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A maximum principle for fully coupled controlled forward–backward stochastic difference systems of mean-field type

Abstract

In this paper, we consider the optimal control problem for fully coupled forward–backward stochastic difference equations of mean-field type under weak convexity assumption. By virtue of employing a suitable product rule and formulating a mean-field backward stochastic difference equation, we establish the stochastic maximum principle and also derive, under additional assumptions, that the stochastic maximum principle is also a sufficient condition. As an application, a Stackelberg game of mean-field backward stochastic difference equation is presented to demonstrate our results.

Introduction

Let \(T>0\) be fixed, \((\varOmega ,\mathfrak{F},\{\mathfrak{F}_{t}\}_{0\leq t\leq T}, \mathbb{P})\) be a filtered probability space, on which a martingale process \(W_{t}\) with independent increments is defined, and \(\mathfrak{F}_{t}=\sigma \{W_{l},l=0,1,\ldots ,t-1\}\vee \mathcal{N}_{ \mathbb{P}}\) (the set of all \(\mathbb{P}\)-null subsets). Consider the following discrete-time fully coupled stochastic system:

$$ \textstyle\begin{cases} \Delta X_{t}=b(t,X_{t},Y_{t},Z_{t},\mathbb{E}X_{t},\mathbb{E}Y_{t}, \mathbb{E}Z_{t},u_{t}) \\ \hphantom{\Delta X_{t}={}}{}+\sigma (t,X_{t},Y_{t},Z_{t},\mathbb{E}X_{t}, \mathbb{E}Y_{t},\mathbb{E}Z_{t},u_{t})\Delta W_{t}, \\ -\Delta Y_{t}=f(t+1,X_{t+1},Y_{t+1},Z_{t+1},\mathbb{E}X_{t+1}, \mathbb{E}Y_{t+1},\mathbb{E}Z_{t+1},u_{t+1}) -Z_{t}\Delta W_{t}- \Delta M_{t}, \\ X_{0}=x_{0},\qquad Y_{T}=l(X_{T},\mathbb{E}X_{T}), \end{cases} $$
(1.1)

with the cost functional

$$ J(u)=\mathbb{E} \Biggl\{ \sum_{t=0}^{T-1} g(t,X_{t},Y_{t},Z_{t}, \mathbb{E}X_{t},\mathbb{E}Y_{t},\mathbb{E}Z_{t},u_{t})+h(X_{T}, \mathbb{E}X_{T}) \Biggr\} . $$
(1.2)

Here, we reserve the notation Δ for the backward difference operator \(\Delta X_{t}=X_{t+1}-X_{t}\). W, M are the square integrable martingale processes and M is strongly orthogonal to W. \(\mathbb{E}\) means the expectation operator and f, b, σ, g, h, l are given functions (satisfying some proper conditions to be elaborated later). Then we could present the following stochastic optimal control problem.

Problem \(\mathcal{A}\). Find \(\bar{u}\in \mathscr{U}_{\mathrm{ad}}\) (which shall be defined later) such that

$$ J(\bar{u})=\inf_{u\in \mathscr{U}_{\mathrm{ad}}}J(u). $$

For stochastic optimal control problems (see [1]), one of the main topics is to establish the stochastic maximum principle (SMP). A wide range of concerns have been given to different versions of SMP (see [25]), especially, to forward–backward stochastic control systems (see [69]). It is widely recognized that forward–backward stochastic differential equations (FBSDEs) are extensively studied and there are productive results (see [6, 10, 11]). Nevertheless, discrete-time optimal control problems are more relevant to economic, engineering, biomedical, operation research problems, optimizing complex technological systems, etc. As is known to all, Pontryagin maximum principle in continuous-time framework cannot be extended to discrete-time counterpart, except for some very special cases, due to the nature of admissible control variations. Naturally, it motivated us to formulate discrete analog and even some improper results were deduced. Butkovskii [12] clearly demonstrated some errors in the existing works. The intrinsic reason for the errors is that the significance of convexity has been ignored. Generally speaking, the discrete-time maximum principle fails unless a certain convexity precondition is imposed on the control system. Pshenichnyi [13] elaborated why discrete-time systems require a certain convexity assumption for the effectiveness of the necessary condition while continuous-time systems enjoy it automatically because of the so-called hidden convexity. To the best of our knowledge, the study on the SMP of forward–backward stochastic difference equations (FBSΔEs) is quite rare in the literature. To fill the gap, in this work, we are devoted to considering the SMP of the forward–backward stochastic difference systems.

As for the discrete-time framework, recently, Mahmudov [14] derived the first-order and second-order necessary optimality conditions for discrete-time stochastic optimal control problems by virtue of new discrete-time backward stochastic equation and backward stochastic matrix equation. Lin and Zhang [15] investigated the SMP where the state equation was just on a forward SΔE with the convex control domain. Xu et al. [16] considered the solvability of fully coupled FBSΔEs, in which the BSΔE was given as the conditional expectation form and the coefficients in the backward equation were degenerate. Some representative works in this direction include [1720]. Very recently, Ji and Liu [21] first discussed the SMP for FBSΔEs under the convex control domain, which made substantial progresses in discrete-time forward–backward systems.

In 2009, Buckdahn et al. [22] investigated a special case of backward stochastic differential equations (BSDEs), the so-called mean-field BSDEs, which were derived by a limit of high dimensional FBSDEs, parallel to a large stochastic particle system. From then on, many authors discussed the mean-field system in different frameworks (see [4, 23]).

Motivated by the above discussions, our purpose of this paper is to derive the more general and constructive SMP for mean-field system (1.1)–(1.2) under weaker convexity assumption. From the perspective of the techniques adopted for discrete-time case, the obstacles encountered are twofold. The first issue entails choosing a suitable expression of the product rule

$$ \Delta \langle X_{t},Y_{t}\rangle =\langle X_{t+1},\Delta Y_{t} \rangle +\langle \Delta X_{t},Y_{t}\rangle =\langle X_{t},\Delta Y_{t} \rangle +\langle \Delta X_{t},Y_{t+1} \rangle . $$

In our setting, the Itô formula in continuous-time framework is invalid. In addition, most of the methods applied to discuss continuous-time systems cannot be directly adapted to discrete-time cases. Hence, it is necessary to employ a more characteristic and refined approach for investigating the discrete-time stochastic optimal control problems. The next issue entails formulating the discrete-time counterpart BSΔE as (1.1), which is distinctly different from the continuous-time BSDE. Lately, many authors have been devoted to considering BSΔE (see [2426]). In general, there are two approaches to formulating BSΔE. One is driving by a finite state process (see [25]). In this work, we adopt another formulation as in [24], which is driven by a martingale with independent increments and the generator f in (1.1) relies on time \(t+1\). Based on these arguments, we could obtain the dual principle. It is worth mentioning that our paper differs from [19] in the following aspects. Firstly, our work is based on a weaker convexity assumption. Secondly, our results are obtained in the mean-field framework. Thirdly, we not only establish the SMP, but also derive, under additional assumptions, the SMP, which turns to be a sufficient condition. Finally, as an application, we present a Stackelberg game of mean-field BSΔE to demonstrate our results. To sum up, this is the first paper to discuss the discrete-time forward–backward stochastic optimal control problems of mean-field type under weaker convexity assumption, enabling us to establish the more general and constructive SMP. Our work generalizes and enhances the previously known SMP of [19, 23]. Meanwhile, it extends the classical results of [17, 23] to the mean-field theory as well as forward–backward system. It is interesting to remark that the results of our work also remain for multi-dimensional driving process; in addition, we could also consider a more general system, in which the mean-field terms are allowed to depend on some functional of the law (see [27]). There is no essential difficulty.

The reminder of the paper is organized as follows. The next section states some preliminaries. Section 3 is devoted to considering MF-FBSΔE (1.1). In Sect. 4, we establish the SMP and the sufficient condition for Problem \(\mathcal{A}\). In Sect. 5, a Stackelberg game of mean-field BSΔE is given to illustrate the theoretical results. Section 6 presents some perspectives and open problems.

Preliminaries

Let \(\mathbb{T}=\{0,1,\ldots ,T\}\). For a vector x, \(x'\) stands for its transpose. I represents the unit matrix with appropriate dimension. \(\mathbb{E}_{t}\) means the conditional mathematical expectation \(\mathbb{E}[\cdot |\mathfrak{F}_{t}]\). \(\mathfrak{F}_{0}=\{\emptyset ,\varOmega \}\) and \(\mathfrak{F}=\mathfrak{F}_{T}\). For \(t\in \{0,\ldots ,T-1\}\), \(\mathbb{E}_{t}[\Delta W_{t}]=\mathbb{E}[\Delta W_{t}]=0\), \(\mathbb{E}[\Delta W_{t}\Delta W_{t}']=I\). Now, we shall introduce some spaces to be used frequently in what follows.

$$\begin{aligned}& \mathcal{L}^{2}\bigl(\mathfrak{F}_{t}; \mathbb{R}^{n}\bigr)= \bigl\{ X_{t}: \varOmega \rightarrow \mathbb{R}^{n} \mid X_{t} \mbox{ is } \mathfrak{F}_{t} \mbox{-measurable}, \mathbb{E} \vert X_{t} \vert ^{2}< \infty \bigr\} , \\& \mathcal{S}^{2}\bigl(0,T;\mathbb{R}^{n}\bigr)= \bigl\{ X: \{0,1, \ldots ,T\}\times \varOmega \rightarrow \mathbb{R}^{n} \mid X \mbox{ is } \{ \mathfrak{F}_{t}\}\mbox{-adapted}, \mathbb{E} \vert X_{t} \vert ^{2}< \infty \bigr\} , \\& \mathcal{M}^{2}[0,T]:=\mathcal{S}^{2}\bigl(0,T; \mathbb{R}^{n}\bigr) \times \mathcal{S}^{2}\bigl(0,T; \mathbb{R}^{n}\bigr) \times \mathcal{S}^{2}\bigl(0,T-1; \mathbb{R}^{n}\bigr)\times \mathcal{S}^{2}\bigl(0,T; \mathbb{R}^{n}\bigr), \\& \mathcal{K}^{2}[0,T]:=\mathcal{S}^{2}\bigl(0,T; \mathbb{R}^{n}\bigr) \times \mathcal{S}^{2}\bigl(0,T; \mathbb{R}^{n}\bigr) \times \mathcal{S}^{2}\bigl(0,T-1; \mathbb{R}^{n}\bigr), \\& \mathcal{H}^{2}[0,T]:=\mathcal{S}^{2}\bigl(0,T; \mathbb{R}^{n}\bigr) \times \mathcal{S}^{2}\bigl(0,T-1; \mathbb{R}^{n}\bigr)\times \mathcal{S}^{2}\bigl(0,T; \mathbb{R}^{n}\bigr), \\& \mathcal{N}^{2}[0,T]:=\mathcal{S}^{2}\bigl(0,T-1; \mathbb{R}^{n}\bigr)\times \mathcal{S}^{2}\bigl(0,T-1; \mathbb{R}^{n}\bigr)\times \mathcal{S}^{2}\bigl(1,T; \mathbb{R}^{n}\bigr). \end{aligned}$$

In addition, we introduce the following admissible control set:

$$ \mathscr{U}_{\mathrm{ad}}= \bigl\{ (u_{0},u_{1}, \ldots ,u_{T}):u_{t}\in \mathcal{L}^{2} \bigl(\mathfrak{F}_{t}; \mathbb{R}^{m} \bigr) \mbox{ and } u_{t}\in U_{t}\subseteq \mathbb{R}^{m} \bigr\} . $$

Definition 2.1

([28])

A point \(\bar{y}\in S\subset \mathbb{R}^{m}\) is called a relative interior point of S along the straight line \(l(\bar{y},\tilde{y}):=\{\tilde{y} \mid \tilde{y}=\bar{y}+\kappa ( \tilde{y}-\bar{y}), \tilde{y}\in S\setminus \{\bar{y}\}\neq \emptyset , \kappa \in \mathbb{R}\}\) if there exists \(\gamma =\gamma (\tilde{y})\in (0,1]\) such that \(\bar{y}+\varepsilon (\tilde{y}-\bar{y})\in S\) holds for all \(\varepsilon \in [-\gamma ,\gamma ]\). Besides, ȳ is called a relative interior point of S in a broad sense if ȳ is a relative interior point of S along every straight line in the set \(\{l(\bar{y},y):y\in S\setminus \{\bar{y}\}\}\). The totality of these points is called a relative interior of S in a broad sense and is denoted by riS. S is called relatively open in a broad sense if \(\operatorname{ri}S=S\).

Definition 2.2

([18])

A set \(S\subset \mathbb{R}^{m}\) is called γ-convex relative to a point \(y_{0}\in S\) if, for each point \(y\in S\), there exists \(\gamma =\gamma (y)\in (0,1]\) such that, for all \(\varepsilon \in [0,\gamma ]\), \(y_{0}+\varepsilon (y-y_{0})\in S\) holds. S is γ-convex if S is γ-convex relative to all of its points.

Definition 2.3

A set \(S\subset \mathbb{R}^{m}\) is called ±γ-convex relative to a point \(y_{0}\in S\) if, for each point \(y\in S\), there exists \(\gamma =\gamma (y)\in (0,1]\) such that, for all \(\varepsilon \in [0,\gamma ]\) or for all \(\varepsilon \in [-\gamma ,0]\), \(y_{0}+\varepsilon (y-y_{0})\in S\) holds. S is ±γ-convex if S is ±γ-convex relative to all of its points.

Remark 2.1

It is obvious that a relatively open set in a broad sense, convex and open sets are a γ-convex set. Besides, a γ-convex set is a ±γ-convex set. Nevertheless, the reverse does not always hold. For instance, \(M_{1}=[1,2)\cup (3,4]\) is γ-convex, but it is neither a convex set nor an open set. \(M_{2}=[1,2]\cup [3,4]\) is ±γ-convex, but it is not γ-convex.

We proceed to introducing some notations and basic assumptions which shall be assumed throughout the paper. Denote the usual inner product by \(\langle \cdot ,\cdot \rangle \) and the norm by \(|\cdot |\) of a Euclidean space. For \(\varGamma =(x,y,z,\tilde{x},\tilde{y},\tilde{z})\), define \(F(t,\varGamma ,u)=(-f(t,\varGamma ,u),b(t,\varGamma ,u), \sigma (t,\varGamma ,u))\) and \(\varLambda =b, \sigma , f, g\).

  1. (A1)

    \(f(t,y,z,\tilde{y},\tilde{z})\) is uniformly Lipschitz continuous and independent of z, at \(t=T\), i.e., for any \(y, y_{1}, z, z_{1}\in \mathbb{R}^{n}\), there exists a constant \(c>0\) such that

    $$ \textstyle\begin{cases} \vert f(T,y,z,\tilde{y},\tilde{z})-f(T,y_{1},z_{1},\tilde{y}_{1},\tilde{z}_{1}) \vert \leq c( \vert y-y_{1} \vert + \vert \tilde{y}-\tilde{y}_{1} \vert ),\quad t=T,\mathbb{P} \mbox{-a.s.}, \\ \vert f(t,y,z,\tilde{y},\tilde{z})-f(t,y_{1},z_{1},\tilde{y}_{1}, \tilde{z}_{1}) \vert \\ \quad \leq c( \vert y-y_{1} \vert + \vert \tilde{y}-\tilde{y}_{1} \vert + \vert z-z_{1} \vert + \vert \tilde{z}-\tilde{z}_{1} \vert ),\quad t\in \{1,2,\ldots ,T-1\}, \mathbb{P} \mbox{-a.s.}, \\ f(t,0,0,0,0)\in \mathcal{L}^{2}(\mathfrak{F}_{t};\mathbb{R}^{n}),\quad t \in \{1,2,\ldots ,T\},\mathbb{P}\mbox{-a.s.} \end{cases} $$
  2. (A2)

    b, σ, f, g are uniformly Lipschitz continuous and differentiable on Γ, u; l, h are continuously differentiable on x, , and all the derivatives are uniformly bounded. Moreover, f is independent of z, at \(t=T\).

  3. (A3)

    Γ, \(u\in \mathscr{U}_{\mathrm{ad}}\), \(\varLambda (\cdot ,\varGamma ,u)\) is a \(\mathfrak{F}_{t}\)-adapted process, \(l(x,\tilde{x})\in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\), \(h(x, \tilde{x})\in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R})\), and \(F(t,0,0)\in \mathcal{L}^{2}(\mathfrak{F}_{t};\mathbb{R}^{n} \times \mathbb{R}^{n}\times \mathbb{R}^{n})\), \(g(t,0,0) \in \mathcal{L}^{2}(\mathfrak{F}_{t};\mathbb{R})\).

  4. (A4)

    (Monotonic conditions)

    For \(t\in \{1,\ldots ,T-1\}\),

    $$\begin{aligned}& \mathbb{E}\bigl\langle F(t,\varGamma ,u)-F(t,\varGamma _{1},u),\varTheta - \varTheta _{1} \bigr\rangle \leq -\beta \mathbb{E} \vert \varTheta - \varTheta _{1} \vert ^{2},\quad \mathbb{P}\mbox{-a.s.}, \\& \quad \forall \varGamma =(x,y,z,\tilde{x},\tilde{y},\tilde{z}), \varGamma _{1}=(x_{1},y_{1},z_{1}, \tilde{x}_{1},\tilde{y}_{1},\tilde{z}_{1}), \varTheta =(x,y,z), \varTheta _{1}=(x_{1},y_{1},z_{1}). \end{aligned}$$

    For \(t=T\),

    $$ \mathbb{E}\bigl\langle -f(T,x,y,\tilde{x},\tilde{y},u)+f(T,x_{1},y_{1}, \tilde{x}_{1},\tilde{y}_{1},u), x-x_{1}\bigr\rangle \leq -\beta \mathbb{E} \vert x-x_{1} \vert ^{2},\quad \mathbb{P}\mbox{-a.s.} $$

    For \(t=0\),

    $$\begin{aligned} & \mathbb{E}\bigl\langle b(0,\varGamma ,u)-b(0,\varGamma _{1},u), y-y_{1} \bigr\rangle +\mathbb{E}\bigl\langle \sigma (0,\varGamma ,u)- \sigma (0,\varGamma _{1},u), z-z_{1} \bigr\rangle \\ &\quad \leq -\beta \mathbb{E} \bigl( \vert y-y_{1} \vert ^{2}+ \vert z-z_{1} \vert ^{2} \bigr),\quad \mathbb{P}\mbox{-a.s.} \end{aligned}$$

    Besides,

    $$ \mathbb{E}\bigl\langle l(x,\tilde{x})-l(x_{1},\tilde{x}_{1}),x-x_{1} \bigr\rangle \geq c\mathbb{E} \vert x-x_{1} \vert ^{2},\mathbb{P}\mbox{-a.s.}, $$

    where c, β are nonnegative constants.

