# Caputo fractional backward stochastic differential equations driven by fractional Brownian motion with delayed generator

## Abstract

Over the years, the research of backward stochastic differential equations (BSDEs) has come a long way. As a extension of the BSDEs, the BSDEs with time delay have played a major role in the stochastic optimal control, financial risk, insurance management, pricing, and hedging. In this paper, we study a class of BSDEs with time-delay generators driven by Caputo fractional derivatives. In contrast to conventional BSDEs, in this class of equations, the generator is also affected by the past values of solutions. Under the Lipschitz condition and some new assumptions, we present a theorem on the existence and uniqueness of solutions.

## 1 Introduction

The fractional derivative introduces a memory effect, enabling the system to exert a long-term influence on past inputs. Simultaneously, the delay is manifested as the system response persists over time, eventually fully reflecting changes in the input after a certain period has elapsed. Both these concepts effectively capture and describe the enduring memory correlation of a system. Because of their advantages in describing the time behavior of a system or process, backward stochastic differential equations (BSDEs) with fractional derivative or delay are widely used in many fields such as optimal control, finance, biology, and physics.

The theory of BSDEs bridges the gap between randomness and determinism and was first proposed by Bismut [2] in 1973 to explain a stochastic version of the process associated with Pontryagin’s maximum principle, but subsequent work has been carried out in a linear context. In 1978, Bismut [3] introduced a class of nonlinear BSDEs (Riccati equations) and proved the existence and uniqueness of their solutions. In 1990, Peng and Pardoux [19] solved the existence and uniqueness problem of the equations under Lipschitz conditions and proposed a general form of BSDEs. In subsequent work, they established a theoretical system of BSDEs, focusing on solving the structure and diffusion part of the generator in the equation. This led to a series of results and studies (Bahlali et al. [1], Zhang et al. [24]), making this theory more perfect. Further, this theory itself has been developed into many different branches (Ma et al. [17], Zhao et al. [26], Pardoux and Răşcanu [20]).

The fractional Brownian motion $$B^{H}= \{B_{t}^{H}, t \ge 0 \}$$ (fBm) with Hurst parameter H ($$0 < H < 1$$), proposed by Kolmogorov [15], is a central Gaussian process. Due to its favorable autocorrelation and long-memory properties, it can serve as a valuable supplement to the study of BSDEs. Therefore Hu and Peng [13] introduced fractional-order Brownian motion and studied general linear and nonlinear BSDEs driven by a fractional-order Brownian motion. Under some mild assumptions, they established the existence and uniqueness of understanding. In the case of nonlinear, BSDE is solved by obtaining an inequality leading to the fixed point theorem, and the existence and uniqueness of the solution is proved when the Hurst parameter $$H\in ( {\frac{1}{2},1} )$$. Maticiuc and Nie [18] then obtained some general results for fractional-order BSDEs through a rigorous approach, extending the discussion to fractional BSVIs. Borkowska [14] studied generalized BSDEs driven by a fractional Brownian motion. Douissi et al. [9] proposed a new mean-field anticipation BSDE driven by a fractional Brownian motion. Wen and Shi [23] studied the existence and uniqueness of expected backward stochastic differential equations driven by a fractional Brownian motion under the Lipschitz condition and a sufficiently small time horizon T. Li et al. [16] derived a European option pricing formula based on the FSDE model with fractional derivative $$\mathrm{d}^{\alpha} f=\Gamma (1+\alpha ) [\mathrm{d} f-f^{ \prime}(x) \,\mathrm{d} x ]$$. Inspired by Li’s method, Chen et al. [4] combined fractional derivatives and fractional differential equations and obtained a BSDE with Caputo fractional derivative of the form

\begin{aligned} Y_{t} =&Y_{T}+ \int _{t}^{T} \biggl( \frac{\alpha \cdot ( \alpha -1 ) \cdot (T-s)^{\alpha -2}}{\Gamma (1+\alpha )} F(s)-f(s) \biggr) \,\mathrm{d} s \\ &{}- \int _{t}^{T} \frac{z(s, T)}{\Gamma (1+\alpha )} \,\mathrm{d} B_{s}, \quad \alpha =2 H \in (1,2) , \end{aligned}

where Γ is the gamma function, f is the generator related to the present time, and B is a standard Brownian motion. Chen et al. [4] constructed a new norm to obtain the existence and uniqueness of the equation.

The mathematical delay method plays an important role in many fields such as stochastic optimal control, financial risk, insurance management, pricing, and hedging. Delong and Imkeller [7, 8] studied the following class of BSDEs with time-delay generators:

$$Y(t)=\xi + \int _{t}^{T} f (s, Y_{s}, Z_{s} )\,ds- \int _{t}^{T} Z(s)\,dW(s), \quad t \in [0, T],$$

where $$(Y_{s}, Z_{s} )=(Y(s+u), Z(s+u))$$, $${-T \leq u \leq 0}$$, and the generated elements can be time dependent on the past values of solutions and weighted by a time-delay function. Examples of BSDEs with time-delay generators, exhibiting either multiple solutions or no solution, are provided, and some properties of the solutions of these BSDEs with time-delay generators are discussed. Subsequently, Wen [22] extended such fractional-order equations. The existence and uniqueness of various BSDE models and their solutions have been proved due to their excellent mathematical properties, broad application prospects, and deep connection with partial differential equations (Zhang et al. [25], Zhuang [27], Delng et al. [6], Peng and Yang [21]).

As an extension of BSDEs, Caputo fractional backward stochastic differential equations with delay driven by a fractional Brownian motion play an important role in stochastic optimal control, financial risk, insurance management, pricing, and hedging. In particular, for pricing problems in finance, by introducing the Caputo fractional derivative and lag or delay term, the market reaction lag effect can be considered in the stock price model, and the change of stock price can be described and predicted more accurately. However, to the best of our knowledge, there have been no studies on such BSDEs. Therefore in this paper, we mainly study the existence and uniqueness of such BSDEs. Specifically, we investigate the following fractional-order BSDEs with a tiem-delay generator:

$$\textstyle\begin{cases} Y_{t} = Y_{T} + \int _{t}^{T} ( \frac{\alpha (\alpha - 1)(T - s)^{\alpha - 2}}{\Gamma(1 + \alpha )} F(s,\eta _{s},Y_{s},Z_{s},Y_{s - \delta _{1}(s)},Z_{s - \delta _{2}(s)}) \\ \hphantom{Y_{t} =}{} - f(s,{\eta _{s}},{Y_{s}},{Z_{s}},{Y_{s - {{{\delta }}_{3}}(s)}},{Z_{s - {{{\delta }}_{4}}(s)}}) )\,ds \\ \hphantom{Y_{t} =}{}- \int _{t}^{T} {\frac{{{Z_{s}}}}{{{\Gamma{(1 + }}\alpha )}}\,dB_{s}^{H},\quad 0 \le t \le T,} \\ {Y_{t}} = {\varphi _{t}},\qquad {Z_{t}} = {\psi _{t}}, \quad - \delta \le t < 0. \end{cases}$$
(1.1)

This paper is organized as follows: In Sect. 2, we introduce some conditions, theorems, and hypotheses to facilitate the subsequent proof. In Sect. 3, we prove the existence and uniqueness of solutions to the equations. The comparison theorem of the equations is obtained in Sect. 4. In Sect. 5, we conclude our investigation.

## 2 Preliminaries

We first review some relevant definitions, lemma, and propositions and make some necessary assumptions. For more specific content, we refer to Decreusefond and Ustunel [5], Duncan et al. [10], and Hu [11].

### 2.1 Fractional Brownian motion

A fractional Brownian motion with Hurst parameter $$H \in (0,1)$$ is a zero-mean Gaussian process $$B^{H} = \{ B_{t}^{H},t \ge 0\}$$ with covariance

$$E\bigl(B_{t}^{H}B_{s}^{H}\bigr) = \frac{1}{2}\bigl({t^{2H}} + {s^{2H}} + { \vert {t - s} \vert ^{2H}}\bigr).$$

When $$H=\frac{1}{2}$$, this process becomes a standard Brownian motion. When $$H\in (0,\frac{1}{2})$$, time series show a negative long-term dependence, that is, there is little correlation between past and future values, and future changes are difficult to predict. Throughout the paper, we only consider $$H\in (\frac{1}{2},1)$$, and the time series shows a positive long-term dependence.

