ω, F, P ) with filtration F generated by fBm {B H t } t≥0 , its covariance kernel is given by When H = 1 2 , it becomes a standard Brownian motion, when 0 ≤ H ≤ 1 2 , B H t displays negative correlation property while it exhibits a positive correlation and long-range dependence properties when 2 , and to simplify the presentation, we only discuss the one-dimensional case throughout this study.Next, consider the following definitions as Hu [28] shown in 2005.At first, we define where ξ and η are two continuous functions on [0, T ], for x ∈ R, φ(x) = 2H(2H −1)|x| 2H−2 .Then ξ, η t is a Hilbert scalar product.Let Θt be the completion of the continuous functions under this Hilbert norm.Let ξ1, ξ2, . . ., ξ k , . . .be continuous functions on [0, T ], f is a polynomial of n variables.Donate PT is the set of all polynomials of fBm over [0, T ] which contains all elements of the form For F ∈ PT , let D H 1,2 be the completion of PT with respect to the norm .

Assumptions
To simplify the presentation, we only discuss the one-dimensional case in this study, assume (Ω, F, P ) is a complete probability space with natural filtration Ft.Consider the following sets: t ∈ [0, T ] and l times coutinuously differentiable with respect to x ∈ R ; , and all deriratives of ϕ are of polynomial growth ; And then, we let Ṽ[0,T +K] , ṼH [0,T +K] be the completions of V [0,T +K] under the following norm, respectively where β ≥ 0 is a constant, from Lemma 7 [29] we have ṼH In addition, we introduce assumptions about di.Let di(•), i = 1, 2, 3, 4 represent four R + -valued continuous functions defined on [0, T ], and consider the following assumptions: (D1) There exists a constant K ≥ 0, such that for all t ∈ [0, T ], 0 (D2) There exists a constant L ≥ 0, such that for all non-negative and integrable f (•), Next, we present assumptions about the generator f .Assume that f (t, ω, u, v, y, z, φ, ψ)

Comparison Theorem
In this section, we investigate a comparison theorem of fractional DABSDEs of the one-dimensional kind shown below: Firstly, we introduce the classical case of the comparison theorem of fractional BSDEs; Lemma 4.1 refers to Theorem 12.3 of Hu et al. [30].
where bs and σs are bounded deterministic functions and σs > 0. For j = 1, 2, assume ξ j T are continuously differentiable and is of polynomial growth, fj (t, x, y, z) and ∂ ∂y fj (t, x, y, z) are uniformly Lipschitz continuous with respect to y and z.Let(y (1) , z (1) ), (y (2) , z (2) ) be the solutions of the following classical type of fractional BSDE: T .
Next, let (Y • ), (Y • ) be the solutions of the two one-dimensional fractional DABSDEs shown below, respectively, where j = 1, 2. The end outcome is as follows.