  5. (A5)

    The set \(U_{t}\)\((t\in \mathbb{T})\) is ±γ-convex.

    Throughout the paper, we formally denote \(f(0,\varGamma ,u)=b(T,\varGamma ,u)=\sigma (T,\varGamma ,u)=g(T,\varGamma ,u) \equiv 0\). Let \(\pi \in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\), we consider the following MF-BSΔE:

    $$ \textstyle\begin{cases} \Delta Y_{t}=-f(t+1,Y_{t+1},Z_{t+1},\mathbb{E}Y_{t+1},\mathbb{E}Z_{t+1})+Z_{t} \Delta W_{t}+\Delta M_{t}, \\ Y_{T}=\pi . \end{cases} $$
    (2.1)

Definition 2.4

The triple of processes \((Y,Z,M)\in \mathcal{H}^{2}[0,T]\) is called a solution of MF-BSΔE (2.1) if it satisfies (2.1) for any \(t\in \{0,1,\ldots ,T-1\}\) and M is a martingale process strongly orthogonal to W.

Theorem 2.1

Assume that (A1) holds, then for any\(\pi \in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\), MF-BSΔE (2.1) admits a unique adapted solution\((Y,Z,M)\).

Proof

Firstly, we shall show the existence by using the backward induction method. From (A1) and \(\pi \in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\), we have \(f(T,\pi ,\mathbb{E}\pi )\in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\). Then \(\mathbb{E}\{|\mathbb{E}_{T-1}[\pi +f(T,\pi ,\mathbb{E}\pi )]|^{2}\}< \infty \). Hence, \(\pi +f(T,\pi ,\mathbb{E}\pi )-\mathbb{E}_{T-1}[\pi +f(T,\pi , \mathbb{E}\pi )]\) is a square integrable martingale difference. Further, by the Galtchouk–Kunita–Watanabe decomposition in [29], there are \(Z_{T-1}\in \mathfrak{F}_{T-1}\), \(Z_{T-1}\Delta W_{T-1}\in \mathcal{L}^{2}(\mathfrak{F}_{T-1}; \mathbb{R}^{n})\), and \(\Delta M_{T-1}\in \mathcal{L}^{2}(\mathfrak{F}_{T-1};\mathbb{R}^{n})\) such that \(\mathbb{E}_{T-1}[\Delta M_{T-1}]=\mathbb{E}_{T-1}[\Delta M_{T-1} \Delta W_{T-1}']=0\) and

$$ \pi +f(T,\pi ,\mathbb{E}\pi )-\mathbb{E}_{T-1}\bigl[\pi +f(T,\pi , \mathbb{E}\pi )\bigr]=Z_{T-1}\Delta W_{T-1}+\Delta M_{T-1}. $$
(2.2)

Here, \(\Delta M_{T-1}\) is uniquely determined in that decomposition. Multiplying (2.2) by \(\Delta W_{T-1}'\) and then applying \(\mathbb{E}_{T-1}\) to both sides, we derive

$$ \mathbb{E}_{T-1}\bigl[\bigl(\pi +f(T,\pi ,\mathbb{E}\pi )\bigr)\Delta W_{T-1}'\bigr]=Z_{T-1}. $$

We further obtain

$$ \mathbb{E} \vert Z_{T-1} \vert ^{2}\leq \mathbb{E} \bigl\{ \mathbb{E}_{T-1}\bigl[ \bigl\vert \pi +f(T, \pi ,\mathbb{E}\pi ) \bigr\vert ^{2}\bigr]\mathbb{E}_{T-1}\bigl[\Delta W_{T-1}^{2}\bigr] \bigr\} < \infty . $$

This implies \(Y_{T-1}=\mathbb{E}_{T-1}[\pi +f(T,\pi ,\mathbb{E}\pi )]\in \mathcal{L}^{2}(\mathfrak{F}_{T-1};\mathbb{R}^{n})\). Thus, we determine \(Y_{T-1}\), \(Z_{T-1}\), \(\Delta M_{T-1}\).

We continue this backward procedure. Assume that \(Y_{t+1}\in \mathcal{L}^{2}(\mathfrak{F}_{t+1};\mathbb{R}^{n})\), \(t\in \{0,1,\ldots ,T-2\}\). Similarly to the above discussions, we know \(Z_{t}\in \mathfrak{F}_{t}\), \(Z_{t}\Delta W_{t}\in \mathcal{L}^{2}(\mathfrak{F}_{t};\mathbb{R}^{n})\), \(\Delta M_{t}\in \mathcal{L}^{2}(\mathfrak{F}_{t};\mathbb{R}^{n})\) such that \(\mathbb{E}_{t}[\Delta M_{t}]=\mathbb{E}_{t}[\Delta M_{t}\Delta W_{t}']=0\) and

$$ \textstyle\begin{cases} Y_{t+1}+f(t+1,Y_{t+1},\mathbb{E}Y_{t+1},Z_{t+1},\mathbb{E}Z_{t+1}) -\mathbb{E}_{t}[Y_{t+1}+f(t+1,Y_{t+1},\mathbb{E}Y_{t+1}, Z_{t+1}, \mathbb{E}Z_{t+1})] \\ \quad =Z_{t}\Delta W_{t}+\Delta M_{t}, \\ Z_{t}=\mathbb{E}_{t}[(Y_{t+1}+f(t+1,Y_{t+1},\mathbb{E}Y_{t+1},Z_{t+1}, \mathbb{E}Z_{t+1}))\Delta W_{t}'], \\ Y_{t}=\mathbb{E}_{t}[Y_{t+1}+f(t+1,Y_{t+1},\mathbb{E}Y_{t+1},Z_{t+1}, \mathbb{E}Z_{t+1})]. \end{cases} $$

In summary, we deduce \((Y_{t},Z_{t},\Delta M_{t})\in \mathcal{L}^{2}(\mathfrak{F}_{t}; \mathbb{R}^{n})\times \mathcal{L}^{2}(\mathfrak{F}_{t};\mathbb{R}^{n}) \times \mathcal{L}^{2}(\mathfrak{F}_{t};\mathbb{R}^{n})\), \(0\leq t\leq T-2\). Without loss of generality, let \(M_{0}=0\) and \(M_{t}=M_{0}+\sum_{s=0}^{t-1}\Delta M_{s}\), we see that (2.1) holds for \(t\in \{0,1,\ldots ,T-1\}\). In addition, M is a square integrable martingale process. Furthermore, since

$$ \mathbb{E}_{t-1}\bigl[M_{t}W_{t}' \bigr]=\sum_{s=0}^{t-2}\Delta M_{s} \mathbb{E}_{t-1}\bigl[W_{t}' \bigr] +\mathbb{E}_{t-1}\bigl[\Delta M_{t-1}(W_{t-1}+ \Delta W_{t-1})'\bigr]=M_{t-1}W_{t-1}', $$

we get that M is strongly orthogonal to W. The existence is finished.

Next, we shall prove the uniqueness. Assume that there are two solutions \((Y_{t}^{1},Z_{t}^{1},\tilde{Y}_{t}^{1}, \tilde{Z}_{t}^{1},M_{t}^{1} )\) and \((Y_{t}^{2},Z_{t}^{2},\tilde{Y}_{t}^{2},\tilde{Z}_{t}^{2},M_{t}^{2} )\) of MF-BSΔE (2.1). Then

$$\begin{aligned} Y_{T-1}^{1}-Y_{T-1}^{2}&=-f \bigl(T,Y_{T},Z_{T}^{1}, \tilde{Y}_{T}, \tilde{Z}_{T}^{1} \bigr) +f \bigl(T,Y_{T},Z_{T}^{2}, \tilde{Y}_{T}, \tilde{Z}_{T}^{2} \bigr) \\ &\quad {}+Z_{T-1}^{1}\Delta W_{T-1} +\Delta M_{T-1}^{1}-Z_{T-1}^{2} \Delta W_{T-1}-\Delta M_{T-1}^{2}. \end{aligned}$$
(2.3)

Combining \(Z_{T-1}^{1}=\mathbb{E}_{T-1}[(\pi +f(T,\pi ,\mathbb{E}\pi ))\Delta W_{T-1}']=Z_{T-1}^{2}\) with (2.2), we have \(\Delta M_{T-1}^{1}=\Delta M_{T-1}^{2}\). Thus, using (2.3), we can immediately get that \(Y_{T-1}^{1}=Y_{T-1}^{2}\). The inductive method and \(Z_{0}^{1}=Z_{0}^{2}=M_{0}^{1}=M_{0}^{2}=0\) yield \((Y_{t}^{1},Z_{t}^{1},M_{t}^{1} )= (Y_{t}^{2},Z_{t}^{2},M_{t}^{2} )\) for \(t\in \{0,1,\ldots ,T-1\}\). □

Controlled MF-FBSΔEs

In this section, we focus on the fully coupled MF-FBSΔE (1.1). Let \(\bar{u}_{t}\) and \((\bar{X}_{t},\bar{Y}_{t},\bar{Z}_{t} )\) be the optimal control and optimal trajectory of Problem \(\mathcal{A}\), respectively. Assume that (A2)–(A5) hold, firstly we define a multi-valued mapping:

$$ \mathcal{I}(t,v_{t})= \textstyle\begin{cases} 1,&(t,v_{t})\in \mathbb{T}\times U^{+}_{t}, \\ -1,&(t,v_{t})\in \mathbb{T}\times U^{-}_{t}, \\ \pm 1,&(t,v_{t})\in \mathbb{T}\times U^{+}_{t}\cap U^{-}_{t}, \end{cases} $$
(3.1)

where \(U^{+}_{t}\), \(U^{-}_{t}\) represent that the set \(U_{t}\) is γ-convex and −γ-convex, respectively. Notice that the set \(U_{t}\) is ±γ-convex, there exists \(\gamma >0\) such that, for all \(\varepsilon \in (0,\gamma ]\), we could find another admissible control

$$ u^{\varepsilon }_{t}=\bar{u}_{t}+\alpha (t) \varepsilon (v_{t}-\bar{u}_{t}),\quad v_{t} \in U_{t}, $$

where \(\alpha (t)\in \mathcal{I}(t,v_{t})\). We construct a needle variation

$$\begin{aligned} \alpha (t)\varepsilon (v_{t}-\bar{u}_{t})= \textstyle\begin{cases} \alpha \varepsilon (v_{\theta }-\bar{u}_{\theta }),&t= \theta , \\ 0,&t\in \mathbb{T}\setminus \{\theta \}, \end{cases}\displaystyle \end{aligned}$$

where \(\alpha \in \mathcal{I}(\theta ,v_{\theta })\), \((\theta ,v_{\theta }) \in \mathbb{T}\times U_{\theta }\), and \(\varepsilon \in (0,\gamma ^{\ast }]\) with \(\gamma ^{\ast }=\frac{\gamma (v_{\theta })}{1+\alpha }\). Denote that \((X^{\varepsilon }_{t},Y^{\varepsilon }_{t},Z^{\varepsilon }_{t})\) is the state trajectory corresponding to the admissible control \(u^{\varepsilon }_{t}\). Now, we give the following existence and uniqueness theorem.

Theorem 3.1

Assume that (A2)(A4) hold, then there exists a unique adapted solution\((X,Y,Z,M)\in \mathcal{M}^{2}[0,T]\)for mean-field system (1.1).

We shall apply the following two technical lemmas to prove the existence part of Theorem 3.1, and the proof of these lemmas shall be presented in the sequel.

Lemma 3.1

Suppose\((r,\phi ,\varphi )\in \mathcal{N}^{2}[0,T]\), \(\lambda \in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\), then the following linear MF-FBSΔE

$$ \textstyle\begin{cases} \Delta X_{t}=-Y_{t}-\mathbb{E}Y_{t}+r_{t} +(-Z_{t}-\mathbb{E}Z_{t}+ \phi _{t})\Delta W_{t}, \\ \Delta Y_{t}=-X_{t+1}-\mathbb{E}X_{t+1}-\varphi _{t+1}+Z_{t} \Delta W_{t}+\Delta M_{t}, \\ X_{0}=x_{0},\qquad Y_{T}=X_{T}+\mathbb{E}X_{T}+\lambda , \end{cases} $$
(3.2)

has a unique solution\((X,Y,Z,M)\in \mathcal{M}^{2}[0,T]\).

Now, we define a family of MF-FBSΔEs parameterized by\(\mu \in [0,1]\)as follows:

$$ \textstyle\begin{cases} \Delta X_{t}=b^{\mu }(t,\varGamma _{t},u_{t})+r_{t}+[\sigma ^{\mu }(t, \varGamma _{t},u_{t})+\phi _{t}]\Delta W_{t}, \\ \Delta Y_{t}=-f^{\mu }(t+1,\varGamma _{t+1},u_{t+1})-\varphi _{t+1}+Z_{t} \Delta W_{t}+\Delta M_{t}, \\ X_{0}=x_{0},\qquad Y_{T}=l^{\mu }(X_{T},\mathbb{E}X_{T})+\lambda , \end{cases} $$
(3.3)

where

$$ \textstyle\begin{cases} b^{\mu }(t,\varGamma _{t},u_{t})=\mu b(t,\varGamma _{t},u_{t})+(1-\mu )(-Y_{t}- \mathbb{E}Y_{t}), \\ \sigma ^{\mu }(t,\varGamma _{t},u_{t})=\mu \sigma (t,\varGamma _{t},u_{t})+(1- \mu )(-Z_{t}-\mathbb{E}Z_{t}), \\ f^{\mu }(t,\varGamma _{t},u_{t})=\mu f(t,\varGamma _{t},u_{t})+( \mu -1)(-X_{t}-\mathbb{E}X_{t}), \\ l^{\mu }(X_{t},\mathbb{E}X_{t})=\mu l(X_{t},\mathbb{E}X_{t})+( \mu -1)(-X_{t}-\mathbb{E}X_{t}), \\ \varGamma _{t}=(X_{t},Y_{t},Z_{t},\mathbb{E}X_{t},\mathbb{E}Y_{t}, \mathbb{E}Z_{t}). \end{cases} $$

Lemma 3.2

For a given\(\mu _{0}\in [0,1)\)and any\((r,\phi ,\varphi )\in \mathcal{N}^{2}[0,T]\), \(\lambda \in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\), MF-FBSΔEs (3.3) have a unique solution. Then there exists\(\delta _{0}\in (0,1)\)such that, for any\(\mu \in [\mu _{0},\mu _{0}+\delta _{0}]\)and\((r,\phi ,\varphi )\in \mathcal{N}^{2}[0,T]\), \(\lambda \in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\), MF-FBSΔEs (3.3) have a unique solution.