Let $$(\Omega ,{\mathcal{F}},{\mathcal{P}})$$ be a complete probability space with filtration $$\mathcal{F}$$ is generated by an fBm $$B^{H}$$. For two continuous functions ξ and η on $$[ {0,T} ]$$, we define the Hilbert scalar product and norm

$$\langle \xi , \eta \rangle _{t}= \int _{0}^{t} \int _{0}^{t} \phi (u-v) \xi _{u} \eta _{v} \,\mathrm{d} u \,\mathrm{d} v \quad \text{and}\quad \Vert \xi \Vert _{t}= \sqrt{\langle \xi , \xi \rangle _{t}},$$

where $$\phi (x) = 2H(2H - 1){ \vert x \vert ^{2H - 2}}$$ for $$x \in R$$. Under this Hilbert scalar product, we use $${{{\Theta }}_{t}}$$ to represent the completion of a continuous function. Let $${{\mathcal{P}}_{T}}$$ be the set of all fractional Brownian motion polynomials over $$[ {0,T} ]$$. The elements therein are represented as

$$F ( \omega ) = f \biggl( { \int _{0}^{T} {{\xi _{1}} ( t )\,dB_{t}^{H}, \dots , \int _{0}^{T} {{\xi _{n}} ( t )\,dB_{t}^{H}} } } \biggr),$$

where $${{\xi _{i}}}$$ ($$i = 1,2, \dots ,n$$) are continuous functions on $$[ {0,T} ]$$, and f is a polynomial of n variables.

The Malliavin derivative $$D_{s}^{H}$$ of a polynomial functional F is defined by

$$D_{s}^{H}F = \sum_{j = 1}^{n} { \frac{{\partial f}}{{\partial {x_{j}}}}} \biggl( { \int _{0}^{T} {{ \xi _{1}} ( t )\,dB_{t}^{H}, \dots , \int _{0}^{T} {{\xi _{n}} ( t )\,dB_{t}^{H}} } } \biggr){\xi _{j}}(s),\quad 0 \le s \le T.$$

The completion of $${{\mathcal{P}}_{T}}$$ represented by $$\mathbb{D}_{1,2}^{H}$$ has the norm

$$\Vert F \Vert _{H,1,2}^{2} = E{ \Vert F \Vert _{T}}^{2} + E \bigl\Vert {D_{s}^{H}F} \bigr\Vert ^{2}_{T}.$$

We also have the derivative

$$\mathbb{D}_{t}^{H}F = \int _{0}^{T} {\phi (t - u)D_{v}^{H}} F\,dv.$$

### Proposition 2.1

(Hu [11], Proposition 6.25) Denote by $$\mathbb{L} _{H}^{1,2}$$ the space of all continuous processes $$F_{s}:(\Omega , \mathcal{F}, P) \rightarrow \Theta _{t}$$ satisfying $$E [ { \Vert F \Vert _{T}^{2} + \int _{0}^{T} {\int _{0}^{T} {{{ \vert {\mathbb{D}_{s}^{H}{F_{t}}} \vert }^{2}}\,ds\,dt} } } ] \le \infty$$. If $$F_{s} \in \mathbb{L}_{H}^{1,2}$$, then the Itô-type stochastic integral $$\int _{0}^{T} {F(t)\,dB_{t}^{H}}$$ exists in $${L^{2}}(\Omega {{,}}{\mathcal{F}}{{,}}{\mathcal{P}})$$, and

\begin{aligned}& E \biggl( { \int _{0}^{T} {F(t)\,dB_{t}^{H}} } \biggr) = 0,\\& E{ \biggl( { \int _{0}^{T} {F(t)\,dB_{t}^{H}} } \biggr)^{2}} = E \bigl( { \Vert F \Vert _{{\Theta _{t}}}^{2}} \bigr) + E \int _{0}^{T} { \int _{0}^{T} {\mathbb{D}_{s}^{H}{F_{t}} \mathbb{D}_{t}^{H}{F_{s}}\,ds\,dt} } . \end{aligned}

### Proposition 2.2

(Hu [11], Theorem 10.3) For $$i = 1,2$$, let $${g_{i}}(s) \in \mathbb{D}_{H}^{1,2}$$, and l et $${f_{i}}$$ and $${g_{i}}$$ be two real-valued processes satisfying $$E\int _{0}^{T} {({{ \vert {{f_{i}}(s)} \vert }^{2}} + {{ \vert {{g_{i}}(s)} \vert }^{2}})}\,ds < \infty$$. We assume that $$\mathbb{D}_{t}^{H}{f_{i}}(s)$$ and $$\mathbb{D}_{t}^{H}{g_{i}}(s)$$ are continuously differentiable for almost all $$(s,t) \in { [ {0,T} ]^{2}}$$ and $$\omega \in \Omega$$. Suppose that $$E [ {\int _{0}^{T} {\int _{0}^{T} {{{ \vert {\mathbb{D}_{t}^{H}{g_{i}}(s)} \vert }^{2}}\,ds\,dt} } } ] < \infty$$. Denote

$${G_{i}}(t) = \int _{0}^{T} {f_{i}}(s)\,ds + \int _{0}^{T} {{g_{i}}(s)\,dB_{s}^{H}} ,\quad 0 \le t \le T.$$

Then

\begin{aligned} {G_{1}}(t){G_{2}}(t) =& \int _{0}^{t} {G_{1}}(s){f_{2}}(s)\,ds + \int _{0}^{t} {G_{1}}(s){g_{2}}(s)\,dB_{s}^{H} + \int _{0}^{t} {G_{2}}(s){f_{1}}(s)\,ds \\ &{}+ \int _{0}^{t} {{G_{2}}(s){g_{1}}(s)\,dB_{s}^{H}} + \int _{0}^{t} \mathbb{D}_{s}^{H}{G_{1}}(s){g_{2}}(s)\,ds + \int _{0}^{t} \mathbb{D}_{s}^{H}{G_{2}}(s){g_{1}}(s)\,ds. \end{aligned}

### Proposition 2.3

(Hu and Peng [13] and Maticiuc and Nie [18]) Let $$(Y_{t}, Z_{t} ) \in \widetilde{\mathcal{V}}_{[-\delta , T]} \times \widetilde{\mathcal{V}}^{H}_{[-\delta , T]}$$ be the solution to the fractional BSDE (2.1), we have the relation

$$\mathbb{D}_{t}^{H} Y_{t}=\frac{\widehat{\sigma}(t)}{\sigma (t)} Z_{t}, \quad 0 \leq t \leq T .$$

Therefore there exists a constant $$M > 0$$ such that

$$\frac{t^{2 H-1}}{M} Z_{t} \leq \mathbb{D} _{t}^{H} Y_{t}= \frac{\hat{\sigma}_{t}}{\sigma _{t}} Z_{t} \leq M t^{2 H-1} Z_{t}, \quad 0 \leq t \leq T .$$

### 2.2 Caputo fractional derivative

In the studies of real financial markets, it appeared that the stochastic differential equation of fractional Brownian motion can only describe the noise memory but cannot be used to study the trend memory effect of the stock price. Subsequently, fractional derivatives were introduced to describe the trend memory process, which is another effective tool to describe the memory effect, for example, the Caputo fractional derivative [16] with Hurst index $$H\in ( \frac{1}{2} ,1 )$$:

$$d^{\alpha} f=\Gamma (1+\alpha ) \bigl[d f-f^{\prime}(x)\,dx \bigr], \quad \alpha =2 H.$$

Some of results for this Caputo fractional derivative are given in [16, Lemma 5]:

\begin{aligned} (1)\quad \int _{0}^{t} f(x) \bigl(d x^{\alpha} \bigr) & = \alpha \int _{0}^{t}(t-x)^{ \alpha -1} f(x)\,dx, \quad 0< \alpha < 1 , \\ (2)\quad \int _{0}^{t} f(x) (d x)^{\alpha} & = \alpha ( \alpha -1) \int _{0}^{t}(t-x)^{ \alpha -2} F(x)\,dx,\quad 1< \alpha < 2 , \qquad F(t) = \int _{0}^{t} f(x)\,dx . \end{aligned}

Combined with a linear stochastic differential equation, stochastic differential equation with Caputo fractional derivative is derived:

$$\mathrm{d} Y_{t}= \frac{f (t, Y_{t}, Z_{t} )}{\Gamma (1+\alpha )}(\mathrm{d} t)^{ \alpha}+ \frac{Z_{t}}{\Gamma (1+\alpha )} \,\mathrm{d} B_{t}+f (t, Y_{t}, Z_{t} ) \,\mathrm{d} t, \quad 1< \alpha < 2, 0.5< H< 1 .$$

Integrating this equation in the interval $$[t,T ]$$ and then by combining the derivation method of Caputo SDE and BSDE [4], we can get a Caputo fractional BSDE driven by a fractional Brownian motion with time delay:

$$\textstyle\begin{cases} Y_{t}=Y_{T}+\int _{t}^{T} ( \frac{\alpha (\alpha -1)(T-s)^{\alpha -2}}{\Gamma (1+\alpha )} F (s, \eta _{s}, Y_{s}, Z_{s}, Y_{s-\delta _{1}(s)}, Z_{s-\delta _{2}(s)} ) \\ \hphantom{Y_{t}=}{}-f (s, \eta _{s}, Y_{s}, Z_{s}, Y_{s-\delta _{3}(s)}, Z_{s- \delta _{4}(s)} ) )\,ds \\ \hphantom{Y_{t}=}{}-\int _{t}^{T} \frac{Z_{s}}{\Gamma (1+\alpha )}\,dB_{s}^{H},\quad 0 \leq t \leq T , \\ Y_{t}=\varphi _{t},\qquad Z_{t}=\psi _{t}, \quad -\delta \leq t< 0 .\end{cases}$$
(2.1)

