On Forward–Backward Stochastic Differential Equations in a Domination-Monotonicity Framework

Abstract

In this paper, inspired by various stochastic linear-quadratic (LQ, for short) problems, we develop the method of continuation to study nonlinear forward–backward stochastic differential equations (FBSDEs, for short) in a kind of domination-monotonicity frameworks. The coupling of FBSDEs is in a general form, i.e., it not only appears in integral terms and terminal terms, but also in initial terms. By virtue of introducing various matrices, matrix-valued random variables and matrix-valued stochastic processes, we present the domination-monotonicity framework carefully and rigorously. A unique solvability result and a pair of estimates for coupled FBSDEs are obtained (see Theorem 3.5 in the case of simple domination-monotonicity conditions and Theorem 5.2 in the case of multi-level self-similar domination-monotonicity structures). As applications of theoretical results, the related stochastic Hamiltonian systems of several LQ problems are discussed.

Appendix A: A Sketch of the Proof of Theorem 5.2

As we stated at the end of Sect. 5, the induction method and the continuation method will be combined together to give the proof of Theorem 5.2. Clearly, Theorem 3.5 means that Theorem 5.2 holds true when $p=1$. Next, for any $p\ge 2$, we assume that Theorem 5.2 holds for $p-1$, then we need prove the conclusions also hold for p.

We notice that there are two cases in the monotonicity conditions at each qth level ($q=1,2,\dots ,p$). Overall, we have $2^p$ situations. In the following, we only consider one of them: All of the monotonicity conditions take Cases 1. The other $2^p-1$ situations can be treated in the same way or in the viewpoint of transformation (see Remark 3.3).

As the start of the method of continuation, now we define a set of coefficients $(\Psi ^0,\Phi ^0,\Gamma ^0)$. In detail, let

$$\begin{aligned} \begin{aligned} {\mathcal {M}}_1 =&\mu _1{{\widetilde{M}}}_1^\top {{\widetilde{M}}}_1, \qquad {\mathcal {G}}_1 = \nu _1{{\widetilde{G}}}_1^\top {{\widetilde{G}}}_1, \\ {\mathcal {L}}_1(\cdot ) =&\big ( \sqrt{\nu _1} {{\widetilde{A}}}_1(\cdot ),\ \sqrt{\mu _1}{{\widetilde{B}}}_1(\cdot ),\ \sqrt{\mu _1}\widetilde{C}_{1,1}(\cdot ),\ \cdots ,\ \sqrt{\mu _1}{{\widetilde{C}}}_{1,d}(\cdot ) \big )^\top \\&\quad \times \big ( \sqrt{\nu _1} {{\widetilde{A}}}_1(\cdot ),\ \sqrt{\mu _1}{{\widetilde{B}}}_1(\cdot ),\ \sqrt{\mu _1}\widetilde{C}_{1,1}(\cdot ),\ \cdots ,\ \sqrt{\mu _1}{{\widetilde{C}}}_{1,d}(\cdot ) \big ), \end{aligned} \end{aligned}$$

(A.1)

and

$$\begin{aligned} {\mathcal {M}}_q= & {} \mu _q\Lambda M_q^\top M_q,\quad {\mathcal {G}}_q = \mu _q\Lambda G_q^\top G_q,\quad {\mathcal {L}}_q(\cdot ) =\mu _q \Lambda L_q(\cdot )^\top L_q(\cdot ),\nonumber \\&q=2,3,\dots ,p. \end{aligned}$$

(A.2)

Moreover, we denote

$$\begin{aligned}&\left\{ \begin{aligned}&{\mathbb {M}}_1 = \text{ diag }\ \{ -{\mathcal {M}}_1,\ {\mathcal {M}}_1,\ \cdots ,\ -{\mathcal {M}}_1,\ {\mathcal {M}}_1 \},\\&{\mathbb {M}}_q = \text{ diag }\ \left\{ \begin{pmatrix} O &{} -{\mathcal {M}}_q \\ O &{} O \end{pmatrix},\ \begin{pmatrix} O &{} {\mathcal {M}}_q \\ O &{} O \end{pmatrix},\ \cdots ,\ \begin{pmatrix} O &{} -{\mathcal {M}}_q \\ O &{} O \end{pmatrix},\ \begin{pmatrix} O &{} {\mathcal {M}}_q \\ O &{} O \end{pmatrix} \right\} ,\\&\quad q=2,3,\dots ,p-1,\\&{\mathbb {M}}_p = \begin{pmatrix} O &{} -{\mathcal {M}}_p \\ O &{} O \end{pmatrix}, \end{aligned} \right. \end{aligned}$$