Proof of Theorem 3.1

Uniqueness. Suppose that \((X,Y,Z,M)\) and \((\check{X},\check{Y},\check{Z},\check{M} )\) are two solutions of (1.1), we denote

$$ \textstyle\begin{cases} \hat{\varGamma }= (\hat{X},\hat{Y},\hat{Z},\mathbb{E}\hat{X}, \mathbb{E}\hat{Y},\mathbb{E}\hat{Z} ) =\varGamma -\check{\varGamma } \\ \hphantom{\hat{\varGamma }}= (X-\check{X},Y-\check{Y},Z-\check{Z},\mathbb{E}X-\mathbb{E} \check{X},\mathbb{E}Y-\mathbb{E}\check{Y}, \mathbb{E}Z-\mathbb{E} \check{Z} ), \\ \hat{b}(t)=b (t,\varGamma _{t},u_{t} )-b (t, \check{\varGamma }_{t},u_{t} ), \qquad \hat{\sigma }(t)=\sigma (t, \varGamma _{t},u_{t} )-\sigma (t,\check{\varGamma }_{t},u_{t} ), \\ \hat{f}(t)=f (t,\varGamma _{t},u_{t} )-f (t, \check{\varGamma }_{t},u_{t} ), \\ \hat{\varTheta }= (\hat{X},\hat{Y},\hat{Z} )=\varTheta - \check{\varTheta } = (X-\check{X},Y-\check{Y},Z-\check{Z} ),\qquad \hat{M}=M-\check{M}. \end{cases} $$

For \(t\in \{0,1,\ldots ,T-1\}\), it yields that

$$\begin{aligned} \Delta \langle \hat{X}_{t},\hat{Y}_{t} \rangle &= \langle \hat{X}_{t+1},\Delta \hat{Y}_{t} \rangle + \langle \Delta \hat{X}_{t},\hat{Y}_{t} \rangle \\ &= \bigl\langle \hat{b}(t),\hat{Y}_{t} \bigr\rangle - \bigl\langle \hat{X}_{t+1}, \hat{f}(t+1) \bigr\rangle + \bigl\langle \hat{\sigma }(t)\Delta W_{t}, \hat{Z}_{t}\Delta W_{t} \bigr\rangle +\varPsi _{t}, \end{aligned}$$

where

$$ \varPsi _{t}= \bigl\langle \hat{\sigma }(t)\Delta W_{t}, \hat{Y}_{t} \bigr\rangle + \langle \Delta \hat{X}_{t}+ \hat{X}_{t},\Delta \hat{M}_{t} \rangle + \bigl\langle \hat{X}_{t}+\hat{b}(t), \hat{Z}_{t}\Delta W_{t} \bigr\rangle . $$

Notice that W, M, are square integrable martingale processes and M, are strongly orthogonal to W, we obtain \(\mathbb{E}[\varPsi _{t}]=0\). Furthermore,

$$\begin{aligned} & \mathbb{E} \bigl\langle X_{T}-\check{X}_{T},l(X_{T}, \mathbb{E}X_{T})-l (\check{X}_{T},\mathbb{E} \check{X}_{T} ) \bigr\rangle \\ &\quad = \mathbb{E} \langle \hat{X}_{T},\hat{Y}_{T} \rangle=\mathbb{E}\sum_{t=0}^{T-1}\Delta \langle \hat{X}_{t}, \hat{Y}_{t} \rangle =\mathbb{E}\sum _{t=0}^{T-1} \bigl\{ \bigl\langle \hat{X}_{t+1},-\hat{f}(t+1) \bigr\rangle + \bigl\langle \hat{\sigma }(t),\hat{Z}_{t} \bigr\rangle + \bigl\langle \hat{b}(t), \hat{Y}_{t} \bigr\rangle \bigr\} \\ &\quad =\mathbb{E} \Biggl\{ \sum_{t=1}^{T-1} \bigl\langle F (t, \varGamma _{t},u_{t} )-F (t,\check{ \varGamma }_{t},u_{t} ), \varTheta _{t}-\check{ \varTheta }_{t} \bigr\rangle + \bigl\langle \hat{b}(0), \hat{Y}_{0} \bigr\rangle + \bigl\langle \hat{\sigma }(0), \hat{Z}_{0} \bigr\rangle - \bigl\langle \hat{X}_{T}, \hat{f}(T) \bigr\rangle \Biggr\} . \end{aligned}$$

Using the monotonic conditions, it follows that

$$\begin{aligned} & c\mathbb{E} \vert X_{T}-\check{X}_{T} \vert ^{2} \\ &\quad \leq \mathbb{E} \bigl\langle X_{T}- \check{X}_{T},l (X_{T},\mathbb{E}X_{T} )-l (\check{X}_{T},\mathbb{E}\check{X}_{T} ) \bigr\rangle \\ &\quad =\mathbb{E} \Biggl\{ \sum_{t=1}^{T-1} \bigl\langle F (t, \varGamma _{t},u_{t} )-F (t,\check{ \varGamma }_{t},u_{t} ), \varTheta _{t}-\check{ \varTheta }_{t} \bigr\rangle + \bigl\langle \hat{b}(0), \hat{Y}_{0} \bigr\rangle + \bigl\langle \hat{\sigma }(0), \hat{Z}_{0} \bigr\rangle - \bigl\langle \hat{X}_{T}, \hat{f}(T) \bigr\rangle \Biggr\} \\ &\quad \leq -c_{1}\mathbb{E} \Biggl\{ \sum_{t=0}^{T} \vert X_{t}- \check{X}_{t} \vert ^{2} + \sum_{t=0}^{T-1} \vert Y_{t}- \check{Y}_{t} \vert ^{2} +\sum _{t=0}^{T-1} \vert Z_{t}- \check{Z}_{t} \vert ^{2} \Biggr\} , \end{aligned}$$

which further implies

$$ \mathbb{E} \Biggl\{ \sum_{t=0}^{T} \vert X_{t}-\check{X}_{t} \vert ^{2} + \sum_{t=0}^{T-1} \vert Y_{t}-\check{Y}_{t} \vert ^{2} +\sum _{t=0}^{T-1} \vert Z_{t}- \check{Z}_{t} \vert ^{2} \Biggr\} =0. $$

Besides, it is easy to see \(\mathbb{E} \vert Y_{T}-\check{Y}_{T} \vert ^{2}=0\) and \(\mathbb{E}\sum_{t=0}^{T} \vert M_{t}-\check{M}_{t} \vert ^{2}=0\). Thereby, \(\varTheta =\check{\varTheta }\).

Existence. By Lemma 3.1, we can immediately get that, when \(\mu =0\), for any \((r,\phi ,\varphi )\in \mathcal{N}^{2}[0,T]\), \(\lambda \in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\), MF-FBSΔEs (3.3) have a unique solution. By Lemma 3.2, for any \((r,\phi ,\varphi )\in \mathcal{N}^{2}[0,T]\), \(\lambda \in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\), (3.3) can be solved successively for \(\mu \in [0,\delta _{0}], [\delta _{0},2\delta _{0}], \ldots \) . Hence, we can deduce that when \(\mu =1\), for any \((r,\phi ,\varphi )\in \mathcal{N}^{2}[0,T]\), \(\lambda \in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\), MF-FBSΔEs (3.3) have a unique solution. Let \(r_{t}=\phi _{t}=\varphi _{t}=\lambda =0\), we conclude that MF-FBSΔE (1.1) has a solution. □

Proof of Lemma 3.1

We consider the following BSΔE:

$$ \textstyle\begin{cases} \Delta Y_{t}^{\ast }=Y_{t}^{\ast }+\mathbb{E}Y_{t}^{\ast }-r_{t}- \varphi _{t+1} +(2Z_{t}^{\ast }+\mathbb{E}Z_{t}^{\ast }-\phi _{t}) \Delta W_{t}+\Delta M_{t}^{\ast }, \\ Y_{T}^{\ast }=\lambda . \end{cases} $$

Using Theorem 2.1, the above equation admits a unique solution \((Y^{\ast },Z^{\ast },M^{\ast })\). Then we solve the following forward equation:

$$ \textstyle\begin{cases} \Delta X_{t}=-X_{t+1}-\mathbb{E}X_{t+1}-Y_{t}^{\ast }-\mathbb{E}Y_{t}^{ \ast }+r_{t}+(-Z_{t}^{\ast }-\mathbb{E}Z_{t}^{\ast } +\phi _{t}) \Delta W_{t}, \\ X_{0}=x_{0}. \end{cases} $$

Let \(Y=Y^{\ast }+X\), \(Z=Z^{\ast }\), and \(M=M^{\ast }\), we can see that \((X,Y,Z,M)\) is a solution of (3.2). Thus, the existence is finished. With regards to the uniqueness, it suffices to apply the method of the proof of uniqueness in Theorem 3.1; here, we omit it. □

Proof of Lemma 3.2

Notice that

$$ \textstyle\begin{cases} b^{\mu _{0}+\delta }(t,\varGamma _{t},u_{t})=b^{\mu _{0}}(t,\varGamma _{t},u_{t})+ \delta [b(t,\varGamma _{t},u_{t})+Y_{t}+\mathbb{E}Y_{t}], \\ \sigma ^{\mu _{0}+\delta }(t,\varGamma _{t},u_{t})=\sigma ^{ \mu _{0}}(t,\varGamma _{t},u_{t})+\delta [\sigma (t,\varGamma _{t},u_{t}) +Z_{t}+ \mathbb{E}Z_{t}], \\ f^{\mu _{0}+\delta }(t,\varGamma _{t},u_{t})=f^{\mu _{0}}(t, \varGamma _{t},u_{t})+\delta [f(t,\varGamma _{t},u_{t})-X_{t}-\mathbb{E}X_{t}], \\ l^{\mu _{0}+\delta }(X_{t},\mathbb{E}X_{t})=l^{\mu _{0}}(X_{t}, \mathbb{E}X_{t})+\delta [l(X_{t},\mathbb{E}X_{t}) -X_{t}-\mathbb{E}X_{t}]. \end{cases} $$

Set \(\varLambda ^{i}= (X^{i},Y^{i},\mathbb{E}X^{i},\mathbb{E}Y^{i} )\), \(\varTheta ^{i}= (X^{i},Y^{i},Z^{i} )\), \(\varGamma ^{i}= (X^{i},Y^{i},Z^{i},\mathbb{E}X^{i},\mathbb{E}Y^{i}, \mathbb{E}Z^{i} )\), and \(\varGamma ^{0}=0\) to solve iteratively the following equations:

$$ \textstyle\begin{cases} \Delta X_{t}^{i+1}=b^{\mu _{0}}(t,\varGamma _{t}^{i+1},u_{t})+\delta \{ b(t,\varGamma _{t}^{i},u)+Y_{t}^{i}+\mathbb{E}Y_{t}^{i} \} +r_{t} \\ \hphantom{\Delta X_{t}^{i+1}={}}{} + \{\sigma ^{\mu _{0}}(t,\varGamma _{t}^{i+1},u_{t})+ \delta [\sigma (t,\varGamma _{t}^{i},u)+Z_{t}^{i} +\mathbb{E}Z_{t}^{i} ]+\phi _{t} \}\Delta W_{t}, \\ \Delta Y_{t}^{i+1}=-f^{\mu _{0}}(t+1,\varGamma _{t+1}^{i+1},u_{t+1}) -\delta \{f(t+1,\varGamma _{t+1}^{i},u_{t+1})-X_{t+1}^{i}-\mathbb{E}X_{t+1}^{i} \} \\ \hphantom{ \Delta Y_{t}^{i+1}={}}{}-\varphi _{t+1}+Z_{t}^{i+1}\Delta W_{t}+\Delta M_{t}^{i+1}, \\ X_{0}=x_{0},Y_{T}^{i+1}=l^{\mu _{0}} (X_{T}^{i+1},\mathbb{E}X_{T}^{i+1} ) +\delta \{l (X_{T}^{i},\mathbb{E}X_{T}^{i} ) -X_{T}^{i}- \mathbb{E}X_{T}^{i} \}+\lambda . \end{cases} $$
(3.4)

Then we apply the product rule to \(\hat{X}_{t}^{i+1}\hat{Y}_{t}^{i+1}\) yielding

$$\begin{aligned} & \mathbb{E} \bigl\langle \hat{X}_{T}^{i+1},l^{\mu _{0}} \bigl(X_{T}^{i+1}, \mathbb{E}X_{T}^{i+1} \bigr) -l^{\mu _{0}} \bigl(X_{T}^{i}, \mathbb{E}X_{T}^{i} \bigr) \bigr\rangle +\mathbb{E} \bigl\langle \hat{X}_{T}^{i+1},\delta \bigl[X_{T}^{i-1}-X_{T}^{i}+ \mathbb{E}X_{T}^{i-1}- \mathbb{E}X_{T}^{i} \bigr] \bigr\rangle \\ &\qquad {}+\mathbb{E} \bigl\langle \hat{X}_{T}^{i+1},\delta \bigl[l \bigl(X_{T}^{i}, \mathbb{E}X_{T}^{i} \bigr) -l \bigl(X_{T}^{i-1},\mathbb{E}X_{T}^{i-1} \bigr) \bigr] \bigr\rangle \\ &\quad =\mathbb{E} \bigl\langle \hat{X}_{T}^{i+1}, \hat{Y}_{T}^{i+1} \bigr\rangle \\ &\quad =\mathbb{E}\sum_{t=1}^{T-1} \bigl\langle F^{\mu _{0}} \bigl(t, \varGamma _{t}^{i+1},u_{t} \bigr)-F^{\mu _{0}} \bigl(t,\varGamma _{t}^{i},u_{t} \bigr), \hat{\varTheta }_{t}^{i+1} \bigr\rangle \\ &\qquad {}+\mathbb{E} \sum_{t=1}^{T-1}\delta \bigl\langle F \bigl(t,\varGamma _{t}^{i},u_{t} \bigr)-F \bigl(t,\varGamma _{t}^{i-1},u_{t} \bigr)+\hat{ \varTheta }_{t}^{i}, \hat{\varTheta }_{t}^{i+1} \bigr\rangle \\ &\qquad {} +\mathbb{E} \bigl\langle b^{\mu _{0}} \bigl(0,\varGamma _{0}^{i+1},u_{0} \bigr)-b^{\mu _{0}} \bigl(0,\varGamma _{0}^{i},u_{0} \bigr), \hat{Y}_{0}^{i+1} \bigr\rangle \\ &\qquad {}+\mathbb{E}\delta \bigl\langle b \bigl(0,\varGamma _{0}^{i},u_{0} \bigr)-b \bigl(0,\varGamma _{0}^{i-1},u_{0} \bigr)+\hat{Y}_{0}^{i}, \hat{Y}_{0}^{i+1} \bigr\rangle \\ &\qquad {} +\mathbb{E} \bigl\langle \sigma ^{\mu _{0}} \bigl(0,\varGamma _{0}^{i+1},u_{0} \bigr) -\sigma ^{\mu _{0}} \bigl(0,\varGamma _{0}^{i},u_{0} \bigr), \hat{Z}_{0}^{i+1} \bigr\rangle \\ &\qquad {}+\mathbb{E}\delta \bigl\langle \sigma \bigl(0,\varGamma _{0}^{i},u_{0} \bigr)-\sigma \bigl(0,\varGamma _{0}^{i-1},u_{0} \bigr) +\hat{Z}_{0}^{i},\hat{Z}_{0}^{i+1} \bigr\rangle \\ &\qquad {} -\mathbb{E} \bigl\langle f^{\mu _{0}} \bigl(T,\varLambda _{T}^{i+1},u_{T} \bigr) -f^{\mu _{0}} \bigl(T,\varLambda _{T}^{i},u_{T} \bigr), \hat{X}_{T}^{i+1} \bigr\rangle \\ &\qquad {}-\mathbb{E}\delta \bigl\langle f \bigl(T,\varLambda _{T}^{i},u_{T} \bigr) -f \bigl(T,\varLambda _{T}^{i-1},u_{T} \bigr)+\hat{X}_{T}^{i}, \hat{X}_{T}^{i+1} \bigr\rangle , \end{aligned}$$

where \(\hat{\varGamma }_{t}^{i}=\varGamma _{t}^{i}-\varGamma _{t}^{i-1}\) and \(\hat{\varTheta }_{t}^{i}=\varTheta _{t}^{i}-\varTheta _{t}^{i-1}\). Set \(\beta _{0}=\min \{1,\beta \}\), we claim that

$$ \mathbb{E} \Biggl\{ 2 \bigl\vert \hat{X}_{T}^{i+1} \bigr\vert ^{2}+\sum_{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i+1} \bigr\vert ^{2} \Biggr\} \leq \frac{\delta (1+c)}{\beta _{0}} \Biggl\{ 2\mathbb{E} \bigl\vert \hat{X}_{T}^{i} \bigr\vert \bigl\vert \hat{X}_{T}^{i+1} \bigr\vert + \mathbb{E}\sum _{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i} \bigr\vert \bigl\vert \hat{\varTheta }_{t}^{i+1} \bigr\vert \Biggr\} . $$

Let \(\varepsilon =\frac{\beta _{0}}{\delta (1+c)}\), by means of \(ab\leq \frac{a^{2}}{2\varepsilon }+\frac{\varepsilon b^{2}}{2}\), we have

$$\begin{aligned} \mathbb{E} \Biggl\{ 2 \bigl\vert \hat{X}_{T}^{i+1} \bigr\vert ^{2}+\sum_{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i+1} \bigr\vert ^{2} \Biggr\} \leq& \frac{1}{2} \biggl(\frac{\delta (1+c)}{\beta _{0}} \biggr)^{2} \Biggl\{ 2\mathbb{E} \bigl\vert \hat{X}_{T}^{i} \bigr\vert ^{2}+ \mathbb{E} \sum_{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i} \bigr\vert ^{2} \Biggr\} \\ &{}+\frac{1}{2} \Biggl\{ 2\mathbb{E} \bigl\vert \hat{X}_{T}^{i+1} \bigr\vert ^{2} + \mathbb{E}\sum_{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i+1} \bigr\vert \Biggr\} , \end{aligned}$$

which indicates

$$ 2\mathbb{E} \bigl\vert \hat{X}_{T}^{i+1} \bigr\vert ^{2}+\mathbb{E}\sum_{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i+1} \bigr\vert ^{2} \leq \biggl(\frac{\delta (1+\beta )}{\beta _{0}} \biggr)^{2} \Biggl\{ 2 \mathbb{E} \bigl\vert \hat{X}_{T}^{i} \bigr\vert ^{2}+ \mathbb{E}\sum_{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i} \bigr\vert ^{2} \Biggr\} . $$

Then

$$ \mathbb{E} \bigl\vert \hat{X}_{T}^{i} \bigr\vert ^{2}=\mathbb{E} \Biggl\vert \sum_{t=0}^{T-1} \Delta \hat{X}_{t}^{i} \Biggr\vert ^{2} \leq T \bigl\vert \Delta \hat{X}_{t}^{i} \bigr\vert ^{2}\leq \beta _{1}\mathbb{E}\sum _{t=0}^{T-1} \bigl( \bigl\vert \hat{\varTheta }_{t}^{i} \bigr\vert ^{2} + \bigl\vert \hat{\varTheta }_{t}^{i-1} \bigr\vert ^{2} \bigr), $$

where \(\beta _{1}>0\) and it only relies on c and T. Thus, there exists \(\beta _{2}>0\) relying on c, β, and T such that

$$ \mathbb{E}\sum_{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i+1} \bigr\vert ^{2}\leq \beta _{2}\delta ^{2} \Biggl\{ \mathbb{E}\sum _{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i} \bigr\vert ^{2} + \mathbb{E} \sum_{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i-1} \bigr\vert ^{2} \Biggr\} . $$

Furthermore, there exists \(\bar{\delta }\in (0,1)\) relying on c, β, and T such that, for \(0<\delta \leq \bar{\delta }\),

$$ \mathbb{E}\sum_{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i+1} \bigr\vert ^{2} \leq \frac{1}{4}\mathbb{E}\sum_{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i} \bigr\vert ^{2} +\frac{1}{8}\mathbb{E}\sum _{t=0}^{T-1} \bigl\vert \hat{\varTheta }_{t}^{i-1} \bigr\vert ^{2},\quad \forall i\geq 1. $$