### 2.3 Assumptions and conditions

To prove our results, we make the following statement. First, we have

$${\eta _{t}} = {\eta _{0}} + \int _{0}^{t} {{b_{s}}\,ds + \int _{0}^{t} {{ \sigma _{s}}} }\,dB_{s}^{H},\quad 0 \le t \le T,$$
(2.2)

where $${\eta _{0}}$$ is a constant, $${b_{t}}$$ and $${\sigma _{t}}$$ are two bounded deterministic continuously differentiable functions from $$[ {0,T} ]$$ to $$\mathbb{R}$$, and $$\sigma _{s} > 0$$. Then we have

$$\Vert \sigma \Vert _{t}^{2} = \int _{0}^{t} { \int _{0}^{t} { \phi (u - v)} } {\sigma _{u}} {\sigma _{v}}\,du\,dv = H(2H - 1) \int _{0}^{t} { \int _{0}^{t} { \vert u-v \vert ^{2H-2}} } {\sigma _{u}} {\sigma _{v}}\,du\,dv .$$

Denote $$\frac{d}{{dt}}( \Vert \sigma \Vert _{t}^{2}) = 2{{\hat{\sigma}}_{t}}{ \sigma _{t}}$$ and $${{\hat{\sigma}}_{u}} = \int _{0}^{t} {\phi (u - v)} {\sigma _{v}}\,dv$$. In addition, consider the following sets:

Denote the completions of $${{\mathcal{V}}_{ [ {{t_{1}},{t_{2}}} ]}}$$ by $${{\tilde {\mathcal{V}}}_{ [ {{t_{1}},{t_{2}}} ]}}$$ and $${{\tilde {\mathcal{V}}}^{H}}_{ [ {{t_{1}},{t_{2}}} ]}$$ with the corresponding β-norms

$${ \Vert {\varphi } \Vert _{\beta }} = { \biggl( {E \int _{{t_{1}}}^{{t_{2}}} {{e^{\beta t}} {{ \bigl\vert { \varphi (t)} \bigr\vert }^{2}}\,dt} } \biggr)^{ \frac{1}{2}}} , \qquad { \Vert {\mu } \Vert _{\beta }} = { \biggl( {E \int _{{t_{1}}}^{{t_{2}}} {{e^{\beta t}} {t^{2H - 1}} {{ \bigl\vert {\mu (t)} \bigr\vert }^{2}}\,dt} } \biggr)^{\frac{1}{2}}},$$

where $$\beta \ge 0$$ is a constant; apparently, they are Banach spaces. In addition, we need to make the following assumptions.

### Assumption 2.1

$${\delta _{i}}$$, $$i=1,2,3,4$$, are $$R^{+}$$-valued continuous functions on $$[ { 0 ,T} ]$$ such that $$0\le s-\delta _{i}(s) \le s$$, where δ is a constant.

1. (A1)

There exists $$L \ge 0$$ such that for all $$t \in [{0,T} ]$$ and for all nonnegative integrable g,

$$\int _{t}^{T} {g\bigl(s - {\delta _{i}} ( s )\bigr)}\,ds \le L \int _{t - \delta }^{T} {g ( s )}\,ds,\quad i = 1,2,3,4.$$
2. (A2)

The generator $$({F,f} )$$: $$({F,f} ) ({t,x,y,z,y',z' } ) : [{0,T} ] \times \mathbb{R}^{5} \to \mathbb{R}$$ is a $${\mathcal{C}}_{\mathrm{pol}}^{0,1}$$-continuous function, there exist $$C_{1} , C_{2} \ge 0$$ such that $$({F,f} )$$ satisfies the following conditions: for all $$t \in [{0,T} ]$$ and $$i =1 , 2 , 3 , 4$$, we have

\begin{aligned}& \bigl\vert {F ( {t,x,y,z,{y_{t - {\delta _{i}} ( \cdot )}},{z_{t - {\delta _{i}} ( \cdot )}}} ) - F \bigl( {t,x,y',z',{{y'}_{t - {\delta _{i}} ( \cdot )}},{{z'}_{t - {\delta _{i}} ( \cdot )}}} \bigr)} \bigr\vert ^{2} \\& \quad \le C_{1} \bigl( \bigl\vert {y - y'} \bigr\vert ^{2} + t^{2H - 1} \bigl\vert {z - z'} \bigr\vert ^{2} + \bigl\vert y_{t - \delta ( \cdot )} - y'_{t - \delta ( \cdot )} \bigr\vert ^{2} \\& \qquad {} + ( t - \delta )^{2H - 1} \bigl\vert z_{t - \delta ( \cdot )} - z'_{t - \delta ( \cdot )} \bigr\vert ^{2} \bigr) \\& \bigl\vert {f ( {t,x,y,z,{y_{t - {\delta _{i}} ( \cdot )}},{z_{t - {\delta _{i}} ( \cdot )}}} ) - f \bigl( {t,x,y',z',{{y'}_{t - {\delta _{i}} ( \cdot )}},{{z'}_{t - {\delta _{i}} ( \cdot )}}} \bigr)} \bigr\vert ^{2} \\& \quad \le C_{2} \bigl( {{ \bigl\vert {y - y'} \bigr\vert }^{2}} + {t^{2H - 1}} {{ \bigl\vert {z - z'} \bigr\vert }^{2}} + {{ \bigl\vert {{y_{t - \delta ( \cdot )}} - {{y'}_{t - \delta ( \cdot )}}} \bigr\vert }^{2}} \\& \qquad {}+ {{ ( {t - \delta } )}^{2H - 1}} {{ \bigl\vert {{z_{t - \delta ( \cdot )}} - {{z'}_{t - \delta ( \cdot )}}} \bigr\vert }^{2}} \bigr). \end{aligned}

## 3 Existence and uniqueness

We first investigate properties of the solutions. If, under hypothetical conditions, there exists a pair of processes $$( Y_{t},Z_{t} )$$ satisfying equation (2.1), then we can derive that there exists a solution to equation (2.1).

### Theorem 3.1

Let $$({F,f} )$$ satisfy (A1) and (A2). Suppose $$\varphi _{t} \in \mathcal{\widetilde{V} } _{ [ - \delta ,T ] }$$ and $$\psi _{t} \in \mathcal{\widetilde{V} } _{ [ - \delta ,T ] }$$. Then BSDEs (2.1) admit a unique solution $$( {{Y_{t}},{Z_{t}}} )\in \mathcal{\widetilde{V} } _{ [ - \delta ,T ] }\times \mathcal{\widetilde{V} }^{H} _{ [ - \delta ,T ] }$$.

### Proof

For given pair $${ ( {y,z} )} \in \mathcal{\widetilde{V} } _{ [ - \delta ,T ] }\times \mathcal{\widetilde{V} } _{ [ - \delta ,T ] }^{H}$$, we consider the following BSDEs:

$$\textstyle\begin{cases} Y_{t} = Y_{T} + \int _{t}^{T} \frac{\alpha ( {\alpha - 1} ){{ ( {T - s} )}^{\alpha - 2}}}{{\Gamma ( {1 + \alpha } )}} F ( {s,{\eta _{s}},{Y_{s}},{Z_{s}},{Y_{s - {\delta _{1}} ( s )}},{Z_{s - {\delta _{2}} ( s )}}} ) \\ \hphantom{Y_{t} =}{} - f ( {s,{\eta _{s}},{Y_{s}},{Z_{s}},{Y_{s - {\delta _{3}} ( s )}},{Z_{s - {\delta _{4}} ( s )}}} )\,ds \\ \hphantom{Y_{t} =}{}- \int _{t}^{T} \frac{Z_{s}}{\Gamma ( 1+\alpha ) }\,dB_{s}^{H},\quad 0 \le t \le T; \\ {Y_{t}} = \varphi ( t ),\qquad {Z_{t}} = \psi ( t ),\qquad {Y_{T}} = g ( {{X_{T}}} ),\quad - \delta \le t < 0.\end{cases}$$
(3.1)

We define a mapping Λ: $$\mathcal{\widetilde{V} } _{ [ - \delta ,T ] } \times \mathcal{\widetilde{V} } _{ [ - \delta ,T ] }^{H} \to \mathcal{\widetilde{V} } _{ [ - \delta ,T ] }\times \mathcal{\widetilde{V} } _{ [ - \delta ,T ] }^{H}$$ such that $$\Lambda [ ( y_{.},z_{.} ) ] = ( Y_{.},Z_{.} )$$. If we can prove that Λ is a compressed map, then we get the desired result. So, let us complete the proof.