(A.3)

$$\begin{aligned}&\left\{ \begin{aligned}&{\mathbb {G}}_1 = \text{ diag }\ \{ {\mathcal {G}}_1,\ -{\mathcal {G}}_1,\ \cdots ,\ {\mathcal {G}}_1,\ -{\mathcal {G}}_1 \},\\&{\mathbb {G}}_q = \text{ diag }\ \left\{ \begin{pmatrix} O &{} {\mathcal {G}}_q \\ O &{} O \end{pmatrix},\ \begin{pmatrix} O &{} -{\mathcal {G}}_q \\ O &{} O \end{pmatrix},\ \cdots ,\ \begin{pmatrix} O &{} {\mathcal {G}}_q \\ O &{} O \end{pmatrix},\ \begin{pmatrix} O &{} -{\mathcal {G}}_q \\ O &{} O \end{pmatrix} \right\} ,\\&\quad q=2,3,\dots ,p-1,\\&{\mathbb {G}}_p = \begin{pmatrix} O &{} {\mathcal {G}}_p \\ O &{} O \end{pmatrix}, \end{aligned} \right. \end{aligned}$$

(A.4)

and

$$\begin{aligned} \left\{ \begin{aligned}&{\mathbb {L}}_1(\cdot ) = \text{ diag }\ \{ -{\mathcal {L}}_1(\cdot ),\ {\mathcal {L}}_1(\cdot ),\ \cdots ,\ -{\mathcal {L}}_1(\cdot ),\ {\mathcal {L}}_1(\cdot ) \},\\&{\mathbb {L}}_q(\cdot ) = \text{ diag }\ \left\{ \begin{pmatrix} O &{} -{\mathcal {L}}_q(\cdot ) \\ O &{} O \end{pmatrix},\ \begin{pmatrix} O &{} {\mathcal {L}}_q(\cdot ) \\ O &{} O \end{pmatrix},\ \cdots ,\ \begin{pmatrix} O &{} -{\mathcal {L}}_q(\cdot ) \\ O &{} O \end{pmatrix},\ \begin{pmatrix} O &{} {\mathcal {L}}_q(\cdot ) \\ O &{} O \end{pmatrix} \right\} ,\\&\quad q=2,3,\dots ,p-1,\\&{\mathbb {L}}_p(\cdot ) = \begin{pmatrix} O &{} -{\mathcal {L}}_p(\cdot ) \\ O &{} O \end{pmatrix}. \end{aligned} \right. \end{aligned}$$

(A.5)

We notice that there are $2^{p-q}$ non-zero submatrices involved in the above defined ${\mathbb {M}}_q$, ${\mathbb {G}}_q$ and $\mathbb L_q(\cdot )$ respectively for any $q=1,2,\dots ,p$. Furthermore, we define

$$\begin{aligned} \Psi ^0(y) = \bigg [ \sum _{q=1}^p {\mathbb {M}}_q \bigg ] y,\qquad \Phi ^0(x) = \bigg [ \sum _{q=1}^p {\mathbb {G}}_q \bigg ] x,\qquad \Gamma ^0(s,\theta ) = \bigg [ \sum _{q=1}^p {\mathbb {L}}_q(s) \bigg ] \theta \end{aligned}$$

(A.6)

for any $(\omega ,s,\theta ) \in \Omega \times [0,T] \times \mathbb R^{2^{p-1}n(2+d)}$. For the sake of intuition, let us give the form of the matrix $\sum _{q=1}^4 {\mathbb {M}}_q$ when $p=4$ as follows:

where the blanks mean being filled by zeros. The matrix-valued random variable $\sum _{q=1}^4 {\mathbb {G}}_q$ and the matrix-valued stochastic process $\sum _{q=1}^4 {\mathbb {L}}_q(\cdot )$ are in similar forms. Similar to [7, Lemma 2.7], we can directly verify that the coefficients $(\Psi ^0, \Phi ^0, \Gamma ^0)$ defined by (A.6) satisfy Assumptions (H$^p_0$)—(H$^p_p$) with all Cases 1.