By [30, Lemma 4.1], we see that \(\{\varTheta _{t}^{i}\}_{t=0}^{T-1}\) is a Cauchy sequence in \(\mathcal{K}^{2}[0,T]\). Denote its limit by \(\varTheta =(X,Y,Z)\). Taking the limit in (3.4), we can derive that, when \(0<\delta \leq \bar{\delta }\), \(\varTheta =(X,Y,Z)\) solves (3.3) for \(\mu =\mu _{0}+\delta \). The proof is completed. □

For simplicity, for \(\rho =b, \sigma , f, g\) and \(a=x, y, z, \tilde{x}, \tilde{y}, \tilde{z}, u\), we use the following abbreviations:

$$ \textstyle\begin{cases} \bar{\varGamma }_{t}= (\bar{X}_{t},\bar{Y}_{t},\bar{Z}_{t}, \mathbb{E}\bar{X}_{t},\mathbb{E}\bar{Y}_{t}, \mathbb{E}\bar{Z}_{t} ),\qquad \varGamma _{t}^{\varepsilon }= (X_{t}^{\varepsilon },Y_{t}^{ \varepsilon },Z_{t}^{\varepsilon },\mathbb{E}X_{t}^{\varepsilon }, \mathbb{E}Y_{t}^{\varepsilon },\mathbb{E}Z_{t}^{\varepsilon } ), \\ \bar{\rho }(t)=\rho (t,\bar{\varGamma }_{t},\bar{u}_{t} ),\qquad \rho ^{\varepsilon }(t)=\rho (t,\varGamma _{t}^{\varepsilon },u_{t}^{ \varepsilon } ), \\ \tilde{\rho }^{\varepsilon }(t)=\rho (t, \bar{\varGamma }_{t},u_{t}^{\varepsilon } ),\qquad \rho _{a}(t)=\rho _{a} (t,\bar{\varGamma }_{t},\bar{u}_{t} ). \end{cases} $$

Let \((k,m,n,N)\) be a solution of the following variational equations:

$$ \textstyle\begin{cases} \Delta k_{t}=\bar{b}_{x}(t)k_{t}+\bar{b}_{y}(t)m_{t}+\bar{b}_{z}(t)n_{t}+ \bar{b}_{\tilde{x}}(t)\mathbb{E}k_{t}+\bar{b}_{\tilde{y}}(t) \mathbb{E}m_{t} +\bar{b}_{\tilde{z}}(t)\mathbb{E}n_{t} \\ \hphantom{\Delta k_{t}={}}{} +\alpha (t) \bar{b}_{u}(t)\varepsilon (v_{t}- \bar{u}_{t}) \\ \hphantom{\Delta k_{t}={}}{}+ \{\bar{\sigma }_{x}(t)k_{t}+\bar{\sigma }_{y}(t)m_{t}+ \bar{\sigma }_{z}(t)n_{t}+\alpha (t)\bar{\sigma }_{u}(t) \varepsilon (v_{t}- \bar{u}_{t}) \}\Delta W_{t} \\ \hphantom{\Delta k_{t}={}}{} + \{\bar{\sigma }_{\tilde{x}}(t)\mathbb{E}k_{t}+ \bar{\sigma }_{\tilde{y}}(t)\mathbb{E}m_{t}+\bar{\sigma }_{\tilde{z}}(t) \mathbb{E}n_{t} \}\Delta W_{t}, \\ \Delta m_{t}=-\bar{f}_{x}(t+1)k_{t+1}-\bar{f}_{y}(t+1)m_{t+1}- \bar{f}_{z}(t+1)n_{t+1} \\ \hphantom{\Delta m_{t}={}}{}-\alpha (t+1)\bar{f}_{u}(t+1)\varepsilon (v_{t+1}- \bar{u}_{t+1}) \\ \hphantom{\Delta m_{t}={}}{} -\bar{f}_{\tilde{x}}(t+1)\mathbb{E}k_{t+1}- \bar{f}_{\tilde{y}}(t+1)\mathbb{E}m_{t+1}-\bar{f}_{\tilde{z}}(t+1) \mathbb{E}n_{t+1}+n_{t}\Delta W_{t}+\Delta \bar{N}_{t}, \\ k_{0}=0,\qquad m_{T}= \{l_{x} (\bar{X}_{T},\mathbb{E}\bar{X}_{T} ) +\mathbb{E}l_{\tilde{x}} (\bar{X}_{T},\mathbb{E}\bar{X}_{T} ) \}k_{T}. \end{cases} $$
(3.5)

We proceed to introducing the following adjoint equations:

$$ \textstyle\begin{cases} \Delta \eta _{t}=-\bar{b}_{x}'(t+1)\eta _{t+1}-\bar{\sigma }_{x}'(t+1) \zeta _{t+1}+\bar{f}_{x}'(t+1)\xi _{t+1}+\bar{g}_{x}(t+1) +\zeta _{t} \Delta W_{t} \\ \hphantom{\Delta \eta _{t}={}}{} +\Delta \bar{V}_{t}+\mathbb{E} \{-\bar{b}_{\tilde{x}}'(t+1)\eta _{t+1}- \bar{\sigma }_{\tilde{x}}'(t+1)\zeta _{t+1} +\bar{f}_{\tilde{x}}'(t+1) \xi _{t+1}+\bar{g}_{\tilde{x}}(t+1) \}, \\ \Delta \xi _{t}= \{\bar{f}_{z}'(t)\xi _{t}-\bar{b}_{z}'(t) \eta _{t}-\bar{\sigma }_{z}'(t)\zeta _{t}+\bar{g}_{z}(t) \\ \hphantom{ \Delta \xi _{t}= {}}{}+\mathbb{E} (\bar{f}_{\tilde{z}}'(t)\xi _{t}-\bar{b}_{\tilde{z}}'(t)\eta _{t} - \bar{\sigma }_{\tilde{z}}'(t)\zeta _{t}+\bar{g}_{\tilde{z}}(t) ) \}\Delta W_{t} \\ \hphantom{ \Delta \xi _{t}= {}}{} +\bar{f}_{y}'(t)\xi _{t}-\bar{b}_{y}'(t)\eta _{t}- \bar{\sigma }_{y}'(t)\zeta _{t}+\bar{g}_{y}(t) \\ \hphantom{ \Delta \xi _{t}= {}}{}+\mathbb{E} \{ \bar{f}_{\tilde{y}}'(t)\xi _{t}-\bar{b}_{\tilde{y}}'(t)\eta _{t}- \bar{\sigma }_{\tilde{y}}'(t)\zeta _{t} +\bar{g}_{\tilde{y}}(t) \}, \\ \eta _{T}=-h_{x} (\bar{X}_{T},\mathbb{E}\bar{X}_{T} )- \mathbb{E}h_{\tilde{x}} (\bar{X}_{T},\mathbb{E}\bar{X}_{T} ) -l_{x} (\bar{X}_{T},\mathbb{E}\bar{X}_{T} )\xi _{T} -\mathbb{E} \{l_{ \tilde{x}} (\bar{X}_{T},\mathbb{E}\bar{X}_{T} )\xi _{T} \}, \\ \xi _{0}=0. \end{cases} $$
(3.6)

Here, W, N, V are square integrable martingale processes and N, V are strongly orthogonal to W. Set

$$ \textstyle\begin{cases} \hat{X}_{t}=X_{t}^{\varepsilon }-\bar{X}_{t},\qquad \hat{Y}_{t}=Y_{t}^{ \varepsilon }-\bar{Y}_{t},\qquad \hat{Z}_{t}=Z_{t}^{\varepsilon }-\bar{Z}_{t},\qquad \hat{M}_{t}=M_{t}^{\varepsilon }-\bar{M}_{t}, \\ \tilde{X}_{t}=\hat{X}_{t}-k_{t},\qquad \tilde{Y}_{t}=\hat{Y}_{t}-m_{t},\qquad \tilde{Z}_{t}=\hat{Z}_{t}-n_{t},\qquad \tilde{M}_{t}=\hat{M}_{t}-N_{t}, \end{cases} $$

then we get

$$ \textstyle\begin{cases} \Delta \hat{X}_{t}=b^{\varepsilon }(t)-\bar{b}(t)+ (\sigma ^{ \varepsilon }(t)-\bar{\sigma }(t) )\Delta W_{t}, \\ \Delta \hat{Y}_{t}=-f^{\varepsilon }(t+1)+\bar{f}(t+1)+\hat{Z}_{t} \Delta W_{t}+\Delta \hat{M}_{t}, \\ \hat{X}_{0}=0,\qquad \hat{Y}_{T}=0. \end{cases} $$
(3.7)

For MF-FBSΔE (1.1), we give the following estimates.

Lemma 3.3

Assume that (A2)(A5) hold, we get

$$ \mathbb{E} \Biggl\{ \sum_{t=0}^{T} \vert \hat{X}_{t} \vert ^{2}+ \sum _{t=0}^{T} \vert \hat{Y}_{t} \vert ^{2} +\sum_{t=0}^{T-1} \vert \hat{Z}_{t} \vert ^{2} \Biggr\} \leq c \varepsilon ^{2} \mathbb{E} \vert v_{\theta }- \bar{u}_{\theta } \vert ^{2}. $$

Proof

According to (3.7), we have

$$\begin{aligned} 0&=\mathbb{E} \langle \hat{X}_{T},\hat{Y}_{T} \rangle - \mathbb{E} \langle \hat{X}_{0},\hat{Y}_{0} \rangle = \mathbb{E} \sum_{t=0}^{T-1}\Delta \langle \hat{X}_{t},\hat{Y}_{t} \rangle \\ &=\mathbb{E}\sum_{t=0}^{T} \bigl\{ \bigl\langle \hat{X}_{t}, \tilde{f}^{\varepsilon }(t)-f^{\varepsilon }(t) \bigr\rangle + \bigl\langle \hat{Y}_{t},b^{\varepsilon }(t)- \tilde{b}^{\varepsilon }(t) \bigr\rangle + \bigl\langle \hat{Z}_{t}, \sigma ^{\varepsilon }(t)- \tilde{\sigma }^{\varepsilon }(t) \bigr\rangle \\ &\quad {}+ \bigl\langle \hat{X}_{t},-\tilde{f}^{\varepsilon }(t)+ \bar{f}(t) \bigr\rangle + \bigl\langle \hat{Y}_{t}, \tilde{b}^{\varepsilon }(t)- \bar{b}(t) \bigr\rangle + \bigl\langle \hat{Z}_{t},\tilde{\sigma }^{ \varepsilon }(t)-\bar{\sigma }(t) \bigr\rangle \bigr\} \\ &=\mathbb{E}\sum_{t=1}^{T-1} \bigl\langle F\bigl(t,\varGamma _{t}^{ \varepsilon },u_{t}^{\varepsilon } \bigr)-F \bigl(t,\bar{\varGamma }_{t},u_{t}^{ \varepsilon } \bigr) ,\hat{\varTheta }_{t} \bigr\rangle \\ &\quad {}+\mathbb{E}\sum _{t=0}^{T} \bigl\{ \bigl\langle \hat{X}_{t},\bar{f}(t)-\tilde{f}^{ \varepsilon }(t) \bigr\rangle + \bigl\langle \hat{Y}_{t},\tilde{b}^{ \varepsilon }(t)-\bar{b}(t) \bigr\rangle + \bigl\langle \hat{Z}_{t},\tilde{\sigma }^{\varepsilon }(t)- \bar{\sigma }(t) \bigr\rangle \bigr\} \\ &\quad {} +\mathbb{E} \bigl\{ \bigl\langle \hat{X}_{T},-f^{\varepsilon }(T)+\tilde{f}^{\varepsilon }(T) \bigr\rangle + \bigl\langle \hat{Y}_{0},b^{\varepsilon }(0)- \tilde{b}^{ \varepsilon }(0) \bigr\rangle + \bigl\langle \hat{Z}_{0}, \sigma ^{ \varepsilon }(0)-\tilde{\sigma }^{\varepsilon }(0) \bigr\rangle \bigr\} . \end{aligned}$$

Combining the above equation with the monotonic conditions, we obtain

$$\begin{aligned} & \mathbb{E}\sum_{t=0}^{T} \bigl\{ \bigl\langle \hat{X}_{t},- \tilde{f}^{\varepsilon }(t)+\bar{f}(t) \bigr\rangle + \bigl\langle \hat{Y}_{t},\tilde{b}^{\varepsilon }(t)- \bar{b}(t) \bigr\rangle + \bigl\langle \hat{Z}_{t},\tilde{\sigma }^{\varepsilon }(t)- \bar{\sigma }(t) \bigr\rangle \bigr\} \\ &\quad =-\mathbb{E}\sum_{t=1}^{T-1} \bigl\langle F\bigl(t,\varGamma _{t}^{ \varepsilon },u_{t}^{\varepsilon } \bigr)-F \bigl(t,\bar{\varGamma }_{t},u_{t}^{ \varepsilon } \bigr) ,\hat{\varTheta }_{t} \bigr\rangle \\ &\qquad {} -\mathbb{E} \bigl\{ \bigl\langle \hat{X}_{T},-f^{\varepsilon }(T)+\tilde{f}^{\varepsilon }(T) \bigr\rangle + \bigl\langle \hat{Y}_{0},b^{\varepsilon }(0)- \tilde{b}^{ \varepsilon }(0) \bigr\rangle + \bigl\langle \hat{Z}_{0}, \sigma ^{ \varepsilon }(0)-\tilde{\sigma }^{\varepsilon }(0) \bigr\rangle \bigr\} \\ &\quad \geq \beta \mathbb{E} \Biggl\{ \sum_{t=0}^{T} \vert \hat{X}_{t} \vert ^{2}+\sum _{t=0}^{T} \vert \hat{Y}_{t} \vert ^{2} + \sum_{t=0}^{T-1} \vert \hat{Z}_{t} \vert ^{2} \Biggr\} . \end{aligned}$$
(3.8)

On the other hand, there is a constant \(c_{1}>0\) such that

$$\begin{aligned} \mathbb{E}\sum_{t=0}^{T} \bigl\langle \hat{X}_{t},-\tilde{f}^{ \varepsilon }(t)+\bar{f}(t) \bigr\rangle \leq& \mathbb{E}\sum_{t=0}^{T} \biggl( \frac{\beta }{2} \vert \hat{X}_{t} \vert ^{2} + \frac{1}{2\beta } \bigl\vert \bar{f}(t)-\tilde{f}^{\varepsilon }(t) \bigr\vert ^{2} \biggr) \\ \leq& \mathbb{E}\sum_{t=0}^{T} \biggl(\frac{\beta }{2} \vert \hat{X}_{t} \vert ^{2} +\frac{c_{1}}{2\beta }\varepsilon ^{2} \vert v_{ \theta }-\bar{u}_{\theta } \vert ^{2} \biggr). \end{aligned}$$

Similarly, we can deduce

$$\begin{aligned} \begin{aligned}[b] & \mathbb{E}\sum_{t=0}^{T} \bigl\{ \bigl\langle \hat{X}_{t},- \tilde{f}^{\varepsilon }(t)+\bar{f}(t) \bigr\rangle + \bigl\langle \hat{Y}_{t},\tilde{b}^{\varepsilon }(t)- \bar{b}(t) \bigr\rangle + \bigl\langle \hat{Z}_{t},\tilde{\sigma }^{\varepsilon }(t)- \bar{\sigma }(t) \bigr\rangle \bigr\} \\ &\quad \leq \frac{\beta }{2}\mathbb{E} \Biggl\{ \sum _{t=0}^{T} \vert \hat{X}_{t} \vert ^{2} +\sum_{t=0}^{T} \vert \hat{Y}_{t} \vert ^{2} +\sum _{t=0}^{T-1} \vert \hat{Z}_{t} \vert ^{2} \Biggr\} +\frac{3c_{1}}{2\beta }\varepsilon ^{2} \mathbb{E} \vert v_{ \theta }-\bar{u}_{\theta } \vert ^{2}. \end{aligned} \end{aligned}$$
(3.9)

Using (3.8)–(3.9), we finally get

$$ \mathbb{E} \Biggl\{ \sum_{t=0}^{T} \vert \hat{X}_{t} \vert ^{2}+ \sum _{t=0}^{T} \vert \hat{Y}_{t} \vert ^{2} +\sum_{t=0}^{T-1} \vert \hat{Z}_{t} \vert ^{2} \Biggr\} \leq c \varepsilon ^{2} \mathbb{E} \vert v_{\theta }- \bar{u}_{\theta } \vert ^{2}. $$

The proof is completed. □

Remark 3.1

Under (A2)–(A5), we have the following results.