For two arbitrary elements $$(y.,z.)$$ and $$(y^{\prime}., z^{\prime}. )$$ in $$\widetilde{\mathcal{V}}_{[-\delta , T]} \times \widetilde{\mathcal{V}}^{H}_{[-\delta , T]}$$, we have $$\Lambda [(y ., z .)]=(Y ., Z .)$$ and $$\Lambda [({y}' ., {z}' .)]=({Y}' ., {Z}' .)$$. Define

\begin{aligned}& {{\hat{y}}_{.}} \triangleq {y_{.}} - {{y'}_{.}} ,\qquad {{ \hat{z}}_{.}} \triangleq {z_{.}} - {{z'}_{.}}\\& {{\hat{Y}}_{.}} \triangleq {Y_{.}} - {{Y'}_{.}} , \qquad {{ \hat{Z}}_{.}} \triangleq {Z_{.}} - {{Z'}_{.}} \end{aligned}

Then applying Itô s formula to $$e^{\beta t} \vert \hat{Y}_{t} \vert ^{2}$$, by Proposition 2.2 we obtain

\begin{aligned} d\bigl({e^{\beta t}} { \vert {{{\hat{Y}}_{t}}} \vert ^{2}}\bigr) ={}& \beta {e^{ \beta t}} { \vert {{{\hat{Y}}_{t}}} \vert ^{2}}\,dt + 2{e^{\beta t}} \vert {{{\hat{Y}}_{t}}} \vert d \vert {{{\hat{Y}}_{t}}} \vert + {e^{ \beta t}}d{ \vert {{{\hat{Y}}_{t}}} \vert ^{2}} \\ ={}& \beta {e^{\beta t}} { \vert {{{\hat{Y}}_{t}}} \vert ^{2}}\,dt - 2{e^{ \beta t}} \vert {{{\hat{Y}}_{t}}} \vert \biggl\vert { \frac{{\alpha (\alpha - 1){{(T - t)}^{\alpha - 2}}}}{{{{(1 + }}\alpha )}}} \biggr\vert \vert {{{\hat{F}}_{t}}} \vert \,dt + 2{e^{\beta t}} \vert {{{ \hat{Y}}_{t}}} \vert \vert {{{\hat{f}}_{t}}} \vert \\ & {}+ \frac{{2{e^{\beta t}} \vert {{{\hat{Y}}_{t}}} \vert \vert {{{\hat{Z}}_{t}}} \vert }}{{{{(1 + }}\alpha )}}\,dB_{t}^{H} + \frac{{2{e^{\beta t}}{\mathbb{D}_{t}^{H}} \vert {{{\hat{Y}}_{t}}} \vert \vert {{{\hat{Z}}_{t}}} \vert }}{{{\Gamma{(1 + \alpha}} )}}\,dt. \end{aligned}

Integrating over interval $$[t,T ]$$ and converting by deformation, we easily have

\begin{aligned}& e^{\beta t} \vert \hat{Y}_{t} \vert ^{2}+\beta \int _{t}^{T} e^{ \beta s} \vert \hat{Y}_{s} \vert ^{2}\,ds+ \frac{2}{\Gamma (1+\alpha )} \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert \vert \hat{Z}_{s} \vert \,dB_{s}^{H} \\& \qquad {}+ \frac{2}{\Gamma (1+\alpha )} \int _{t}^{T} e^{\beta s} \mathbb{D}_{s}^{H} \vert \hat{Y}_{s} \vert \vert \hat{Z}_{s} \vert \,ds \\& \quad =e^{\beta T} \vert \hat{Y}_{T} \vert ^{2}+2 \int _{t}^{T} e^{\beta s} \biggl\vert \frac{\alpha (\alpha -1)(T-s)^{\alpha -2}}{\Gamma (1+\alpha )} \biggr\vert \vert \hat{Y}_{s} \vert \vert \hat{F}_{s} \vert \,ds-2 \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert \vert \hat{f}_{s} \vert \,ds. \end{aligned}

Here, because of the difficulty of integrating the term of the Malliavin derivative $$\mathbb{D} _{s}^{H} \vert \hat{Y} _{s} \vert$$, we need to transform it. According to the above equation and Proposition 2.3, we have

\begin{aligned}& e^{\beta t} \vert \hat{Y}_{t} \vert ^{2}+\beta \int _{t}^{T} e^{ \beta s} \vert \hat{Y}_{s} \vert ^{2}\,ds+ \frac{2}{\Gamma (1+\alpha )} \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert \vert \hat{Z}_{s} \vert \,dB_{s}^{H} \\& \qquad {}+ \frac{2}{M \Gamma (1+\alpha )} \int _{t}^{T} e^{\beta s} s^{2 H-1} \vert \hat{Z}_{s} \vert ^{2}\,ds \\& \quad \le e^{\beta T} \vert \hat{Y}_{T} \vert ^{2}+2 \int _{t}^{T} e^{\beta s} \biggl\vert \frac{\alpha (\alpha -1)(T-s)^{\alpha -2}}{\Gamma (1+\alpha )} \biggr\vert \vert \hat{Y}_{s} \vert \vert \hat{F}_{s} \vert \,ds-2 \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert \vert \hat{f}_{s} \vert \,ds. \end{aligned}

Taking the expectation on both sides and applying the inequality $$2AB \le A^{2} + B^{2}$$, we have

\begin{aligned}& E \biggl[e^{\beta t} \vert \hat{Y}_{t} \vert ^{2}-e^{\beta T} \vert \hat{Y}_{T} \vert ^{2}+\beta \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert ^{2}\,ds+\frac{2}{\Gamma (1+\alpha )} \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert \vert \hat{Z}_{s} \vert \,dB_{s}^{H} \\& \qquad {}+ \frac{2}{M \Gamma (1+\alpha )} \int _{t}^{T} e^{\beta s} s^{2 H-1} \vert \hat{Z}_{s} \vert ^{2}\,ds \biggr] \\& \quad \leq E \biggl[2 \int _{t}^{T} e^{\beta s} \biggl\vert \frac{\alpha (\alpha -1)(T-s)^{\alpha -2}}{\Gamma (1+\alpha )} \biggr\vert \vert \hat{Y}_{s} \vert \vert \hat{F}_{s} \vert \,ds-2 \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert \vert \hat{f}_{s} \vert \,ds \biggr] \\& \quad =E \biggl[2 \int _{t}^{T} e^{\beta s} \biggl\vert \frac{\alpha (\alpha -1)(T-s)^{\alpha -2}}{\Gamma (1+\alpha )} \biggr\vert \vert \hat{Y}_{s} \vert \bigl\vert F (s, \eta _{s}, y_{s}, z_{s}, y_{s- \delta _{1}(s)}, z_{s-\delta _{2}(s)} ) \\& \qquad {}-F \bigl(s, \eta _{s}, y_{s}^{ \prime}, z_{s}^{\prime}, y_{s-\delta _{1}(s)}^{\prime}, z_{s-\delta _{2}(s)}^{ \prime} \bigr) \bigr\vert \,ds \\& \qquad {} -2 \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert \bigl\vert f (s, \eta _{s}, y_{s}, z_{s}, y_{s-\delta _{3}(s)}, z_{s-\delta _{4}(s)} )-f \bigl(s, \eta _{s}, y_{s}^{\prime}, z_{s}^{\prime}, y_{s- \delta _{3}(s)}^{\prime}, z_{s-\delta _{4}(s)}^{\prime} \bigr) \bigr\vert \,ds \biggr] \\& \quad \leq E \biggl[\frac{\beta}{4} \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert ^{2}\,ds+\frac{4}{\beta} \int _{t}^{T} e^{\beta s} \biggl\vert \frac{\alpha (\alpha -1)(T-s)^{\alpha -2}}{\Gamma (1+\alpha )} \biggr\vert ^{2} \bigl\vert F (s, \eta _{s}, y_{s}, z_{s}, y_{s-\delta _{1}(s)}, z_{s- \delta _{2}(s)} ) \\& \qquad {}- F \bigl(s, \eta _{s}, y_{s}^{\prime}, z^{\prime}, y_{s- \delta _{1}(s)}^{\prime}, z^{\prime}{ }_{s-\delta _{2}(s)} \bigr) \bigr\vert ^{2}+\frac{\beta}{4} \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert ^{2}\,ds \\& \qquad {}+\frac{4}{\beta} \int _{t}^{T} e^{\beta s} \bigl\vert f (s, \eta _{s}, y_{s}, z_{s}, y_{s-\delta _{3}(s)}, z_{s-\delta _{4}(s)} ) \\& \qquad {} - f \bigl(s, \eta _{s}, y_{s}^{\prime}, z_{s}^{\prime}, y_{s- \delta _{3}(s)}^{\prime}, z_{s-\delta _{4}(s)}^{\prime} \bigr) \bigr\vert ^{2}\,ds \biggr] \\& \quad =E \biggl[\frac{\beta}{2} \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert ^{2}\,ds+\frac{4}{\beta} \int _{t}^{T} e^{\beta s} \biggl\vert \frac{\alpha (\alpha -1)(T-s)^{\alpha -2}}{\Gamma (1+\alpha )} \biggr\vert ^{2} \bigl\vert F (s, \eta _{s}, y_{s}, z_{s}, y_{s-\delta _{1}(s)}, z_{s- \delta _{2}(s)} ) \\& \qquad {}- F \bigl(s, \eta _{s}, y_{s}^{\prime}, z_{s}^{\prime}, y_{s- \delta _{1}(s)}^{\prime}, z^{\prime}{ }_{s-\delta _{2}(s)} \bigr) \bigr\vert ^{2}+ \frac{4}{\beta} \int _{t}^{T} e^{\beta s} \bigl\vert f (s, \eta _{s}, y_{s}, z_{s}, y_{s-\delta _{3}(s)}, z_{s-\delta _{4}(s)} ) \\& \qquad {} - f \bigl(s, \eta _{s}, y_{s}^{\prime}, z_{s}^{\prime}, y_{s- \delta _{3}(s)}^{\prime}, z^{\prime}{ }_{s-\delta _{4}(s)} \bigr) \bigr\vert ^{2}\,ds \biggr]. \end{aligned}
(3.2)