Now, for any $(\xi ,\eta ,\rho (\cdot )) \in {\mathcal {H}}[0,T;p]$ with $\rho (\cdot ) = (\varphi (\cdot )^\top ,\psi (\cdot )^\top ,\gamma (\cdot )^\top )^\top$, we introduce a family of FBSDEs parameterized by $\alpha \in [0,1]$ on the interval [0, T] as follows:

$$\begin{aligned} \left\{ \begin{aligned}&\mathrm dx_i^\alpha (s) = \big [ b_i^\alpha (s,\theta ^\alpha (s)) +\psi _i(s) \big ]\, \mathrm ds +\sum _{j=1}^d \big [ \sigma _{ij}^\alpha (s,\theta ^\alpha (s)) +\gamma _{ij}(s) \big ]\, \mathrm dW_j(s),\\&\mathrm dy_i^\alpha (s) = \big [ g_i^\alpha (s,\theta ^\alpha (s)) +\varphi _i(s) \big ]\, \mathrm ds +\sum _{j=1}^d z_{ij}^\alpha (s)\, \mathrm dW_j(s),\\&x^\alpha _i(0) = \Psi ^\alpha _i(y^\alpha (0)) +\xi _i, \qquad y^\alpha _i(T) = \Phi ^\alpha _i(x^\alpha (T)) +\eta _i,\\&i=1,2,\dots , 2^{p-1}, \end{aligned} \right. \end{aligned}$$

(A.7)

where the coefficients $(\Psi ^\alpha , \Phi ^\alpha , \Gamma ^\alpha )$ are defined as a convex combination of $(\Psi ^0,\Phi ^0,\Gamma ^0)$ (see (A.6)) and the original set of coefficients $(\Psi ,\Phi ,\Gamma )$:

$$\begin{aligned} (\Psi ^\alpha (\cdot ), \Phi ^\alpha (\cdot ), \Gamma ^\alpha (\cdot ,\cdot )) = (1-\alpha ) (\Psi ^0(\cdot ), \Phi ^0(\cdot ), \Gamma ^0(\cdot ,\cdot )) +\alpha (\Psi (\cdot ), \Phi (\cdot ), \Gamma (\cdot ,\cdot )). \end{aligned}$$

(A.8)

Since both $(\Psi ^0,\Phi ^0,\Gamma ^0)$ and $(\Psi ,\Phi ,\Gamma )$ satisfy Assumptions (H$^p_0$)—(H$^p_p$) with all Cases 1, then it is clear that $(\Psi ^\alpha , \Phi ^\alpha , \Gamma ^\alpha )$ (and then $(\Psi ^\alpha +\xi , \Phi ^\alpha +\eta , \Gamma ^\alpha +\rho )$) also satisfy Assumptions (H$^p_0$)—(H$^p_p$) with all Cases 1.

When $\alpha =0$, due to the delicate setting (A.6) of $(\Psi ^0,\Phi ^0,\Gamma ^0)$, FBSDE (A.7) is actually in a decoupled form with any given $(\xi ,\eta ,\rho (\cdot )) \in {\mathcal {H}}[0,T;p]$, therefore it is unique solvable. The detailed verification is very similar to [7, Lemma 3.3] and is omitted. Another fact is that, when $\alpha =1$ and $(\xi ,\eta ,\rho (\cdot ))$ vanish, FBSDE (A.7) coincides with FBSDE (5.1). From the known $\alpha =0$ case to the unknown $\alpha =1$ case, we need to prove a continuity lemma similar to Lemma 3.7. In the establishing of the continuity lemma, the following a priori estimate will play a key role.