If \(t=1,2,\ldots ,T-1\),

$$ \begin{pmatrix} -\bar{f}_{x}(t) & -\bar{f}_{y}(t) & -\bar{f}_{z}(t) \\ \bar{b}_{x}(t) & \bar{b}_{y}(t) & \bar{b}_{z}(t) \\ \bar{\sigma }_{x}(t) & \bar{\sigma }_{y}(t) & \bar{\sigma }_{z}(t) \end{pmatrix}\leq -\beta I_{3n},\qquad \begin{pmatrix} -\bar{f}_{\tilde{x}}(t) & -\bar{f}_{\tilde{y}}(t) & -\bar{f}_{ \tilde{z}}(t) \\ \bar{b}_{\tilde{x}}(t) & \bar{b}_{\tilde{y}}(t) & \bar{b}_{\tilde{z}}(t) \\ \bar{\sigma }_{\tilde{x}}(t) & \bar{\sigma }_{\tilde{y}}(t) & \bar{\sigma }_{\tilde{z}}(t) \end{pmatrix}\leq -\beta I_{3n}. $$

If \(t=0\),

$$ \begin{pmatrix} \bar{b}_{y}(0) & \bar{b}_{z}(0) \\ \bar{\sigma }_{y}(0) & \bar{\sigma }_{z}(0) \end{pmatrix}\leq -\beta I_{2n},\qquad \begin{pmatrix} \bar{b}_{\tilde{y}}(0) & \bar{b}_{\tilde{z}}(0) \\ \bar{\sigma }_{\tilde{y}}(0) & \bar{\sigma }_{\tilde{z}}(0) \end{pmatrix}\leq -\beta I_{2n}. $$

If \(t=T\),

$$ -\bar{f}_{x}(T)\leq -\beta I_{n},\qquad -\bar{f}_{\tilde{x}}(T) \leq - \beta I_{n}. $$

Consequently, the coefficients of (3.5) satisfy the monotonic conditions and there exists a unique solution \((k,m,n,N)\) to (3.5). Following the proof of Lemma 3.3, it is easy to check

$$ \mathbb{E} \Biggl\{ \sum_{t=0}^{T} \vert k_{t} \vert ^{2}+\sum _{t=0}^{T} \vert m_{t} \vert ^{2} +\sum_{t=0}^{T-1} \vert n_{t} \vert ^{2} \Biggr\} \leq c\varepsilon ^{2} \mathbb{E} \vert v_{\theta }-\bar{u}_{\theta } \vert ^{2}. $$

Lemma 3.4

Assume that (A2)(A5) hold, we get

$$ \mathbb{E} \Biggl\{ \sum_{t=0}^{T} \vert \tilde{X}_{t} \vert ^{2}+ \sum _{t=0}^{T} \vert \tilde{Y}_{t} \vert ^{2} +\sum_{t=0}^{T-1} \vert \tilde{Z}_{t} \vert ^{2} \Biggr\} =o\bigl( \varepsilon ^{2}\bigr). $$

Proof

Observe that

$$\begin{aligned} \varLambda ^{\varepsilon }(t)-\bar{\varLambda }(t) =&\tilde{\varLambda }_{x}(t) \hat{X}_{t}+\tilde{\varLambda }_{y}(t)\hat{Y}_{t}+ \tilde{\varLambda }_{z}(t) \hat{Z}_{t}+\tilde{\varLambda }_{\tilde{x}}(t)\mathbb{E}\hat{X}_{t} \\ &{}+ \tilde{\varLambda }_{\tilde{y}}(t)\mathbb{E}\hat{Y}_{t}+ \tilde{\varLambda }_{\tilde{z}}(t)\mathbb{E}\hat{Z}_{t}+\alpha (t) \tilde{\varLambda }_{u}(t)\varepsilon (v_{t}-\bar{u}_{t}), \end{aligned}$$

where

$$ \tilde{\varLambda }_{a}(t)= \int _{0}^{1}\varLambda _{a} \bigl(t, \bar{\varGamma }_{t}+ \lambda \hat{\varGamma }_{t}, \bar{u}_{t}+\lambda \bigl(u_{t}^{\varepsilon }- \bar{u}_{t}\bigr) \bigr)\,d\lambda , $$

with \(a=x, y, z, \tilde{x}, \tilde{y}, \tilde{z}, u\), then we have

$$ \textstyle\begin{cases} \Delta \tilde{X}_{t}=\bar{b}_{x}(t)\tilde{X}_{t}+\bar{b}_{y}(t) \tilde{Y}_{t}+\bar{b}_{z}(t)\tilde{Z}_{t}+\kappa _{1}(t) +\bar{b}_{ \tilde{x}}(t)\mathbb{E}\tilde{X}_{t} \\ \hphantom{\Delta \tilde{X}_{t}={}}{}+\bar{b}_{\tilde{y}}(t)\mathbb{E} \tilde{Y}_{t}+\bar{b}_{\tilde{z}}(t)\mathbb{E}\tilde{Z}_{t} + \tilde{\kappa }_{1}(t) \\ \hphantom{\Delta \tilde{X}_{t}={}}{} + \{\bar{\sigma }_{\tilde{x}}(t)\mathbb{E} \tilde{X}_{t}+\bar{\sigma }_{\tilde{y}}(t)\mathbb{E}\tilde{Y}_{t} + \bar{\sigma }_{\tilde{z}}(t)\mathbb{E}\tilde{Z}_{t}+\tilde{\kappa }_{2}(t) \}\Delta W_{t} \\ \hphantom{\Delta \tilde{X}_{t}={}}{} + \{\bar{\sigma }_{x}(t)\tilde{X}_{t}+ \bar{\sigma }_{y}(t)\tilde{Y}_{t}+\bar{\sigma }_{z}(t)\tilde{Z}_{t}+ \kappa _{2}(t) \}\Delta W_{t}, \\ \Delta \tilde{Y}_{t}=-\bar{f}_{x}(t+1)\tilde{X}_{t+1}- \bar{f}_{y}(t+1)\tilde{Y}_{t+1}-\bar{f}_{z}(t+1)\tilde{Z}_{t+1}- \kappa _{3}(t+1) +\tilde{Z}_{t}\Delta W_{t} \\ \hphantom{\Delta \tilde{Y}_{t}={}}{} +\Delta \tilde{M}_{t}-\bar{f}_{\tilde{x}}(t+1)\mathbb{E}\tilde{X}_{t+1}- \bar{f}_{\tilde{y}}(t+1)\mathbb{E}\tilde{Y}_{t+1} -\bar{f}_{\tilde{z}}(t+1) \mathbb{E}\tilde{Z}_{t+1}-\tilde{\kappa }_{3}(t+1), \\ \tilde{X}_{0}=0,\qquad \tilde{Y}_{T}=0, \end{cases} $$
(3.10)

where

$$ \textstyle\begin{cases} \kappa _{1}(t)= \{ \tilde{b}_{x}(t)-\bar{b}_{x}(t) \} \hat{X}_{t}+ \{ \tilde{b}_{y}(t)-\bar{b}_{y}(t) \} \hat{Y}_{t} + \{ \tilde{b}_{z}(t)-\bar{b}_{z}(t) \} \hat{Z}_{t} \\ \hphantom{\kappa _{1}(t)={}}{}+\alpha (t) \{ \tilde{b}_{u}(t)-\bar{b}_{u}(t) \} \varepsilon (v_{t}- \bar{u}_{t}), \\ \tilde{\kappa }_{1}(t)= \{ \tilde{b}_{\tilde{x}}(t)- \bar{b}_{\tilde{x}}(t) \} \mathbb{E}\hat{X}_{t} + \{ \tilde{b}_{\tilde{y}}(t)-\bar{b}_{\tilde{y}}(t) \} \mathbb{E} \hat{Y}_{t} + \{ \tilde{b}_{\tilde{z}}(t)-\bar{b}_{\tilde{z}}(t) \} \mathbb{E}\hat{Z}_{t}, \\ \kappa _{2}(t)= \{\tilde{\sigma }_{x}(t)-\bar{\sigma }_{x}(t) \}\hat{X}_{t} + \{\tilde{\sigma }_{y}(t)-\bar{\sigma }_{y}(t) \}\hat{Y}_{t} + \{\tilde{\sigma }_{z}(t)-\bar{\sigma }_{z}(t) \}\hat{Z}_{t} \\ \hphantom{\kappa _{2}(t)={}}{}+\alpha (t) \{\tilde{\sigma }_{u}(t)- \bar{\sigma }_{u}(t) \}\varepsilon (v_{t}-\bar{u}_{t}), \\ \tilde{\kappa }_{2}(t)= \{\tilde{\sigma }_{\tilde{x}}(t)- \bar{\sigma }_{\tilde{x}}(t) \}\mathbb{E}\hat{X}_{t} + \{ \tilde{\sigma }_{\tilde{y}}(t)-\bar{\sigma }_{\tilde{y}}(t) \} \mathbb{E}\hat{Y}_{t} + \{\tilde{\sigma }_{\tilde{z}}(t)- \bar{\sigma }_{\tilde{z}}(t) \}\mathbb{E}\hat{Z}_{t}, \\ \kappa _{3}(t)=- \{ \tilde{f}_{x}(t)-\bar{f}_{x}(t) \} \hat{X}_{t}- \{ \tilde{f}_{y}(t)-\bar{f}_{y}(t) \} \hat{Y}_{t} - \{ \tilde{f}_{z}(t)-\bar{f}_{z}(t) \} \hat{Z}_{t} \\ \hphantom{\kappa _{3}(t)={}}{}- \alpha (t) \{ \tilde{f}_{u}(t)-\bar{f}_{u}(t) \} \varepsilon (v_{t}-\bar{u}_{t}), \\ \tilde{\kappa }_{3}(t)=- \{ \tilde{f}_{\tilde{x}}(t)-\bar{f}_{ \tilde{x}}(t) \} \mathbb{E}\hat{X}_{t} - \{ \tilde{f}_{ \tilde{y}}(t)-\bar{f}_{\tilde{y}}(t) \} \mathbb{E}\hat{Y}_{t} - \{ \tilde{f}_{\tilde{z}}(t)-\bar{f}_{\tilde{z}}(t) \} \mathbb{E}\hat{Z}_{t}. \end{cases} $$

From (3.10), it yields that

$$\begin{aligned} 0&=\mathbb{E} \langle \tilde{X}_{T},\tilde{Y}_{T} \rangle - \mathbb{E} \langle \tilde{X}_{0},\tilde{Y}_{0} \rangle = \mathbb{E}\sum_{t=0}^{T-1}\Delta \langle \tilde{X}_{t}, \tilde{Y}_{t} \rangle \\ &=\mathbb{E}\sum_{t=0}^{T} \bigl\{ \bigl\langle \tilde{X}_{t},- \bar{f}_{\varGamma }(t)\tilde{\varGamma }_{t} \bigr\rangle + \bigl\langle \tilde{Y}_{t} , \bar{b}_{\varGamma }(t)\tilde{\varGamma }_{t} \bigr\rangle + \bigl\langle \tilde{Z}_{t} ,\bar{\sigma }_{\varGamma }(t) \tilde{ \varGamma }_{t} \bigr\rangle \\ &\quad {}- \bigl\langle \tilde{X}_{t},\kappa _{3}(t)+\tilde{ \kappa }_{3}(t) \bigr\rangle + \bigl\langle \tilde{Y}_{t} ,\kappa _{1}(t)+ \tilde{\kappa }_{1}(t) \bigr\rangle + \bigl\langle \tilde{Z}_{t} , \kappa _{2}(t)+\tilde{\kappa }_{2}(t) \bigr\rangle \bigr\} , \end{aligned}$$

where

$$\begin{aligned}& \tilde{\varGamma }_{t}= \bigl(\tilde{X}_{t}', \tilde{Y}_{t}',\tilde{Z}_{t}', \mathbb{E}\tilde{X}_{t}',\mathbb{E} \tilde{Y}_{t}', \mathbb{E} \tilde{Z}_{t}' \bigr)', \\& \bar{\rho }_{\varGamma }(t)= \bigl(\bar{\rho }_{x}(t), \bar{\rho }_{y}(t),\bar{\rho }_{z}(t),\bar{\rho }_{\tilde{x}}(t), \bar{\rho }_{\tilde{y}}(t), \bar{\rho }_{\tilde{z}}(t) \bigr). \end{aligned}$$

Noticing Remark 3.1, we derive

$$\begin{aligned} & \mathbb{E}\sum_{t=0}^{T} \bigl\{ \bigl\langle \tilde{X}_{t},- \kappa _{3}(t)-\tilde{ \kappa }_{3}(t) \bigr\rangle + \bigl\langle \tilde{Y}_{t}, \kappa _{1}(t)+\tilde{\kappa }_{1}(t) \bigr\rangle + \bigl\langle \tilde{Z}_{t},\kappa _{2}(t)+\tilde{\kappa }_{2}(t) \bigr\rangle \bigr\} \\ &\quad \geq \beta \mathbb{E} \Biggl\{ \sum_{t=0}^{T} \vert \tilde{X}_{t} \vert ^{2} +\sum _{t=0}^{T} \vert \tilde{Y}_{t} \vert ^{2}+ \sum_{t=0}^{T-1} \vert \tilde{Z}_{t} \vert ^{2} \Biggr\} . \end{aligned}$$

On the other hand,

$$\begin{aligned} \mathbb{E} \bigl\langle \tilde{X}_{t},-\kappa _{3}(t)- \tilde{\kappa }_{3}(t) \bigr\rangle &=\mathbb{E} \bigl\langle \tilde{X}_{t}, \bigl[ \tilde{f}_{x}(t)- \bar{f}_{x}(t) \bigr]\hat{X}_{t} \bigr\rangle + \mathbb{E} \bigl\langle \tilde{X}_{t}, \bigl[\tilde{f}_{y}(t)- \bar{f}_{y}(t) \bigr]\hat{Y}_{t} \bigr\rangle \\ &\quad {}+\mathbb{E} \bigl\langle \tilde{X}_{t}, \bigl[ \tilde{f}_{z}(t)- \bar{f}_{z}(t) \bigr] \hat{Z}_{t} \bigr\rangle +\mathbb{E} \bigl\langle \tilde{X}_{t}, \bigl[\tilde{f}_{\tilde{z}}(t)- \bar{f}_{ \tilde{z}}(t) \bigr]\mathbb{E}\hat{Z}_{t} \bigr\rangle \\ &\quad {}+\mathbb{E} \bigl\langle \tilde{X}_{t}, \bigl[ \tilde{f}_{ \tilde{x}}(t)-\bar{f}_{\tilde{x}}(t) \bigr]\mathbb{E} \hat{X}_{t} \bigr\rangle +\mathbb{E} \bigl\langle \tilde{X}_{t}, \bigl[ \tilde{f}_{\tilde{y}}(t)- \bar{f}_{\tilde{y}}(t) \bigr]\mathbb{E} \hat{Y}_{t} \bigr\rangle \\ &\quad {}+\mathbb{E} \bigl\langle \tilde{X}_{t},\alpha (t) \bigl[ \tilde{f}_{u}(t)-\bar{f}_{u}(t) \bigr]\varepsilon (v_{t}-\bar{u}_{t}) \bigr\rangle . \end{aligned}$$

Let \(\varepsilon =\frac{1}{\beta }\), by virtue of \(ab\leq \frac{a^{2}}{8\varepsilon }+2\varepsilon b^{2}\), we have

$$\begin{aligned} & \mathbb{E} \bigl\langle \tilde{X}_{t},-\kappa _{3}(t)- \tilde{\kappa }_{3}(t) \bigr\rangle \\ &\quad \leq \frac{7\beta }{8}\mathbb{E} \vert \tilde{X}_{t} \vert ^{2}+ \frac{2}{\beta }\mathbb{E} \bigl\{ \bigl\Vert \tilde{f}_{x}(t)-\bar{f}_{x}(t) \bigr\Vert ^{2} \vert \hat{X}_{t} \vert ^{2} + \bigl\Vert \tilde{f}_{y}(t)- \bar{f}_{y}(t) \bigr\Vert ^{2} \vert \hat{Y}_{t} \vert ^{2} + \bigl\Vert \tilde{f}_{z}(t)-\bar{f}_{z}(t) \bigr\Vert ^{2} \vert \hat{Z}_{t} \vert ^{2} \\ &\qquad {}+\varepsilon ^{2} \bigl\Vert \tilde{f}_{u}(t)- \bar{f}_{u}(t) \bigr\Vert ^{2} \vert v_{t}-\bar{u}_{t} \vert ^{2} + \bigl\Vert \tilde{f}_{ \tilde{x}}(t)-\bar{f}_{\tilde{x}}(t) \bigr\Vert ^{2} \vert \mathbb{E} \hat{X}_{t} \vert ^{2} \\ &\qquad {}+ \bigl\Vert \tilde{f}_{\tilde{y}}(t)-\bar{f}_{\tilde{y}}(t) \bigr\Vert ^{2} \vert \mathbb{E}\hat{Y}_{t} \vert ^{2} + \bigl\Vert \tilde{f}_{ \tilde{z}}(t)- \bar{f}_{\tilde{z}}(t) \bigr\Vert ^{2} \vert \mathbb{E} \hat{Z}_{t} \vert ^{2} \bigr\} . \end{aligned}$$

Combining \(\lim_{\varepsilon \rightarrow 0} \Vert \tilde{f}_{\varGamma }(t)- \bar{f}_{\varGamma }(t) \Vert =0\) with Lemma 3.3, we obtain

$$ \mathbb{E} \bigl\langle \tilde{X}_{t},-\kappa _{3}(t)- \tilde{\kappa }_{3}(t) \bigr\rangle \leq \frac{7\beta }{8} \mathbb{E} \vert \tilde{X}_{t} \vert ^{2} +o\bigl( \varepsilon ^{2}\bigr). $$

In a similar way, we have

$$\begin{aligned}& \mathbb{E} \bigl\langle \tilde{Y}_{t},\kappa _{1}(t)+ \tilde{\kappa }_{1}(t) \bigr\rangle \leq \frac{7\beta }{8} \mathbb{E} \vert \tilde{Y}_{t} \vert ^{2} +o\bigl( \varepsilon ^{2}\bigr), \\& \mathbb{E} \bigl\langle \tilde{Z}_{t}, \kappa _{2}(t)+\tilde{\kappa }_{2}(t) \bigr\rangle \leq \frac{7\beta }{8}\mathbb{E} \vert \tilde{Z}_{t} \vert ^{2} +o\bigl( \varepsilon ^{2}\bigr). \end{aligned}$$

Thus,

$$ \mathbb{E} \Biggl\{ \sum_{t=0}^{T} \vert \tilde{X}_{t} \vert ^{2}+ \sum _{t=0}^{T} \vert \tilde{Y}_{t} \vert ^{2}+ \sum_{t=0}^{T-1} \vert \tilde{Z}_{t} \vert ^{2} \Biggr\} \leq o\bigl( \varepsilon ^{2}\bigr). $$