Since $$\int _{t}^{T} { \frac{{\alpha (\alpha - 1){{(T - s)}^{\alpha - 2}}}}{{\Gamma{{(1 + }}\alpha )}}}\,ds = \frac{{\alpha {{(T - t)}^{\alpha - 1}}}}{{{\Gamma{{(1 + }}}\alpha )}} - (T - t)$$, it is a continuous bounded function on $$[t,T ]$$. By assumptions (A1) and (A2) and Jensen’s inequality there exists a constant $$k > 0$$ such that

\begin{aligned}& E \biggl[e^{\beta t} \vert \hat{Y}_{t} \vert ^{2}-e^{\beta T} \vert \hat{Y}_{T} \vert ^{2}+\frac{\beta}{2} \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert ^{2}\,ds+\frac{2}{\Gamma (1+\alpha )} \int _{t}^{T} e^{\beta s} \vert \hat{Y}_{s} \vert \vert \hat{Z}_{s} \vert \,dB_{s}^{H} \\& \qquad {}+ \frac{2}{M \Gamma (1+\alpha )} \int _{t}^{T} e^{\beta s} s^{2 H-1} \vert \hat{Z}_{s} \vert ^{2}\,ds \biggr] \\& \quad \leq E \biggl[\frac{4 k}{\beta} \int _{t}^{T} e^{\beta s} \bigl\vert F (s, \eta _{s}, y_{s}, z_{s}, y_{s-\delta _{1}(s)}, z_{s-\delta _{2}(s)} )-F \bigl(s, \eta _{s}, y_{s}^{\prime}, z_{s}^{\prime}, y_{s- \delta _{1}(s)}^{\prime}, z_{s-\delta _{2}(s)}^{\prime} \bigr) \bigr\vert ^{2}\,ds \\& \qquad {} +\frac{4}{\beta} \int _{t}^{T} e^{\beta s} \bigl\vert f (s, \eta _{s}, y_{s}, z_{s}, y_{s-\delta _{3}(s)}, z_{s-\delta _{4}(s)} )-f \bigl(s, \eta _{s}, y_{s}^{\prime}, z_{s}^{\prime}, y_{s- \delta _{3}(s)}^{\prime}, z_{s-\delta _{4}(s)}^{\prime} \bigr) \bigr\vert ^{2}\,ds \biggr] \\& \quad \leq E \biggl[ \biggl(\frac{4 k}{\beta} C_{1}+ \frac{4}{\beta} C_{2} \biggr) \int _{t-\delta}^{T} e^{\beta s} \bigl( \vert \hat{y}_{s} \vert ^{2}+s^{2 H-1} \vert \hat{z}_{s} \vert ^{2}+ \vert \hat{y}_{s- \delta} \vert ^{2}+(s-\delta )^{2 H-1} \vert \hat{z}_{s-\delta} \vert ^{2} \bigr)\,ds \biggr] \\& \quad \le E \biggl[ { \biggl( {\frac{{4k}}{\beta }} } {C_{1}} + { {\frac{4}{\beta }{C_{2}}} \biggr) ( {L + 1} ) \int _{t - \delta }^{T} {{e^{\beta s}}} {{{ \bigl( { \vert {{{\hat{y}}_{s}}} \vert } }^{2}} + {s^{2H - 1}} {{ \vert {{{\hat{z}}_{s}}} \vert }^{2}}} \bigr)\,ds} \biggr]. \end{aligned}
(3.3)

Let $$t=0$$ and multiply both sides by $$\frac{M\Gamma ( 1+\alpha ) }{2}$$; since $$\Gamma ( 1+\alpha ) \in ( {1,2} )$$, we get

\begin{aligned}& E \biggl[ \int _{0}^{T} e^{\beta s} \biggl( \frac{\beta M}{4} \vert \hat{Y}_{s} \vert ^{2}+s^{2 H-1} \vert \hat{Z}_{s} \vert ^{2} \biggr)\,ds \biggr] \\& \quad \leq E \biggl[ \int _{0}^{T} e^{\beta s} \biggl( \frac{\beta M \Gamma (1+\alpha )}{4} \vert \hat{Y}_{s} \vert ^{2}+s^{2 H-1} \vert \hat{Z}_{s} \vert ^{2} \biggr)\,ds \biggr] \\& \quad \leq E \biggl[ \biggl(\frac{4 k}{\beta} C_{1}+ \frac{4}{\beta} C_{2} \biggr) (L+1) M \int _{-\delta}^{T} e^{\beta s} \bigl( \vert \hat{y} \vert ^{2}+s^{2 H-1} \bigl\vert \hat{z}^{2} \bigr\vert \bigr)\,ds \biggr]. \end{aligned}

Denoting $$\beta = ( {4k{C_{1}} + 4{C_{2}}} ) ( {L + 1} )M + \frac{4}{M}$$, we have

\begin{aligned}& {E \biggl[ { \int _{0}^{T} {{e^{\beta s}} \bigl( {{{ \vert {{{ \hat{Y}}_{s}}} \vert }^{2}} + {s^{2H - 1}} {{ \vert {{{\hat{Z}}_{s}}} \vert }^{2}}} \bigr)\,ds} } \biggr]} \\& \quad \le \frac{1}{2}{E} \biggl[ \int _{ - \delta }^{T} {{e^{ \beta s}}} {{{ \bigl( { \vert {{{\hat{y}}_{s}}} \vert } }^{2}} + {s^{2H - 1}} {{ \vert {{{\hat{z}}_{s}}} \vert }^{2}}} \bigr)\,ds \biggr], \end{aligned}

that is,

$${ \Vert {\hat{Y},\hat{Z}} \Vert _{\beta }} \le \frac{1}{{\sqrt {2} }}{ \Vert {\hat{y},\hat{z}} \Vert _{\beta }} .$$

Therefore the map Λ is a strictly compressed mapping, which means that equation (3.1) has a unique solution. □

### Remark

Note that in this paper, we assume that $$H\in ( \frac{1}{2},1 )$$. However, in this paper the generators in fractional-order backward stochastic differential equations only consider present and past time and do not include future time. Next, we make a supplement to the scope of this study and expand the scope of the generator. Consider the following generator for equations that include not only present and past time, but also future time:

$$\textstyle\begin{cases} Y_{t}=\xi _{t}+\int _{t}^{T} \frac{\alpha (\alpha -1)(T-s)^{\alpha -2}}{\Gamma (1+\alpha )} F (s, Y_{s-\delta _{1}(s)}, Z_{s-\delta _{2}(s)}, Y_{s}, Z_{s}, Y_{s+ \delta _{3}(s)}, Z_{s+\delta _{4}(s)} ) \\ \hphantom{Y_{t}=}{}-f (s, Y_{s-\delta _{5}(s)}, Z_{s-\delta _{6}(s)}, Y_{s}, Z_{s}, Y_{s+ \delta _{7}(s)}, Z_{s+\delta _{s}(s)} )\,ds \\ \hphantom{Y_{t}=}{}-\int _{t}^{T} \frac{Z_{t}}{\Gamma (1+\alpha )}\,dB_{t}^{H}, \quad 0 \leq t \leq T ; \\ Y(t)=\xi (t),\qquad Z(t)=\phi (t), \quad T \leq t \leq T+\delta .\end{cases}$$
(3.4)

Here we make the necessary assumptions:

1. (A3)

$$\delta _{i}$$, $$i=1,2,\dots ,8$$, are $$R^{+}$$-valued continuous functions on $$[0,T ]$$ such that $$\delta _{i} ( \cdot ) \le \delta$$, where δ is a normal constant. There exist $${L}' ,{L}'' \ge 0$$ such that for all $$t\in [ 0,T ]$$ and all nonnegative and integrable g,

\begin{aligned}& \int _{t}^{T} g \bigl(s - \delta _{i}(s) \bigr)\,ds \leq L^{\prime} \int _{t-\delta}^{T} g(s)\,ds,\quad i=1,2, \dots , 8 ,\\& \int _{t}^{T} g \bigl(s + \delta _{i}(s) \bigr)\,ds \leq {L}'' \int _{t}^{T+ \delta } g(s)\,ds,\quad i=1,2, \dots , 8 . \end{aligned}