The proof is completed. □

Remark 3.2

From Lemma 3.4, we see that \(k_{t}\) is the first-order variation of \(X_{t}\), and \(m_{t}\) is the first-order variation of \(Y_{t}\). It is easy to derive

$$\begin{aligned} J\bigl(u^{\varepsilon }_{t}\bigr)-J(\bar{u}_{t}) &= \mathbb{E}\sum_{t=0}^{T-1} \bigl\{ \bigl\langle \bar{g}_{x}(t)+\mathbb{E}\bar{g}_{\tilde{x}}(t),k_{t} \bigr\rangle +\bigl\langle \bar{g}_{y}(t)+\mathbb{E} \bar{g}_{\tilde{y}}(t),m_{t} \bigr\rangle +\bigl\langle \bar{g}_{z}(t)+\mathbb{E}\bar{g}_{\tilde{z}}(t),n_{t} \bigr\rangle \\ &\quad {}+\alpha (t)\varepsilon \bigl\langle \bar{g}_{u}(t),v_{t}- \bar{u}_{t} \bigr\rangle \bigr\} +\mathbb{E} \bigl\langle h_{x} (\bar{X}_{T}, \mathbb{E}\bar{X}_{T} ) +\mathbb{E}h_{\tilde{x}} (\bar{X}_{T}, \mathbb{E} \bar{X}_{T} ),k_{T} \bigr\rangle \\ &\quad {}+o(\varepsilon ). \end{aligned}$$
(3.11)

Stochastic maximum principle

In this section, we are devoted to establishing the stochastic maximum principle for fully coupled MF-FBSΔE (1.1). Define the following Hamiltonian function:

$$\begin{aligned} \mathscr{H}(t,x,y,z,\tilde{x},\tilde{y},\tilde{z},\eta ,\zeta ,\xi ,u)&= \bigl\langle \eta , b(t,x,y,z,\tilde{x},\tilde{y},\tilde{z},u)\bigr\rangle + \bigl\langle \zeta , \sigma (t,x,y,z,\tilde{x},\tilde{y},\tilde{z},u) \bigr\rangle \\ &\quad {}-\bigl\langle \xi ,f(t,x,y,z,\tilde{x},\tilde{y},\tilde{z},u)\bigr\rangle -g(t,x,y,z, \tilde{x},\tilde{y},\tilde{z},u). \end{aligned}$$

Theorem 4.1

(Stochastic maximum principle)

Assume that (A2)(A5) hold. Let\(\bar{u}_{t}\)be the optimal control and\((\bar{X}_{t},\bar{Y}_{t},\bar{Z}_{t} )\)be the corresponding optimal trajectory of Problem\(\mathcal{A}\), then for\(v_{t}\in U_{t}\)and\(\alpha (t)\in \mathcal{I}(t,v_{t})\), one has

$$ \alpha (t) \bigl\langle \mathscr{H}_{u} (t,\bar{X}_{t}, \bar{Y}_{t}, \bar{Z}_{t},\mathbb{E}\bar{X}_{t}, \mathbb{E}\bar{Y}_{t},\mathbb{E} \bar{Z}_{t}, \eta _{t},\zeta _{t},\xi _{t}, \bar{u}_{t} ),v_{t}- \bar{u}_{t} \bigr\rangle \leq 0,\quad \mathbb{P}\textit{-a.s.} $$
(4.1)

Proof

By (3.5)–(3.6), for \(t\in \{0,1,\ldots ,T-1\}\), we obtain

$$\begin{aligned} \Delta \langle k_{t},\eta _{t}\rangle &=\langle k_{t+1},\Delta \eta _{t} \rangle +\langle \Delta k_{t},\eta _{t}\rangle \\ &= \bigl\langle k_{t+1},-\bar{b}_{x}'(t+1) \eta _{t+1}-\bar{\sigma }_{x}'(t+1) \zeta _{t+1}+\bar{f}_{x}'(t+1)\xi _{t+1}+\bar{g}_{x}(t+1) \bigr\rangle \\ &\quad {}+ \bigl\langle k_{t+1},\mathbb{E} \bigl[-\bar{b}_{\tilde{x}}'(t+1) \eta _{t+1}-\bar{\sigma }_{\tilde{x}}'(t+1)\zeta _{t+1} +\bar{f}_{ \tilde{x}}'(t+1)\xi _{t+1}+\bar{g}_{\tilde{x}}(t+1) \bigr] \bigr\rangle \\ &\quad {}+ \bigl\langle \bar{b}_{x}(t)k_{t}+ \bar{b}_{y}(t)m_{t}+\bar{b}_{z}(t)n_{t} +\bar{b}_{\tilde{x}}(t)\mathbb{E}k_{t}+\bar{b}_{\tilde{y}}(t) \mathbb{E}m_{t} \\ &\quad {}+\bar{b}_{\tilde{z}}(t)\mathbb{E}n_{t} +\alpha (t) \varepsilon \bar{b}_{u}(t) (v_{t}- \bar{u}_{t}),\eta _{t} \bigr\rangle \\ &\quad {}+ \bigl\langle \bigl[\bar{\sigma }_{x}(t)k_{t}+\bar{ \sigma }_{y}(t)m_{t}+ \bar{\sigma }_{z}(t)n_{t}+ \alpha (t)\varepsilon \bar{\sigma }_{u}(t) (v_{t}- \bar{u}_{t})\bigr]\Delta W_{t},\zeta _{t} \Delta W_{t} \bigr\rangle \\ &\quad {}+ \bigl\langle \bigl[\bar{\sigma }_{\tilde{x}}(t)\mathbb{E}k_{t}+ \bar{\sigma }_{\tilde{y}}(t)\mathbb{E}m_{t}+\bar{\sigma }_{\tilde{z}}(t) \mathbb{E}n_{t}\bigr]\Delta W_{t},\zeta _{t}\Delta W_{t} \bigr\rangle + \varPhi _{t}, \end{aligned}$$

where

$$\begin{aligned} \varPhi _{t}&= \bigl\langle k_{t}+\bar{b}_{x}(t)k_{t}+ \bar{b}_{y}(t)m_{t}+ \bar{b}_{z}(t)n_{t}+ \bar{b}_{\tilde{x}}(t)\mathbb{E}k_{t} +\bar{b}_{ \tilde{y}}(t) \mathbb{E}m_{t} \\ &\quad {}+\bar{b}_{\tilde{z}}(t)\mathbb{E}n_{t} + \alpha (t)\varepsilon \bar{b}_{u}(t) (v_{t}- \bar{u}_{t}),\zeta _{t} \Delta W_{t} \bigr\rangle \\ &\quad {}+ \bigl\langle \bigl[\bar{\sigma }_{x}(t)k_{t}+\bar{ \sigma }_{y}(t)m_{t}+ \bar{\sigma }_{z}(t)n_{t}+ \bar{\sigma }_{\tilde{x}}(t)\mathbb{E}k_{t} +\bar{\sigma }_{\tilde{y}}(t)\mathbb{E}m_{t} \\ &\quad {}+\bar{\sigma }_{ \tilde{z}}(t)\mathbb{E}n_{t}+\alpha (t)\varepsilon \bar{ \sigma }_{u}(t) (v_{t}-\bar{u}_{t})\bigr] \Delta W_{t},\eta _{t} \bigr\rangle \\ &\quad {}+ \bigl\langle k_{t}+\bar{b}_{x}(t)k_{t}+ \bar{b}_{y}(t)m_{t}+ \bar{b}_{z}(t)n_{t}+ \bar{b}_{\tilde{x}}(t)\mathbb{E}k_{t}+\bar{b}_{ \tilde{y}}(t) \mathbb{E}m_{t} \\ &\quad {}+\bar{b}_{\tilde{z}}(t)\mathbb{E}n_{t} + \alpha (t)\varepsilon \bar{b}_{u}(t) (v_{t}- \bar{u}_{t}),\Delta \bar{V}_{t} \bigr\rangle \\ &\quad {}+ \bigl\langle \bigl[\bar{\sigma }_{x}(t)k_{t}+\bar{ \sigma }_{y}(t)m_{t} +\bar{\sigma }_{z}(t)n_{t}+ \bar{\sigma }_{\tilde{x}}(t) \mathbb{E}k_{t} +\bar{\sigma }_{\tilde{y}}(t)\mathbb{E}m_{t} + \bar{\sigma }_{\tilde{z}}(t)\mathbb{E}n_{t} \\ &\quad {} +\alpha (t)\varepsilon \bar{ \sigma }_{u}(t) (v_{t}-\bar{u}_{t})\bigr] \Delta W_{t},\Delta \bar{V}_{t} \bigr\rangle . \end{aligned}$$

Since W, V are martingale processes and V is strongly orthogonal to W, then \(\mathbb{E}[\varPhi _{t}]=0\). Similarly,

$$\begin{aligned} \Delta \langle m_{t},\xi _{t}\rangle =&\langle m_{t},\Delta \xi _{t} \rangle +\langle \Delta m_{t},\xi _{t+1}\rangle \\ =&- \bigl\langle \bar{f}_{\tilde{x}}(t+1)\mathbb{E}k_{t+1}+ \bar{f}_{ \tilde{y}}(t+1)\mathbb{E}m_{t+1}+\bar{f}_{\tilde{z}}(t+1) \mathbb{E}n_{t+1} ,\xi _{t+1} \bigr\rangle \\ &{}- \bigl\langle \bar{f}_{x}(t+1)k_{t+1}+ \bar{f}_{y}(t+1)m_{t+1}+ \bar{f}_{z}(t+1)n_{t+1} \\ &{}+ \alpha (t+1) \bar{f}_{u}(t+1)\varepsilon (v_{t+1}- \bar{u}_{t+1}),\xi _{t+1} \bigr\rangle \\ &{}+ \bigl\langle n_{t}\Delta W_{t}, \bar{f}_{z}'(t)\xi _{t}- \bar{b}_{z}'(t) \eta _{t}-\bar{\sigma }_{z}'(t)\zeta _{t}+ \bar{g}_{z}(t) \\ &{}+\mathbb{E} \bigl[\bar{f}_{\tilde{z}}'(t) \xi _{t}-\bar{b}_{\tilde{z}}'(t)\eta _{t}- \bar{\sigma }_{\tilde{z}}'(t)\zeta _{t} +\bar{g}_{\tilde{z}}(t) \bigr]\Delta W_{t} \bigr\rangle \\ &{}+ \bigl\langle m_{t},\bar{f}_{y}'(t) \xi _{t}-\bar{b}_{y}'(t) \eta _{t}-\bar{\sigma }_{y}'(t)\zeta _{t}+\bar{g}_{y}(t) \\ &{}+\mathbb{E} \bigl[ \bar{f}_{\tilde{y}}'(t)\xi _{t}- \bar{b}_{\tilde{y}}'(t)\eta _{t} -\bar{\sigma }_{\tilde{y}}'(t)\zeta _{t}+ \bar{g}_{\tilde{y}}(t) \bigr] \bigr\rangle +\varUpsilon _{t}, \end{aligned}$$

where

$$\begin{aligned} \varUpsilon _{t}&= \bigl\langle n_{t}\Delta W_{t},\xi _{t}+\bar{f}_{y}'(t) \xi _{t}-\bar{b}_{y}'(t)\eta _{t}-\bar{\sigma }_{y}(t)\zeta _{t}+ \bar{g}_{y}(t) \\ &\quad {}+\mathbb{E} \bigl[\bar{f}_{\tilde{y}}'(t) \xi _{t}- \bar{b}_{\tilde{y}}'(t)\eta _{t}-\bar{\sigma }_{\tilde{y}}(t)\zeta _{t} + \bar{g}_{\tilde{y}}(t) \bigr] \bigr\rangle \\ &\quad {}+ \bigl\langle m_{t}, \bigl\{ \bar{f}_{z}'(t) \xi _{t}-\bar{b}_{z}'(t) \eta _{t}-\bar{\sigma }_{z}'(t)\zeta _{t}+\bar{g}_{z}(t) \\ &\quad {}+\mathbb{E} \bigl[ \bar{f}_{\tilde{z}}'(t)\xi _{t}- \bar{b}_{\tilde{z}}'(t)\eta _{t}- \bar{\sigma }_{\tilde{z}}'(t)\zeta _{t} + \bar{g}_{\tilde{z}}(t) \bigr] \bigr\} \Delta W_{t} \bigr\rangle \\ &\quad {}+ \bigl\langle \xi _{t}+\bar{f}_{y}'(t) \xi _{t}-\bar{b}_{y}'(t) \eta _{t}-\bar{\sigma }_{y}'(t)\zeta _{t}+\bar{g}_{y}(t) \\ &\quad {}+\mathbb{E} \bigl[ \bar{f}_{\tilde{y}}'(t)\xi _{t}- \bar{b}_{\tilde{y}}'(t)\eta _{t}- \bar{\sigma }_{\tilde{y}}'(t)\zeta _{t} + \bar{g}_{\tilde{y}}(t) \bigr], \Delta \bar{N}_{t} \bigr\rangle \\ &\quad {}+ \bigl\langle \bigl\{ \bar{f}_{z}'(t)\xi _{t}-\bar{b}_{z}'(t) \eta _{t}-\bar{\sigma }_{z}'(t)\zeta _{t}+\bar{g}_{z}(t) \\ &\quad {}+\mathbb{E} \bigl[ \bar{f}_{\tilde{z}}'(t)\xi _{t}- \bar{b}_{\tilde{z}}'(t)\eta _{t}- \bar{\sigma }_{\tilde{z}}(t)\zeta _{t} +\bar{g}_{\tilde{z}}(t) \bigr] \bigr\} \Delta W_{t},\Delta \bar{N}_{t} \bigr\rangle . \end{aligned}$$

We further derive

$$\begin{aligned} & \mathbb{E}\Delta \bigl\{ \langle k_{t},\eta _{t} \rangle +\langle m_{t}, \xi _{t}\rangle \bigr\} \\ &\quad =\mathbb{E} \bigl\{ \bigl\langle k_{t+1},-\bar{b}_{x}'(t+1) \eta _{t+1}- \mathbb{E} \bigl[\bar{b}_{\tilde{x}}'(t+1) \eta _{t+1} \bigr] \bigr\rangle + \bigl\langle \bar{b}_{x}(t)k_{t}+ \bar{b}_{\tilde{x}}(t) \mathbb{E}k_{t},\eta _{t} \bigr\rangle \\ &\qquad {}- \bigl\langle k_{t+1},\bar{\sigma }_{x}'(t+1) \zeta _{t+1}+ \mathbb{E} \bigl[\bar{\sigma }_{\tilde{x}}'(t+1) \zeta _{t+1} \bigr] \bigr\rangle + \bigl\langle \bar{\sigma }_{x}(t)k_{t}+\bar{\sigma }_{ \tilde{x}}(t) \mathbb{E}k_{t},\zeta _{t} \bigr\rangle \\ &\qquad {}- \bigl\langle \bar{f}_{y}(t+1)m_{t+1}+ \bar{f}_{\tilde{y}}(t+1) \mathbb{E}m_{t+1},\xi _{t+1} \bigr\rangle + \bigl\langle m_{t}, \bar{f}_{y}'(t) \xi _{t}+\mathbb{E} \bigl[\bar{f}_{\tilde{y}}'(t) \xi _{t} \bigr] \bigr\rangle \\ &\qquad {}- \bigl\langle \bar{f}_{z}(t+1)n_{t+1}+ \bar{f}_{\tilde{z}}(t+1) \mathbb{E}n_{t+1},\xi _{t+1} \bigr\rangle + \bigl\langle n_{t}, \bar{f}_{z}'(t) \xi _{t}+\mathbb{E} \bigl[\bar{f}_{\tilde{z}}'(t) \xi _{t} \bigr] \bigr\rangle \\ &\qquad {}+\varepsilon \bigl\langle \alpha (t)\bar{b}_{u}(t) (v_{t}- \bar{u}_{t}),\eta _{t} \bigr\rangle +\varepsilon \bigl\langle \alpha (t) \bar{\sigma }_{u}(t) (v_{t}-\bar{u}_{t}),\zeta _{t} \bigr\rangle + \bigl\langle \bar{g}_{x}(t+1)+\mathbb{E} \bar{g}_{\tilde{x}}(t+1),k_{t+1} \bigr\rangle \\ &\qquad {}+ \bigl\langle \bar{g}_{z}(t)+\mathbb{E}\bar{g}_{\tilde{z}}(t),n_{t} \bigr\rangle + \bigl\langle \bar{g}_{y}(t)+\mathbb{E} \bar{g}_{ \tilde{y}}(t),m_{t} \bigr\rangle -\varepsilon \bigl\langle \alpha (t+1) \bar{f}_{u}(t+1) (v_{t+1}- \bar{u}_{t+1}),\xi _{t+1} \bigr\rangle \bigr\} . \end{aligned}$$

Accordingly,

$$\begin{aligned} &{-}\mathbb{E} \bigl\langle h_{x} (\bar{X}_{T}, \mathbb{E} \bar{X}_{T} ) +\mathbb{E}h_{\tilde{x}} ( \bar{X}_{T}, \mathbb{E}\bar{X}_{T} ),k_{T} \bigr\rangle \\ &\quad =\mathbb{E} \bigl\{ \langle k_{T},\eta _{T}\rangle + \langle m_{T},\xi _{T} \rangle -\langle k_{0},\eta _{0}\rangle -\langle m_{0},\xi _{0} \rangle \bigr\} =\mathbb{E}\sum_{t=0}^{T-1} \Delta \bigl\{ \langle k_{t},\eta _{t}\rangle +\langle m_{t},\xi _{t}\rangle \bigr\} \\ &\quad =\mathbb{E}\sum_{t=0}^{T-1} \bigl\{ \bigl\langle \bar{g}_{x}(t)+ \mathbb{E}\bar{g}_{\tilde{x}}(t),k_{t} \bigr\rangle + \bigl\langle \bar{g}_{y}(t)+\mathbb{E} \bar{g}_{\tilde{y}}(t),m_{t} \bigr\rangle + \bigl\langle \bar{g}_{z}(t)+\mathbb{E}\bar{g}_{\tilde{z}}(t),n_{t} \bigr\rangle \bigr\} \\ &\qquad {}+\mathbb{E}\sum_{t=0}^{T}\varepsilon \alpha (t) \bigl\langle \bar{b}_{u}'(t)\eta _{t}+\bar{\sigma }_{u}'(t)\zeta _{t}- \bar{f}_{u}'(t)\xi _{t},v_{t}-\bar{u}_{t} \bigr\rangle + \mathbb{E} \bigl\{ \bigl\langle \bar{b}_{x}(0)k_{0}+ \bar{b}_{\tilde{x}}(0) \mathbb{E}k_{0},\eta _{0} \bigr\rangle \\ &\qquad {}+ \bigl\langle k_{0},\bar{\sigma }_{x}'(0) \mathbb{E}\zeta _{0}+ \bar{\sigma }_{\tilde{x}}'(0) \mathbb{E}\zeta _{0} \bigr\rangle + \bigl\langle \bar{f}_{y}(0)m_{0}+\bar{f}_{\tilde{y}}(0) \mathbb{E}m_{0}, \xi _{0} \bigr\rangle + \bigl\langle \bar{f}_{z}(0)n_{0}+\bar{f}_{ \tilde{z}}(0) \mathbb{E}n_{0},\xi _{0} \bigr\rangle \bigr\} . \end{aligned}$$