Assume that the generator $$(F, f)(\omega , t, a, b, x, y, c, d) : \Omega \times [0, T] \times L^{2} (\mathcal{F}_{s^{\prime}}, \mathbb{R} ) \times L^{2} (\mathcal{F}_{s}, \mathbb{R} ) \times \mathbb{R}^{2} \times L^{2} (\mathcal{F}_{r^{\prime}}, \mathbb{R} ) \times L^{2} (\mathcal{F}_{r}, \mathbb{R} ) \rightarrow L^{2} (\mathcal{F}_{t}, \mathbb{R} ), s^{\prime}$$, $$s \in [- \delta , T], r^{\prime}$$, $$r \in [0, T+\delta ]$$, satisfies the following conditions:

2. (A4)

There exists a constant $$D>0$$ such that for all $$t\in [ 0,T ]$$,

\begin{aligned}& \bigl\vert (F, f) (t, a, b, x, y, c, d)-(F, f) \bigl(t, a^{\prime}, b^{ \prime}, x, y, c^{\prime}, d^{\prime} \bigr) \bigr\vert \\& \quad \leq D \bigl( \bigl\vert a-a^{\prime} \bigr\vert +t^{H-\frac{1}{2}} \bigl\vert b-b^{ \prime} \bigr\vert + \bigl\vert y-y^{\prime} \bigr\vert +t^{H-\frac{1}{2}} \bigl\vert z-z^{ \prime} \bigr\vert \\& \qquad {}+E^{\mathcal{F}_{t}} \bigl\{ \bigl\vert c-c^{\prime} \bigr\vert +t^{H- \frac{1}{2}} \bigl\vert d-d^{\prime} \bigr\vert \bigr\} \bigr), \end{aligned}
3. (A5)

$$E [\int _{t}^{T}|F(t, 0,0,0,0,0,0)|^{2}\,ds ]<\infty$$, $$E [\int _{t}^{T}|f(t, 0,0,0,0,0,0)|^{2}\,ds ]<\infty$$.

With the help of the above method, we have the existence and uniqueness of the solution to equation (3.4).

## 4 Comparison theorem

In this section, we further study the properties of the solutions to the following Caputo-driven fractional backward stochastic differential equations:

$$\textstyle\begin{cases} -d Y_{t}= ( ( \frac{\alpha (\alpha -1)(T-t)^{\alpha -2}}{\Gamma (1+\alpha )} ) F (t, \eta _{t}, Y_{t}, Z_{t}, Y_{t-\delta _{1}(t)} )-f (t, \eta _{t}, Y_{t}, Z_{t}, Y_{t-\delta _{3}(t)} ) )\,dt \\ \hphantom{-d Y_{t}=}{}-\frac{Z_{t}}{\Gamma (1+\alpha )}\,dB_{t}^{H},\quad 0 \leq t \leq T ; \\ Y_{t}=\varphi (t),\qquad Y_{T}=g (X_{T} ),\quad -\delta \leq t< 0 .\end{cases}$$

By Theorem 3.1 this equation has a unique solution in $$\widetilde{\mathcal{V}}_{[-\delta , T]} \times \widetilde{\mathcal{V}}_{[-\delta , T]}^{H}$$. Next, we give a comparison theorem of the equations studied in this paper based on Hu [12, Theorem 12.3].

### Theorem 4.2

For $$i=1,2$$, let $$g_{i} (X_{T} )$$ and $$\varphi _{i} (t )$$ be continuously differentiable functions of polynomial growth together with their derivatives. Assume that $$F_{i} ( t,x,y,z,y_{\delta } )$$, $$\partial F_{i} ( t,x,y,z,y_{\delta } )$$, $$f_{i} ( t,x,y,z,y_{\delta } )$$, and $$\partial f_{i} ( t,x,y,z,y_{\delta } )$$ are uniformly Lipschitz continuous with respect to y and z and satisfy (A1) and (A2) in the interval $$[ 0,T ] \times \mathbb{R}^{4}$$. Moreover, assume that $$F_{2}$$ is increasing with respect to $$y_{\delta }$$ and $$f_{2}$$ is decreasing about $$y_{\delta}$$. If $$\varphi _{1} ( t ) \le \varphi _{2} ( t )$$ for all $$t \in [ -\delta ,0 ]$$ and

\begin{aligned}& \biggl( \biggl( \frac{\alpha ( \alpha -1 ) ( T-t )^{\alpha -2} }{\Gamma ( 1+\alpha ) } \biggr)F_{1} ( t,\eta _{t},Y_{t},Z_{t},Y_{t-\delta _{1} ( t ) } )-f_{1} ( t,\eta _{t},Y_{t},Z_{t},Y_{t-\delta _{3} ( t ) } ) \biggr) \\& \quad \le \biggl( \biggl( \frac{\alpha ( \alpha -1 ) ( T-t )^{\alpha -2} }{\Gamma ( 1+\alpha ) } \biggr)F_{2} ( t,\eta _{t},Y_{t},Z_{t},Y_{t-\delta _{1} ( t ) } )-f_{2} ( t,\eta _{t},Y_{t},Z_{t},Y_{t-\delta _{3} ( t ) } ) \biggr), \\& y_{t-\delta _{1} ( t )} , y_{t-\delta _{3} ( t )}\in L_{\mathcal{F} }^{2} [ - \delta ,t ], \end{aligned}

then $$Y_{1} ( t ) \le Y_{2} ( t )$$.

### Proof

First, consider the following BSDE with terminal condition:

$$\textstyle\begin{cases} -d Y_{1}(t)= (( \frac { \alpha ( \alpha - 1 ) ( T - t ) ^{ \alpha - 2 } }{ \Gamma ( 1 + \alpha ) } ) F _{ 1 } (t, \eta _{t}, Y_{1}(t), Z_{1}(t), Y_{1} (t- \delta _{1}(t) ) \\ \hphantom{-d Y_{1}(t)=}{}-f_{1} (t, \eta _{t}, Y_{1}(t), Z_{1}(t), Y_{1} (t-\delta _{3}(t) ) ) \\ \hphantom{-d Y_{1}(t)=}{}-\frac{Z_{1}(t)}{\Gamma (1+\alpha )}\,dB_{t}^{H},\quad 0 \leq t \leq T, \\ Y_{1}(t)=\varphi _{1}(t),\quad -\delta \leq t \leq 0 .\end{cases}$$

From Theorem 3.1 we know that this equation has a unique solution $$( Y_{1},Z_{1})\in \mathcal{V}_{ [ -\delta , T ] } \times \mathcal{V} _{ [ 0 , T ] }^{H}$$. Next, consider another equation

$$\textstyle\begin{cases} -d Y_{3}(t)= (( \frac { \alpha ( \alpha - 1 ) ( T - t ) ^{ \alpha - 2 } }{ \Gamma ( 1 + \alpha ) } ) F _{ 2 } (t, \eta _{t}, Y_{3}(t), Z_{3}(t), Y_{1} (t- \delta _{1}(t) ) \\ \hphantom{-d Y_{3}(t)=}{}-f_{2} (t, \eta _{t}, Y_{3}(t), Z_{3}(t), Y_{1} (t-\delta _{3}(t) ) ) \\ \hphantom{-d Y_{3}(t)=}{}-\frac{Z_{3}(t)}{\Gamma (1+\alpha )}\,dB_{t}^{H},\quad 0 \leq t \leq T , \\ {Y_{3}(t)=\varphi _{2}(t)},\quad -\delta \leq t \leq 0 .\end{cases}$$

Obviously, the only solution to this equation is $$( Y_{3} ,Z_{3} )$$. Meanwhile, we have the following assumptions: $$\varphi _{1} ( t ) \le \varphi _{2} ( t )$$, and

\begin{aligned}& \biggl( \biggl( \frac{\alpha ( \alpha -1 ) ( T-t )^{\alpha -2} }{\Gamma ( 1+\alpha ) } \biggr)F_{1} ( t,\eta _{t},Y_{t},Z_{t},Y_{t-\delta _{1} ( t ) } )-f_{1} ( t,\eta _{t},Y_{t},Z_{t},Y_{t-\delta _{3} ( t ) } ) \biggr) \\& \quad \le \biggl( \biggl( \frac{\alpha ( \alpha -1 ) ( T-t )^{\alpha -2} }{\Gamma ( 1+\alpha ) } \biggr)F_{2} ( t,\eta _{t},Y_{t},Z_{t},Y_{t-\delta _{1} ( t ) } )-f_{2} ( t,\eta _{t},Y_{t},Z_{t},Y_{t-\delta _{3} ( t ) } ) \biggr) \\& {y_{t-\delta _{1} ( t )} , y_{t-\delta _{3} ( t )}}\in L_{\mathcal{F} }^{2} [ - \delta ,t ]. \end{aligned}