Since \(k_{0}=0\), \(\xi _{0}=0\), then the above equation leads to

$$\begin{aligned} & \mathbb{E}\sum_{t=0}^{T-1} \bigl\{ \bigl\langle \bar{g}_{x}(t)+ \mathbb{E}\bar{g}_{\tilde{x}}(t),k_{t} \bigr\rangle + \bigl\langle \bar{g}_{y}(t)+\mathbb{E} \bar{g}_{\tilde{y}}(t),m_{t} \bigr\rangle + \bigl\langle \bar{g}_{z}(t)+\mathbb{E}\bar{g}_{\tilde{z}}(t),n_{t} \bigr\rangle \bigr\} \\ &\qquad {} +\mathbb{E} \bigl\langle h_{x} (\bar{X}_{T}, \mathbb{E} \bar{X}_{T} )+\mathbb{E}h_{\tilde{x}} ( \bar{X}_{T}, \mathbb{E}\bar{X}_{T} ),k_{T} \bigr\rangle \\ &\quad =-\varepsilon \alpha \mathbb{E} \bigl\langle \bar{b}_{u}'( \theta ) \eta _{\theta }+\bar{\sigma }_{u}'( \theta )\zeta _{\theta }- \bar{f}_{u}'( \theta )\xi _{\theta },v_{\theta }-\bar{u}_{\theta } \bigr\rangle . \end{aligned}$$

Combining \(\lim_{\varepsilon \rightarrow 0}\frac{1}{\varepsilon }[J(u^{ \varepsilon }_{t})-J(\bar{u}_{t})]\geq 0\) with (3.11), we have

$$ \alpha \mathbb{E} \bigl\langle \bar{b}_{u}'(\theta ) \eta _{\theta }+ \bar{\sigma }_{u}'(\theta ) \zeta _{\theta }- \bar{f}_{u}'(\theta ) \xi _{\theta }-\bar{g}_{u}(\theta ),v_{\theta }- \bar{u}_{\theta } \bigr\rangle \leq 0, $$

then (4.1) holds owing to the arbitrariness of θ. □

Remark 4.1

Theorem 4.1 establishes a more general and constructive stochastic maximum principle under weakened convexity assumption. To be specific, if the set \(U_{t}\) is not convex, then the discrete-time stochastic maximum principles in [15, 21] are invalid. In this sense, our work generalizes and strengthens the discrete-time stochastic maximum principle of the existing works.

Corollary 4.1

Assume that (A2)(A4) hold. Let\(\bar{u}_{t}\)be the optimal control and\((\bar{X}_{t},\bar{Y}_{t},\bar{Z}_{t} )\)be the corresponding optimal trajectory of Problem\(\mathcal{A}\). Moreover, assume that the set\(U_{t}\)isγ-convex, \(t\in \mathbb{T}\), then for\(v_{t}\in U_{t}\), one has

$$ \bigl\langle \mathscr{H}_{u} (t,\bar{X}_{t}, \bar{Y}_{t},\bar{Z}_{t}, \mathbb{E}\bar{X}_{t}, \mathbb{E}\bar{Y}_{t},\mathbb{E}\bar{Z}_{t}, \eta _{t},\zeta _{t},\xi _{t}, \bar{u}_{t} ),v_{t}-\bar{u}_{t} \bigr\rangle \leq 0,\quad \mathbb{P}\textit{-a.s.} $$

Corollary 4.2

Assume that (A2)(A4) hold. Let\(\bar{u}_{t}\)be the optimal control and\((\bar{X}_{t},\bar{Y}_{t},\bar{Z}_{t} )\)be the corresponding optimal trajectory of Problem\(\mathcal{A}\). Moreover, assume that\(U_{t}=\operatorname{ri}U_{t}\), \(t\in \mathbb{T}\), then for\(v_{t}\in U_{t}\), one has

$$ \bigl\langle \mathscr{H}_{u} (t,\bar{X}_{t}, \bar{Y}_{t},\bar{Z}_{t}, \mathbb{E}\bar{X}_{t}, \mathbb{E}\bar{Y}_{t},\mathbb{E}\bar{Z}_{t}, \eta _{t},\zeta _{t},\xi _{t}, \bar{u}_{t} ),v_{t}-\bar{u}_{t} \bigr\rangle =0,\quad \mathbb{P}\textit{-a.s.} $$

In what follows, we discuss assumptions, under which the necessary condition (4.1) turns into a sufficient one.

Theorem 4.2

(Sufficient conditions for optimality)

Under (A2)(A5), assume that\(h(\cdot ,\cdot )\)is convex and\(\mathscr{H}(t,\cdot ,\cdot ,\cdot ,\cdot ,\cdot ,\cdot , \eta _{t}, \zeta _{t},\xi _{t},\cdot )\)is convex. Then\(\bar{u}_{t}\)is an optimal control of Problem\(\mathcal{A}\)if (4.1) holds.

Proof

Let \(u_{t}\) be an arbitrary admissible control and \((X_{t},\mathbb{E}X_{t},Y_{t},\mathbb{E}Y_{t},Z_{t},\mathbb{E}Z_{t})\) be the corresponding trajectory. Set \(\acute{X}_{t}=X_{t}-\bar{X}_{t}\) and \(\acute{Y}_{t}=Y_{t}-\bar{Y}_{t}\). Since \(\acute{X}_{0}=\acute{Y}_{T}=\xi _{0}=0\), it is derived that

$$\begin{aligned} -\mathbb{E} \langle \eta _{T},\acute{X}_{T} \rangle &= \mathbb{E} \bigl\{ \langle \eta _{T},\acute{X}_{T} \rangle + \langle \xi _{T},\acute{Y}_{T} \rangle - \langle \eta _{0},\acute{X}_{0} \rangle - \langle \xi _{0}, \acute{Y}_{0} \rangle \bigr\} \\ &=\mathbb{E}\sum _{t=0}^{T-1} \bigl\{ \Delta \langle \eta _{t},\acute{X}_{t} \rangle + \Delta \langle \xi _{t},\acute{Y}_{t} \rangle \bigr\} \\ &=\mathbb{E}\sum_{t=0}^{T-1} \bigl\{ \bigl\langle \xi _{t},-f(t)+ \bar{f}(t) \bigr\rangle + \bigl\langle \eta _{t},b(t)-\bar{b}(t) \bigr\rangle + \bigl\langle \zeta _{t},\sigma (t)-\bar{\sigma }(t) \bigr\rangle \\ &- \bigl\langle \acute{X}_{t},\bar{\mathscr{H}}_{x}(t)+ \mathbb{E} \bar{\mathscr{H}}_{\tilde{x}}(t) \bigr\rangle - \bigl\langle \acute{Y}_{t}, \bar{\mathscr{H}}_{y}(t) +\mathbb{E}\bar{ \mathscr{H}}_{\tilde{y}}(t) \bigr\rangle - \bigl\langle \acute{Z}_{t}, \bar{\mathscr{H}}_{z}(t) + \mathbb{E}\bar{\mathscr{H}}_{\tilde{z}}(t) \bigr\rangle \bigr\} . \end{aligned}$$

Denote by \((\partial _{(x,y,z,\tilde{x},\tilde{y},\tilde{z},u)} \bar{\mathscr{H}} )(t)\), etc., the Clarke generalized gradients of \(\mathscr{H}\) evaluated at \((\bar{X}_{t},\bar{Y}_{t},\bar{Z}_{t}, \mathbb{E}\bar{X}_{t},\mathbb{E}\bar{Y}_{t}, \mathbb{E}\bar{Z}_{t}, \bar{u}_{t} )\). Together with the stochastic maximum principle (4.1), it follows that

$$ 0\in (\partial _{u}\bar{\mathscr{H}} ) (t),\quad \mbox{a.e. }t. $$

By [31, Lemma 2.3], \(( (\partial _{a}\bar{\mathscr{H}} )(t),0 ) \in (\partial _{a,u}\bar{\mathscr{H}} )(t)\), a.e. t. Besides, from [31, Lemma 2.2(4)], we get

$$\begin{aligned} \mathscr{H}(t) -\bar{\mathscr{H}}(t) &\leq (\partial _{x} \bar{ \mathscr{H}} ) (t)\acute{X}_{t} + (\partial _{ \tilde{x}}\bar{ \mathscr{H}} ) (t)\mathbb{E}\acute{X}_{t} + ( \partial _{y}\bar{\mathscr{H}} ) (t)\acute{Y}_{t} \\ &\quad {}+ (\partial _{\tilde{y}}\bar{\mathscr{H}} ) (t) \mathbb{E} \acute{Y}_{t} + (\partial _{z}\bar{\mathscr{H}} ) (t) \acute{Z}_{t} + (\partial _{\tilde{z}}\bar{\mathscr{H}} ) (t) \mathbb{E}\acute{Z}_{t}. \end{aligned}$$

Therefore, along with the Hamiltonian function, it yields that

$$\begin{aligned} -\mathbb{E} \langle \eta _{T},\acute{X}_{T} \rangle & \leq \mathbb{E}\sum_{t=0}^{T-1} \bigl\{ \bigl\langle \xi _{t},\bar{f}(t)-f(t) \bigr\rangle + \bigl\langle \eta _{t},b(t)-\bar{b}(t) \bigr\rangle + \bigl\langle \zeta _{t},\sigma (t)-\bar{\sigma }(t) \bigr\rangle + \bar{ \mathscr{H}}(t)-\mathscr{H}(t) \bigr\} \\ &=\mathbb{E}\sum_{t=0}^{T-1} \bigl\{ g (t, \varGamma _{t}, \bar{u}_{t},v_{t} )- g (t, \bar{\varGamma }_{t},\bar{u}_{t}, \bar{v}_{t} ) \bigr\} . \end{aligned}$$

On the other hand, using the convexity assumption on \(h(\cdot ,\cdot )\), we claim that

$$ \mathbb{E} \bigl\{ h (X_{T},\mathbb{E}X_{T} )-h ( \bar{X}_{T}, \mathbb{E}\bar{X}_{T} ) \bigr\} \geq \mathbb{E} \biggl\{ \biggl( \frac{\partial \bar{h}}{\partial x} \biggr) (t)\acute{X}_{T} + \biggl( \frac{\partial \bar{h}}{\partial \tilde{x}} \biggr) (t)\mathbb{E} \acute{X}_{T} \biggr\} =\mathbb{E} \langle \eta _{T},\acute{X}_{T} \rangle . $$

Thus,

$$\begin{aligned} \mathbb{E} \bigl\{ h (X_{T},\mathbb{E}X_{T} )-h ( \bar{X}_{T}, \mathbb{E}\bar{X}_{T} ) \bigr\} +\mathbb{E} \sum_{t=0}^{T-1} \bigl\{ g (t,\varGamma _{t},u_{t},v_{t} )- g (t,\bar{\varGamma }_{t}, \bar{u}_{t},\bar{v}_{t} ) \bigr\} \geq 0. \end{aligned}$$

The proof is completed. □

A Stackelberg game of MF-BSΔE

As an application, in this section, we consider a Stackelberg game of MF-BSΔE:

$$ \textstyle\begin{cases} \Delta Y_{t}=-f(t+1,Y_{t+1},Z_{t+1},\mathbb{E}Y_{t+1},\mathbb{E}Z_{t+1},u_{t+1},v_{t+1})+Z_{t} \Delta W_{t}+\Delta M_{t}, \\ Y_{T}=\kappa ,\quad u_{t}, v_{t}\in U_{t}\subseteq \mathbb{R}^{m}. \end{cases} $$
(5.1)

The cost functionals for the follower and the leader are given, respectively, as follows:

$$\begin{aligned}& J_{1}(u_{t},v_{t};\kappa )=\mathbb{E} \Biggl\{ \sum_{t=1}^{T} g_{1}(t,Y_{t},Z_{t}, \mathbb{E}Y_{t}, \mathbb{E}Z_{t},u_{t},v_{t})+h_{1}(Y_{0}, \mathbb{E}Y_{0}) \Biggr\} , \\& J_{2}(u_{t},v_{t};\kappa )=\mathbb{E} \Biggl\{ \sum_{t=1}^{T} g_{2}(t,Y_{t},Z_{t}, \mathbb{E}Y_{t}, \mathbb{E}Z_{t},u_{t},v_{t})+h_{2}(Y_{0}, \mathbb{E}Y_{0}) \Biggr\} . \end{aligned}$$

Here, \(u_{t}\), \(v_{t}\) denote the control processes of the follower and leader, respectively. The admissible control sets are given by

$$\begin{aligned}& \mathscr{U}_{1}[0,T]= \bigl\{ u_{t}, t\in \mathbb{T} \mid u_{t}\in \mathcal{L}^{2}\bigl(\mathfrak{F}_{t}; \mathbb{R}^{m}\bigr),u_{t}\in U_{t} \bigr\} , \\& \mathscr{U}_{2}[0,T]= \bigl\{ v_{t}, t\in \mathbb{T} \mid v_{t}\in \mathcal{L}^{2}\bigl(\mathfrak{F}_{t}; \mathbb{R}^{m}\bigr),v_{t}\in U_{t} \bigr\} . \end{aligned}$$

In this section, we make the following assumptions. Set \(\varGamma =(y,z,\tilde{y},\tilde{z})\), \(i=1, 2\).

  1. (H1)
    1. (a)

      Γ, \(u\in \mathscr{U}_{1}[0,T]\), \(v\in \mathscr{U}_{2}[0,T]\), \(f(\cdot ,\varGamma ,u,v)\), \(g_{i}(\cdot ,\varGamma ,u,v)\) are \(\mathfrak{F}_{t}\)-adapted processes.

    2. (b)

      \(g_{i}(t,0,0,0)\in \mathcal{L}^{2}(\mathfrak{F}_{t};\mathbb{R})\), \(f(t,0,0,0)\in \mathcal{L}^{2}(\mathfrak{F}_{t};\mathbb{R}^{n})\), \(h_{i}(y,\tilde{y})\in \mathcal{L}^{2}(\mathfrak{F}_{0};\mathbb{R})\).

    3. (c)

      \(\forall t\in \mathbb{T}\), \(f(t,\cdot ,\cdot ,\cdot )\), \(g_{i}(t,\cdot ,\cdot ,\cdot )\) are uniformly Lipschitz continuous and differentiable on Γ, u, v; \(h_{i}(\cdot ,\cdot )\) are continuously differentiable on y, and all derivatives are uniformly bounded.

    4. (d)

      The function f is independent of z, at \(t=T\).

  2. (H2)

    The functions \(f(t,\cdot ,\cdot ,\cdot )\), \(g_{i}(t,\cdot ,\cdot ,\cdot )\) are twice continuously differentiable on Γ, u, v; \(h_{i}(\cdot ,\cdot )\) are twice continuously differentiable on y, , and all derivatives are uniformly bounded.

  3. (H3)

    The set \(U_{t}\)\((t\in \mathbb{T})\) is ±γ-convex.

Besides, throughout this section, we formally denote \(f(0,\varGamma ,u,v)=g_{i}(0,\varGamma ,u,v)\equiv 0\), \(i=1,2\). The optimal control problem to be solved can be stated in the following definition.

Definition 5.1

The pair \((\bar{u},\bar{v})\in \mathscr{U}_{1}[0,T]\times \mathscr{U}_{2}[0,T]\) is called an optimal solution to the Stackelberg game of MF-BSΔE if it satisfies the following statements:

  1. (a)

    Given \(\kappa \in \mathcal{L}^{2} (\mathfrak{F};\mathbb{R}^{n} )\), there is a mapping \(l:\mathscr{U}_{2}[0,T]\times \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n}) \rightarrow \mathscr{U}_{1}[0,T]\) such that

    $$ J_{1} \bigl(l(v_{t},\xi ),v_{t};\kappa \bigr)=\min_{u_{t}\in \mathscr{U}_{1}[0,T]}J_{1}(u_{t},v_{t}; \kappa ),\quad \forall v_{t}\in \mathscr{U}_{2}[0,T]. $$
  2. (b)

    There exists unique \(\bar{v}_{t}\in \mathscr{U}_{2}[0,T]\) such that

    $$ J_{2} \bigl(l(\bar{v}_{t},\kappa ), \bar{v}_{t};\kappa \bigr)=\min_{v_{t} \in \mathscr{U}_{2}[0,T]}J_{2} \bigl(l(\bar{v}_{t},\kappa ),v_{t}; \kappa \bigr). $$
  3. (c)

    The optimal strategy of the follower is \(\bar{u}_{t}=l(\bar{v}_{t},\kappa )\).