According to Hu et al. [12, Theorem 12.3], we have

$$Y_{1} ( t ) \le Y_{3} ( t )\quad \text{a.e., a.s.}$$

In the same way, we set up the following equation:

$$\textstyle\begin{cases} -d Y_{4}(t)= (( \frac { \alpha ( \alpha - 1 ) ( T - t ) ^{ \alpha - 2 } }{ \Gamma ( 1 + \alpha ) } ) F _{ 2 } (t, \eta _{t}, Y_{4}(t), Z_{4}(t), Y_{3} (t- \delta _{1}(t) ) \\ \hphantom{-d Y_{4}(t)=}{}-f_{2} (t, \eta _{t}, Y_{4}(t), Z_{4}(t), Y_{3} (t-\delta _{3}(t) ) ) \\ \hphantom{-d Y_{4}(t)=}{}-\frac{Z_{4}(t)}{\Gamma (1+\alpha )}\,dB_{t}^{H},\quad 0 \leq t \leq T ; \\ Y_{4}(t)=\varphi _{2}(t),\quad -\delta \leq t \leq 0 .\end{cases}$$

The unique solution to this equation is represented by $$(Y_{4}, Z_{4} ) \in \tilde{\mathcal{V}}_{[-\delta , T]} \times \mathcal{\tilde{V} }_{[0, T]}^{H}$$. As $$F_{2} ( t,x,y,z,y_{\delta } )$$ is increasing with respect to $$y_{\delta}$$, $$f_{2} ( t,x,y,z,y_{\delta } )$$ is decreasing with respect to $$y_{\delta}$$, and $$Y_{1} ( t ) < Y_{3} ( t )$$, it follows that

\begin{aligned}& \biggl( \biggl( \frac{\alpha (\alpha -1)(T-t)^{\alpha -2}}{\Gamma (1+\alpha )} \biggr) F_{2} \bigl(t, \eta _{t}, Y(t), Z(t), Y_{1} \bigl(t-\delta _{1}(t) \bigr) \bigr) \\& \qquad {}-f_{2} \bigl(t, \eta _{t}, Y(t), Z(t), Y_{1} \bigl(t- \delta _{3}(t) \bigr) \bigr) \biggr) \\& \quad \leq \biggl( \biggl( \frac{\alpha (\alpha -1)(T-t)^{\alpha -2}}{\Gamma (1+\alpha )} \biggr) F_{2} \bigl(t, \eta _{t}, Y(t), Z(t), Y_{3} \bigl(t-\delta _{1}(t) \bigr) \bigr) \\& \qquad {}-f_{2} \bigl(t, \eta _{t}, Y(t), Z(t), Y_{3} \bigl(t- \delta _{3}(t) \bigr) \bigr) \biggr) \\& y_{t-\delta _{1} ( t )} , y_{t-\delta _{3} ( t )} \in L_{\mathcal{F}}^{2}[- \delta , t]. \end{aligned}

Similarly,

$$Y_{3} ( t ) \le Y_{4} ( t )\quad \text{a.e., a.s.}$$

Finally, consider the following BSDEs for $$n = 5 ,6 , \dots$$:

$$\textstyle\begin{cases} -d Y_{n}(t)= (( \frac { \alpha ( \alpha - 1 ) ( T - t ) ^{ \alpha - 2 } }{ \Gamma ( 1 + \alpha ) } ) F _{ 2 } (t, \eta _{t}, Y_{n}(t), Z_{n}(t), Y_{n-1} (t- \delta _{1}(t) ) \\ \hphantom{-d Y_{n}(t)=}{}-f_{2} (t, \eta _{t}, Y_{n}(t), Z_{n}(t), Y_{n-1} (t-\delta _{3}(t) ) ) \\ \hphantom{-d Y_{n}(t)=}{}-\frac{Z_{n}(t)}{\Gamma (1+\alpha )}\,dB_{t}^{H},\quad 0 \leq t \leq T , \\ {Y_{n}(t)=\varphi _{2}(t)},\quad -\delta \leq t \leq 0 .\end{cases}$$

From the above we can conclude that

$$Y_{4}(t) \leq Y_{5}(t) \leq Y_{6}(t) \leq Y_{7}(t) \leq \cdots \leq Y_{n}(t) \leq \cdots \quad \text{a.e., a.s. }$$

Let us scale up the result so that we get a Cauchy sequence $$\{ Y_{n},Z_{n} \}_{n \ge 3 }$$. Denote $$\hat{Y} _{n}(t)=Y_{n}(t)-Y_{n-1}(t)$$ and $$\hat{Z} _{n}(t)=Z_{n}(t)-Z_{n-1}(t)$$. From estimate (3.3), assumptions (A1) and (A2), and Jensen’s inequality we have

\begin{aligned}& E \biggl[ {e^{\beta t}} {{ \bigl\vert {{{\hat{Y}}_{n}} ( t )} \bigr\vert }^{2}} - {e^{\beta T}} {{ \bigl\vert {{{\hat{Y}}_{n}} ( T )} \bigr\vert }^{2}} + \frac{\beta }{2} \int _{t}^{T} {{e^{\beta s}}} {{ \bigl\vert {{{\hat{Y}}_{n}} ( s )} \bigr\vert }^{2}}\,ds \\& \qquad {} + \frac{2}{{{\mathrm{{M\Gamma (1 + }}}\alpha )}} \int _{t}^{T} {{e^{ \beta s}}} {s^{2H - 1}} {{ \bigl\vert {{{\hat{Z}}_{n}} ( s )} \bigr\vert }^{2}}\,ds \biggr] \\& \quad \le E [\frac{4}{\beta} \int _{t}^{T} e^{\beta s} \biggl[ \frac{\alpha (\alpha -1)(T-s)^{\alpha -2}}{\Gamma (1+\alpha )} \biggr]^{2} \big| F \bigl(s, \eta _{s}, \tilde{Y}_{n}(s), \tilde{Z}_{n}(s), \tilde{Y}_{n-1} \bigl(s-\delta _{1}(s) \bigr) \bigr) \\& \qquad {}- { {F\bigl(s,{\eta _{s}},{{\tilde{Y}}_{n - 1}} ( s ),{{ \tilde{Z}}_{n - 1}} ( s ),{{\tilde{Y}}_{n - 2}} \bigl( {s - {{ {\delta }}_{1}}(s)} \bigr)\bigr)} \big|^{2}}\,ds + \frac{4}{\beta } \int _{t}^{T} {{e^{\beta s}}} \big| {f(s,{\eta _{s}},{{\tilde{Y}}_{n}}} ( s ),{{\tilde{Z}}_{n}} ( s ), \\& \qquad {} {{ \tilde{Y}}_{n - 1}}\bigl(s - {{{\delta }}_{1}}(s)\bigr)\big) - f\bigl(s,{\eta _{s}},{{\tilde{Y}}_{n - 1}} ( s ),{{\tilde{Z}}_{n - 1}} ( s ),{{\tilde{Y}}_{n - 2}} { \bigl( {s - {{{\delta }}_{1}}(s)} \bigr){{ \bigr) \big|}^{2}}\,ds} ] \\& \quad \le \frac{4}{\beta } ( {k{C_{1}} + {C_{2}}} )E \biggl[ { \int _{t}^{T} {{e^{\beta s}}} } \bigl( {{{ \bigl\vert {{{\hat{Y}}_{n}} ( s )} \bigr\vert }^{2}} + {s^{2H - 1}} {{ \bigl\vert {{{\hat{Z}}_{n}} ( s )} \bigr\vert }^{2}}} \bigr)\,ds + { \int _{t - \delta }^{T} {{e^{\beta s}}} {{ \bigl\vert {{{\hat{Y}}_{n - 1}} ( s )} \bigr\vert }^{2}}\,ds} \biggr]. \end{aligned}

Then letting $$t=0$$, multiplying both sides by $$\frac{M\Gamma ( 1+\alpha ) }{2}$$, and noting that $$\Gamma ( 1+\alpha ) \in ( {1,2} )$$, we have

\begin{aligned}& E \biggl[ { \int _{0}^{T} {{e^{\beta s}} \biggl( { \frac{{\beta M}}{4}{{ \bigl\vert {{{\hat{Y}}_{n}} ( s )} \bigr\vert }^{2}} + {s^{2H - 1}} {{ \bigl\vert {{{\hat{Z}}_{n}} ( s )} \bigr\vert }^{2}}} \biggr)\,ds} } \biggr] \\& \quad \le \frac{{4M\Gamma ( {1 + \alpha } )}}{\beta } ( {k{C_{1}} + {C_{2}}} ) \\& \qquad {}\times E \biggl[ { \int _{0}^{T} {{e^{\beta s}}} } \bigl( {{{ \bigl\vert {{{\hat{Y}}_{n}} ( s )} \bigr\vert }^{2}} + {s^{2H - 1}} {{ \bigl\vert {{{\hat{Z}}_{n}} ( s )} \bigr\vert }^{2}}} \bigr)\,ds + { \int _{ - \delta }^{T} {{e^{\beta s}}} {{ \bigl\vert {{{\hat{Y}}_{n - 1}} ( s )} \bigr\vert }^{2}}\,ds} \biggr]. \end{aligned}