Optimization for the follower

In view of the hierarchy property of the leader-follower game, the follower’s optimization problem is firstly considered in this subsection. Denote by \(\bar{u}_{t}\) and \((\bar{Y}_{t},\bar{Z}_{t} )\), respectively, the optimal control and optimal trajectory. To begin with, we consider the admissible control \(u^{\varepsilon }_{t}=\bar{u}_{t}+\alpha (t)\varepsilon (u_{t}- \bar{u}_{t})\), where \(u_{t}\in U_{t}\), \(\alpha (t)\in \mathcal{I}(t,u_{t})\). We construct a needle variation

$$ \alpha (t)\varepsilon (u_{t}-\bar{u}_{t})= \textstyle\begin{cases} \alpha \varepsilon (u_{\theta }-\bar{u}_{\theta }),&t= \theta , \\ 0,&t\in \mathbb{T}\setminus \{\theta \}, \end{cases} $$

where \(\alpha \in \mathcal{I}(\theta ,u_{\theta })\), \((\theta ,u_{\theta }) \in T\times U_{\theta }\), and \(\varepsilon \in (0,\gamma ^{\ast }_{1}]\). Let \((Y^{\varepsilon }_{t},Z^{\varepsilon }_{t})\) be the state trajectory corresponding to the control \(u^{\varepsilon }_{t}\). Set \(\varGamma _{t}=(Y_{t},\mathbb{E}Y_{t},Z_{t},\mathbb{E}Z_{t})\). We introduce the following variational equation:

$$ \textstyle\begin{cases} \Delta m_{t}=-f_{y} (t+1,\bar{\varGamma }_{t+1},\bar{u}_{t+1},v_{t+1} )m_{t+1} -f_{z}(t+1,\bar{\varGamma }_{t+1},\bar{u}_{t+1},v_{t+1})n_{t+1} \\ \hphantom{\Delta m_{t}={}}{} -f_{\tilde{y}} (t+1,\bar{\varGamma }_{t+1}, \bar{u}_{t+1},v_{t+1} )\mathbb{E}m_{t+1} -f_{\tilde{z}} (t+1, \bar{\varGamma }_{t+1},\bar{u}_{t+1},v_{t+1} )\mathbb{E}n_{t+1} \\ \hphantom{\Delta m_{t}={}}{}-\alpha (t+1)f_{u} (t+1,\bar{\varGamma }_{t+1},\bar{u}_{t+1},v_{t+1} )\varepsilon (u_{t+1}-\bar{u}_{t+1}) +n_{t}\Delta W_{t}+ \Delta \bar{N}_{t}, \\ m_{T}=0, \end{cases} $$

and the adjoint equation

$$ \textstyle\begin{cases} \Delta X_{t}=f_{y}' (t,\bar{\varGamma }_{t},\bar{u}_{t},v_{t} )X_{t} +g_{1y} (t,\bar{\varGamma }_{t},\bar{u}_{t},v_{t} ) +\mathbb{E} \{ f_{\tilde{y}}' (t,\bar{\varGamma }_{t}, \bar{u}_{t},v_{t} )X_{t} +g_{1\tilde{y}} (t,\bar{\varGamma }_{t}, \bar{u}_{t},v_{t} ) \} \\ \hphantom{\Delta X_{t}={}}{} + \{ f_{z}' (t,\bar{\varGamma }_{t}, \bar{u}_{t},v_{t} )X_{t} +g_{1z} (t,\bar{\varGamma }_{t}, \bar{u}_{t},v_{t} ) \\ \hphantom{\Delta X_{t}={}}{}+\mathbb{E} [f_{\tilde{z}}' (t, \bar{\varGamma }_{t},\bar{u}_{t},v_{t} )X_{t} +g_{1\tilde{z}} (t,\bar{\varGamma }_{t},\bar{u}_{t},v_{t} ) ] \} \Delta W_{t}, \\ X_{0}=h_{1y} (\bar{Y}_{0},\mathbb{E}\bar{Y}_{0} ) +\mathbb{E}h_{1 \tilde{y}} (\bar{Y}_{0},\mathbb{E}\bar{Y}_{0} ). \end{cases} $$

Define the Hamiltonian function

$$ \mathscr{H}_{1}(t,y,z,\tilde{y},\tilde{z},u,v,x)=-\bigl\langle x,f(t,y,z, \tilde{y},\tilde{z},u,v)\bigr\rangle -g_{1}(t,y,z, \tilde{y},\tilde{z},u,v). $$

Using Theorems 4.14.2, we can immediately obtain the following statements.

Theorem 5.1

(Stochastic maximum principle)

Let (H1), (H3) hold and\(\kappa \in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\). Given the leader’s strategy\(v_{t}\in \mathscr{U}_{2}[0,T]\), assume that\((\bar{Y}_{t},\bar{Z}_{t} )\)is the optimal trajectory and\(\bar{u}_{t}\)is the optimal control of the follower, then for any\(u_{t}\in U_{t}\), one has

$$ \alpha (t) \bigl\langle \mathscr{H}_{1u} (t,\bar{Y}_{t}, \bar{Z}_{t}, \mathbb{E}\bar{Y}_{t},\mathbb{E} \bar{Z}_{t}, \bar{u}_{t},v_{t},X_{t} ),u_{t}-\bar{u}_{t} \bigr\rangle \leq 0,\quad \mathbb{P} \textit{-a.s.} $$
(5.2)

Theorem 5.2

(Sufficient conditions for optimality)

Let (H1), (H3) hold and\(\kappa \in \mathcal{L}^{2}(\mathfrak{F};\mathbb{R}^{n})\). Given the leader’s strategy\(v_{t}\in \mathscr{U}_{2}[0,T]\), assume that\(h_{1}(\cdot ,\cdot )\)is convex and\(\mathscr{H}_{1}(t,\cdot ,\cdot ,\cdot ,\cdot , \cdot , v_{t},X_{t})\)is concave and Lipschitz continuous. Then\(\bar{u}_{t}\)is an optimal control of the follower’s problem if it satisfies (5.2).

Optimization for the leader

Notice that the follower’s optimal response \(\bar{u}_{t}\) can be determined by the leader, the state equation of the leader turns to be a MF-FBSΔE:

$$ \textstyle\begin{cases} \Delta X_{t}=f_{y}' (t,\bar{\varGamma }_{t},\bar{u}_{t},v_{t} )X_{t} +g_{1y} (t,\bar{\varGamma }_{t},\bar{u}_{t},v_{t} ) \\ \hphantom{\Delta X_{t}={}}{} +\mathbb{E} \{ f_{\tilde{y}}' (t,\bar{\varGamma }_{t}, \bar{u}_{t},v_{t} )X_{t} +g_{1\tilde{y}} (t,\bar{\varGamma }_{t}, \bar{u}_{t},v_{t} ) \} \\ \hphantom{\Delta X_{t}={}}{} + \{ f_{z}' (t,\bar{\varGamma }_{t},\bar{u}_{t},v_{t} )X_{t} +g_{1z} (t,\bar{\varGamma }_{t},\bar{u}_{t},v_{t} ) \\ \hphantom{\Delta X_{t}={}}{} +\mathbb{E} [f_{\tilde{z}}' (t,\bar{\varGamma }_{t}, \bar{u}_{t},v_{t} )X_{t} +g_{1\tilde{z}} (t,\bar{\varGamma }_{t}, \bar{u}_{t},v_{t} ) ] \} \Delta W_{t}, \\ \Delta \bar{Y}_{t}=-f (t+1,\bar{Y}_{t+1},\bar{Z}_{t+1}, \mathbb{E}\bar{Y}_{t+1},\mathbb{E}\bar{Z}_{t+1},\bar{u}_{t+1},v_{t+1} ) +\bar{Z}_{t}\Delta W_{t}+\Delta \bar{M}_{t}, \\ X_{0}=h_{1y} (\bar{Y}_{0},\mathbb{E}\bar{Y}_{0} ) +\mathbb{E}h_{1 \tilde{y}} (\bar{Y}_{0},\mathbb{E}\bar{Y}_{0} ), \bar{Y}_{T}= \kappa . \end{cases} $$
(5.3)

In this subsection, for \(a=y, \tilde{y}, z, \tilde{z}, v, x, \tilde{x}\), we use the following abbreviations:

$$\begin{aligned} &\bar{b}(t)=f_{y}' (t,\bar{\varGamma }_{t},\bar{u}_{t},\bar{v}_{t} ) \bar{X}_{t} +g_{1y} (t,\bar{\varGamma }_{t}, \bar{u}_{t}, \bar{v}_{t} ) +\mathbb{E} \bigl\{ f_{\tilde{y}}' (t, \bar{\varGamma }_{t}, \bar{u}_{t},\bar{v}_{t} )\bar{X}_{t} +g_{1 \tilde{y}} (t,\bar{\varGamma }_{t},\bar{u}_{t}, \bar{v}_{t} ) \bigr\} , \\ &\bar{\sigma }(t)=f_{z}' (t,\bar{\varGamma }_{t},\bar{u}_{t}, \bar{v}_{t} ) \bar{X}_{t} +g_{1z} (t,\bar{\varGamma }_{t}, \bar{u}_{t},\bar{v}_{t} ) +\mathbb{E} \bigl\{ f_{\tilde{z}}' (t,\bar{\varGamma }_{t}, \bar{u}_{t},\bar{v}_{t} )\bar{X}_{t} +g_{1 \tilde{z}} (t,\bar{\varGamma }_{t},\bar{u}_{t}, \bar{v}_{t} ) \bigr\} , \\ &\bar{\mathscr{H}}_{2a}(t)=\mathscr{H}_{2a} (t,\bar{ \varGamma }_{t}, \bar{u}_{t},\bar{v}_{t}, \bar{X}_{t},\mathbb{E}\bar{X}_{t}, p_{t}, \zeta _{t},q_{t} ), \\ &\bar{f}_{a}(t)=f_{a} (t,\bar{\varGamma }_{t}, \bar{u}_{t},\bar{v}_{t} ),\qquad \bar{g}_{2a}(t)=g_{2a} (t, \bar{\varGamma }_{t},\bar{u}_{t},\bar{v}_{t} ). \end{aligned}$$

Likewise, we consider the admissible control

$$ v^{\varepsilon }_{t}= \textstyle\begin{cases} \bar{v}_{\theta }+\alpha \varepsilon (v_{\theta }-\bar{v}_{\theta }),&t= \theta , \\ 0,&t\in \mathbb{T}\setminus \{\theta \}, \end{cases} $$

where \(\alpha \in \mathcal{I}(\theta ,v_{\theta })\), \((\theta ,v_{\theta }) \in \mathbb{T}\times U_{\theta }\), and \(\varepsilon \in (0,\gamma ^{\ast }_{2}]\). Let \((X^{\varepsilon }_{t},Y^{\varepsilon }_{t},Z^{\varepsilon }_{t})\) be the state trajectory corresponding to the admissible control \(v^{\varepsilon }_{t}\). Let \((k,\eta ,\rho ,N)\) be a solution of the following variational equations:

$$ \textstyle\begin{cases} \Delta k_{t}=\bar{b}_{x}(t)k_{t}+\bar{b}_{y}(t)\eta _{t}+\bar{b}_{z}(t) \rho _{t}+\bar{b}_{\tilde{x}}(t)\mathbb{E}k_{t} +\bar{b}_{\tilde{y}}(t) \mathbb{E}\eta _{t} +\bar{b}_{\tilde{z}}(t)\mathbb{E}\rho _{t}+ \alpha (t)\bar{b}_{v}(t)\varepsilon (v_{t}-\bar{v}_{t}) \\ \hphantom{\Delta k_{t}={}}{} +[\bar{\sigma }_{x}(t)k_{t}+\bar{\sigma }_{y}(t) \eta _{t}+\bar{\sigma }_{z}(t)\rho _{t} +\alpha (t)\bar{\sigma }_{v}(t) \varepsilon (v_{t}-\bar{v}_{t})]\Delta W_{t} \\ \hphantom{\Delta k_{t}={}}{} +[\bar{\sigma }_{\tilde{x}}(t)\mathbb{E}k_{t}+ \bar{\sigma }_{\tilde{y}}(t)\mathbb{E}\eta _{t} +\bar{\sigma }_{ \tilde{z}}(t)\mathbb{E}\rho _{t} ]\Delta W_{t}, \\ \Delta \eta _{t}=-\bar{f}_{y}(t+1)\eta _{t+1}-\bar{f}_{z}(t+1) \rho _{t+1} -\alpha (t+1)\bar{f}_{v}(t+1)\varepsilon (v_{t+1}-\bar{v}_{t+1}) \\ \hphantom{\Delta \eta _{t}={}}{} -\bar{f}_{\tilde{y}}(t+1)\mathbb{E}\eta _{t+1}- \bar{f}_{\tilde{z}}(t+1)\mathbb{E}\rho _{t+1} +\rho _{t}\Delta W_{t}+ \Delta N_{t}, \\ k_{0}= \{ h_{1yy} (\bar{Y}_{0},\mathbb{E}\bar{Y}_{0} )+h_{1y \tilde{y}} (\bar{Y}_{0},\mathbb{E}\bar{Y}_{0} )+ \mathbb{E}h_{1y\tilde{y}} (\bar{Y}_{0},\mathbb{E}\bar{Y}_{0} ) +\mathbb{E}h_{1\tilde{y}\tilde{y}} (\bar{Y}_{0}, \mathbb{E}\bar{Y}_{0} ) \} \eta _{0},\qquad \eta _{T}=0. \end{cases} $$

We define the Hamiltonian

$$\begin{aligned} \mathscr{H}_{2}(t,y,z,\tilde{y},\tilde{z},u,v,x,\tilde{x},p,\zeta ,q)&= \bigl\langle p,b(t,y,z,\tilde{y},\tilde{z},u,v,x,\tilde{x})\bigr\rangle + \bigl\langle \zeta , \sigma (t,y,z,\tilde{y},\tilde{z},u,v,x,\tilde{x}) \bigr\rangle \\ &\quad {} -\bigl\langle q, f(t,y,z,\tilde{y},\tilde{z},u,v)\bigr\rangle -g_{2}(t,y,z, \tilde{y},\tilde{z},u,v). \end{aligned}$$

Similarly, we proceed to introducing the following adjoint equations associated with MF-FBSΔE (5.3):

$$ \textstyle\begin{cases} \Delta p_{t}=-\bar{\mathscr{H}}_{2x}(t)-\mathbb{E}\bar{\mathscr{H}}_{2 \tilde{x}}(t)+\zeta _{t}\Delta W_{t}+\Delta V_{t},\qquad p_{T}=0, \\ \Delta q_{t}=-\bar{\mathscr{H}}_{2y}(t)-\mathbb{E} \bar{\mathscr{H}}_{2\tilde{y}}(t)- \{\bar{\mathscr{H}}_{2z}(t) + \mathbb{E}\bar{\mathscr{H}}_{2\tilde{z}}(t) \}\Delta W_{t}, \\ q_{0}=h_{2y} (\bar{Y}_{0},\mathbb{E}\bar{Y}_{0} )+\mathbb{E}h_{2\tilde{y}} (\bar{Y}_{0},\mathbb{E} \bar{Y}_{0} ) - \{h_{1yy} (\bar{Y}_{0},\mathbb{E} \bar{Y}_{0} )p_{0} +\mathbb{E}h_{1\tilde{y}y} (\bar{Y}_{0}, \mathbb{E}\bar{Y}_{0} )p_{0} \} \\ \hphantom{ q_{0}={}}{}-\mathbb{E} \{h_{1y\tilde{y}} (\bar{Y}_{0},\mathbb{E} \bar{Y}_{0} )p_{0} +\mathbb{E}h_{1\tilde{y}\tilde{y}} ( \bar{Y}_{0},\mathbb{E}\bar{Y}_{0} )p_{0} \}. \end{cases} $$

Here, W, N, V are square integrable martingale processes and N, V are strongly orthogonal to W.

The following conclusions are straightforward with the aid of Theorems 4.14.2.

Theorem 5.3

(Stochastic maximum principle)

Assume that (H1)(H3) hold. Let\(\bar{v}_{t}\)and\((\bar{X}_{t},\bar{Y}_{t},\bar{Z}_{t} )\)be the optimal control, optimal trajectory, respectively. Then, for\(v_{t}\in U_{t}\), one has

$$ \alpha (t) \bigl\langle \mathscr{H}_{2v} (t,\bar{Y}_{t}, \bar{Z}_{t}, \mathbb{E}\bar{Y}_{t},\mathbb{E} \bar{Z}_{t}, \bar{u}_{t},\bar{v}_{t}, \bar{X}_{t},\mathbb{E}\bar{X}_{t},p_{t}, \zeta _{t},q_{t} ),v_{t}- \bar{v}_{t} \bigr\rangle \leq 0,\quad \mathbb{P}\textit{-a.s.} $$
(5.4)

Theorem 5.4

(Sufficient conditions for optimality)

Under (H1)(H3), assume that\(h_{2}(\cdot ,\cdot )\)is convex and\(\mathscr{H}_{2}(t,\cdot ,\cdot ,\cdot ,\cdot , \bar{u}_{t},\cdot , \cdot ,\cdot ,p_{t},\zeta _{t},q_{t})\)is convex, then\(\bar{v}_{t}\)is an optimal control of the leader’s problem if (5.4) holds.

Perspectives and open problems

In this section, we give a brief exposition on the prospects that are open to the researchers. The following topics shall be explored in our future works.

Firstly, we see that the effectiveness of optimality conditions obtained in this paper substantially relies on the structure of the set \(U_{t}\). So it is pregnant to discuss more general and essential convexity assumptions for the discrete-time forward–backward stochastic system.

Secondly, there are many more partially observable cases which are more constructive and inevitable for applications and are technologically demanding in their filtering procedure.

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Acknowledgements

The authors would like to thank the anonymous referees and editor very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.

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This work is partially supported by the National Natural Science Foundation of China (Grant No. 11971185).

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Song, T., Liu, B. A maximum principle for fully coupled controlled forward–backward stochastic difference systems of mean-field type. Adv Differ Equ 2020, 188 (2020). https://doi.org/10.1186/s13662-020-02640-x

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MSC

  • 49J21
  • 93E20
  • 93C55
  • 49K45
  • 49M05

Keywords

  • Forward–backward stochastic difference equations
  • Backward stochastic difference equations
  • Mean-field theory
  • Stochastic maximum principle
  • Adjoint difference equation