Denoting $$\beta = 16M\Gamma ( {1 + \alpha } ) ( {k{C_{1}} + {C_{2}}} ) + \frac{4}{M}$$, we have

\begin{aligned}& E \biggl[ { \int _{0}^{T} {{e^{\beta s}} \biggl( { \frac{{\beta M}}{4}{{ \bigl\vert {{{\hat{Y}}_{n}} ( s )} \bigr\vert }^{2}} + {s^{2H - 1}} {{ \bigl\vert {{{\hat{Z}}_{n}} ( s )} \bigr\vert }^{2}}} \biggr)\,ds} } \biggr]\\& \quad \le E { \biggl[ { \int _{0}^{T} {{e^{\beta s}}} } \bigl( {{{ \bigl\vert {{{\hat{Y}}_{n}} ( s )} \bigr\vert }^{2}} + {s^{2H - 1}} {{ \bigl\vert {{{\hat{Z}}_{n}} ( s )} \bigr\vert }^{2}}} \bigr)\,ds} \biggr] + \frac{1}{4}E \biggl[ { \int _{ - \delta }^{T} {{e^{ \beta s}}} {{ \bigl\vert {{{\hat{Y}}_{n - 1}} ( s )} \bigr\vert }^{2}}\,ds} \biggr]. \end{aligned}

Therefore

\begin{aligned}& E \biggl[ { \int _{0}^{T} {{e^{\beta s}} \bigl( {{{ \bigl\vert {{{\hat{Y}}_{n}} ( s )} \bigr\vert }^{2}} + {s^{2H - 1}} {{ \bigl\vert {{{\hat{Z}}_{n}} ( s )} \bigr\vert }^{2}}} \bigr)\,ds} } \biggr] \\& \quad \le \frac{1}{3}E \biggl[ { \int _{ - \delta }^{T} {{e^{\beta s}}} {{ \bigl\vert {{{\hat{Y}}_{n - 1}} ( s )} \bigr\vert }^{2}}\,ds} \biggr] \\& \quad \le \frac{1}{3}E \biggl[ { \int _{ - \delta }^{T} {{e^{\beta s}}} {{ \bigl\vert {{{\hat{Y}}_{n - 1}} ( s )} \bigr\vert }^{2}}\,ds + \int _{0}^{T} {{e^{\beta s}} {s^{2H - 1}} {{ \bigl\vert {{{\hat{Z}}_{n - 1}} ( s )} \bigr\vert }^{2}}} } \biggr]. \end{aligned}

Hence

\begin{aligned}& E \biggl[ { \int _{0}^{T} {{e^{\beta s}} \bigl( {{{ \bigl\vert {{{\hat{Y}}_{n}} ( s )} \bigr\vert }^{2}} + {s^{2H - 1}} {{ \bigl\vert {{{\hat{Z}}_{n}} ( s )} \bigr\vert }^{2}}} \bigr)\,ds} } \biggr] \\& \quad \le { \biggl( {\frac{1}{3}} \biggr)^{n - 4}}E \biggl[ { \int _{ - \delta }^{T} {{e^{\beta s}}} {{ \bigl\vert {{{\hat{Y}}_{4}} ( s )} \bigr\vert }^{2}}\,ds + \int _{0}^{T} {{e^{\beta s}} {s^{2H - 1}} {{ \bigl\vert {{{\hat{Z}}_{4}} ( s )} \bigr\vert }^{2}}} } \biggr]. \end{aligned}

Obviously, $${( {{{\hat{Y}}_{n}}} )_{n \ge 4}}$$ and $$( {{{\hat{Z}}_{n}})_{n \ge 4}}$$ are Cauchy sequences in the Banach spaces $${{\tilde {\mathcal{V}}}_{ [ { - \delta ,T} ]}}$$ and $${{\tilde {\mathcal{V}}}^{H}}_{ [ {0,T} ]}$$, respectively. Their limits are represented by $$( Y_{t}, Z_{t} )$$ for all $$0\le t\le T$$. Therefore $$( Y_{t} ,Z_{t} )$$ is a solution of the following fractional BSDE:

$$\textstyle\begin{cases} -d Y_{t}= ( ( \frac{\alpha (\alpha -1)(T-t)^{\alpha -2}}{\Gamma (1+\alpha )} ) F_{2} (t, \eta _{t}, Y_{t}, Z_{t}, Y_{t-\delta _{1}(t)} )-f_{2} (t, \eta _{t}, Y_{t}, Z_{t}, Y_{t-\delta _{3}(t)} ) )\,dt \\ \hphantom{-d Y_{t}=}{}-\frac{Z_{t}}{\Gamma (1+\alpha )}\,dB_{t}^{H},\quad 0 \leq t \leq T, \\ Y_{t}=\varphi _{2}(t),\quad -\delta \leq t< 0 .\end{cases}$$

At the same time, by Theorem 3.1 on the uniqueness of the solution it is also the solution of the following equation:

$$\textstyle\begin{cases} -d Y_{2}(t)= (( \frac { \alpha ( \alpha - 1 ) ( T - t ) ^{ \alpha - 2 } }{ \Gamma ( 1 + \alpha ) } ) F _{ 2 } (t, \eta _{t}, Y_{2}(t), Z_{2}(t), Y_{2} (t- \delta _{1}(t) ) \\ \hphantom{-d Y_{2}(t)=}{}-f_{2} (t, \eta _{t}, Y_{2}(t), Z_{2}(t), Y_{2} (t-\delta _{3}(t) ) ) \\ \hphantom{-d Y_{2}(t)=}{}-\frac{Z_{2}(t)}{\Gamma (1+\alpha )}\,dB_{t}^{H},\quad 0 \leq t \leq T, \\ Y_{2}(t)=\varphi _{2}(t),\quad -\delta \leq t \leq 0 .\end{cases}$$

Hence

$$Y ( t ) = {Y_{2}} ( t ) \quad \text{a.s.}$$

Since

$$Y_{1}{(t)} \leq Y_{3}{(t)} \leq Y_{4}{(t)} \leq Y(t),$$

we get $$Y_{1}{(t)} \leq Y_{2}{(t)}$$, as desired, □

### Remark

Under assumptions (A3), (A4), and (A5), equation (3.4) is still applicable to the above equation, and the comparison theorem of the equation is also valid.

### Example

To illustrate the results, consider the equation

$$\textstyle\begin{cases} Y_{t}=Y_{T}+\int _{t}^{T} ( \frac{\alpha (\alpha -1)(T-s)^{\alpha -2}}{\Gamma (1+\alpha )} \int _{0}^{s} (Y(u)+Z(u)+Y (u-\delta _{3}(u) )+Z (u-\delta _{4}(u) ) )\,du \\ \hphantom{Y_{t}=}{}- (Y(s)+Z(s)+Y (s-\delta _{3}(s) ) \\ \hphantom{Y_{t}=}{}+Z (s-\delta _{4}(s) )\,ds-\int _{t}^{T} \frac{Z_{s}}{\Gamma (1+\alpha )}\,dB_{s}^{H},\quad 0 \leq t \leq T , \\ Y_{t}=\varphi _{t},\qquad Z_{t}=\psi _{t},\quad -\delta \leq t< 0.\end{cases}$$

where for $$i=1,2,3,4$$, the generator satisfies (A1) and (A2). Suppose $$\varphi _{t} \in \mathcal{\widetilde{V} } _{ [ - \delta ,T ] }$$ and $$\psi _{t} \in \mathcal{\widetilde{V} } _{ [ - \delta ,T ] }$$. Then BSDE (2.1) admits a unique solution $$( {{Y_{t}},{Z_{t}}} )\in \mathcal{\widetilde{V} } _{ [ - \delta ,T ] }\times \mathcal{\widetilde{V} }^{H} _{ [ - \delta ,T ] }$$.

## 5 Conclusions

In this paper, we introduced the fractional derivative and time-delay generator into backward stochastic differential equations. We obtained the existence and uniqueness of a solution and gave an example to illustrate our results. However, the analytical solution to BSDEs is not easy to get and becomes more difficult after the introduction of fractional derivatives and time delays. Our future work will focus on the numerical solution of this equation. At the same time, since fractional derivatives and time delay both provide effective mathematical tools for the study of complex systems and phenomena, it seems interesting to explore their applications in finance.

## Data Availability

No datasets were generated or analysed during the current study.

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## Acknowledgements

The authors would like to thank the editor for handling this manuscript.

## Funding

This research is supported by the National Natural Science Foundation of China (62076039), Natural Science Foundation of Hubei Province (2022CFB023), Education Science Planning Project of Hubei Province (2022GA031), and college student innovation entrepreneurship project from Yangtze University (Yz2022293).

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All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.

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Correspondence to Junjie Du.

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Shao, Y., Du, J., Li, X. et al. Caputo fractional backward stochastic differential equations driven by fractional Brownian motion with delayed generator. Bound Value Probl 2024, 35 (2024). https://doi.org/10.1186/s13661-024-01842-6