Lemma A.1

Let $p\in {\mathbb {N}}$ and a set of coefficients $(\Psi ,\Phi ,\Gamma )$ are given to satisfy Assumptions (H$^p_0$)—(H$^p_p$) with all Cases 1. Let $\alpha \in [0,1]$ and $(\xi ,\eta ,\rho (\cdot ))$, $({{\bar{\xi }}}, {{\bar{\eta }}},{{\bar{\rho }}}(\cdot )) \in {\mathcal {H}}[0,T;p]$. Suppose that $\theta (\cdot )$, ${{\bar{\theta }}}(\cdot ) \in (M^2_{\mathbb F}(0,T;{\mathbb {R}}^{n(2+d)}))^{2^{p-1}}$ satisfy FBSDE (A.7) with the coefficients $(\Psi ^\alpha +\xi ,\Phi ^\alpha +\eta , \Gamma ^\alpha +\rho )$ and $(\Psi ^\alpha +{{\bar{\xi }}},\Phi ^\alpha +{{\bar{\eta }}}, \Gamma ^\alpha +{{\bar{\rho }}})$, respectively. Then the estimate (3.13) holds, where the constant K depending on p, T, the Lipschitz constants, $\mu _1$, $\nu _1$, $\mu _q$, and the bounds of all ${{\widetilde{M}}}_1$, ${{\widetilde{G}}}_1$, $\widetilde{A}_1(\cdot )$, ${{\widetilde{B}}}_1(\cdot )$, ${{\widetilde{C}}}_{1,j}(\cdot )$, $M_q$, $G_q$, $L_q(\cdot )$, $j=1,2,\dots ,d$, $q=2,3,\dots ,p$.

Proof

We split the index set $\{1,2,3,\dots ,2^{p-1}\}$ into two groups. Similar to Sect. 5, we denote

$$\begin{aligned} \kappa _1 =1, \quad \kappa _2 = 2^{p-2}, \quad \kappa _3 = 2^{p-2}+1, \quad \kappa _4 = 2^{p-1}. \end{aligned}$$

(A.9)

Firstly, due to the self-similar property of the domination-monotonicity structure, the following holds true: Being frozen $\theta _{[\kappa _3, \kappa _4]}$, both $(\Psi ^\alpha +\xi ,\Phi ^\alpha +\eta , \Gamma ^\alpha +\rho )_{[\kappa _1,\kappa _2]}$ and $(\Psi ^\alpha +{{\bar{\xi }}},\Phi ^\alpha +{{\bar{\eta }}}, \Gamma ^\alpha +{{\bar{\rho }}})_{[\kappa _1,\kappa _2]}$ satisfy Assumptions (H$^{p-1}_0$)—(H$^{p-1}_{p-1}$) with all Cases 1. This fact, with the help of Theorem 5.2 for $(p-1)$, leads to

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \bigg [ \sup _{s\in [0,T]} \big | \widehat{x}_{[\kappa _1,\kappa _2]}(s) \big |^2 +\sup _{s\in [0,T]} \big | \widehat{y}_{[\kappa _1,\kappa _2]}(s) \big |^2 +\int _0^T \big | \widehat{z}_{[\kappa _1, \kappa _2]}(s) \big |^2\, \mathrm ds \bigg ]\\&\quad \le \ K {\mathbb {E}}\bigg \{ {{\widehat{J}}}_{[\kappa _1, \kappa _2]} +\text{ I } +\text{ II } +\int _0^T \text{ III }\, \mathrm ds \bigg \}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} {{\widehat{J}}}_{[\kappa _1, \kappa _2]} =\,&\big | \widehat{\xi }_{[\kappa _1,\kappa _2]} \big |^2 +\big | \widehat{\eta }_{[\kappa _1,\kappa _2]} \big |^2 +\bigg ( \int _0^T \big | {{\widehat{\varphi }}}_{[\kappa _1,\kappa _2]}(s) \big |\, \mathrm ds \bigg )^2\\&+\bigg ( \int _0^T \big | {{\widehat{\psi }}}_{[\kappa _1,\kappa _2]}(s) \big |\, \mathrm ds \bigg )^2 +\int _0^T \big |{{\widehat{\gamma }}}_{[\kappa _1,\kappa _2]}(s)\big |^2\, \mathrm ds, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{aligned} \text{ I } =\,&\Big | \Psi ^\alpha _{[\kappa _1, \kappa _2]}\big ( \bar{y}_{[\kappa _1,\kappa _2]}(0), y_{[\kappa _3,\kappa _4]}(0) \big ) -\Psi ^\alpha _{[\kappa _1, \kappa _2]}\big ( \bar{y}_{[\kappa _1,\kappa _2]}(0), {{\bar{y}}}_{[\kappa _3,\kappa _4]}(0) \big ) \Big |^2,\\ \text{ II } =\,&\Big | \Phi ^\alpha _{[\kappa _1, \kappa _2]}\big ( {{\bar{x}}}_{[\kappa _1,\kappa _2]}(T), x_{[\kappa _3,\kappa _4]}(T) \big ) -\Phi ^\alpha _{[\kappa _1, \kappa _2]}\big ( {{\bar{x}}}_{[\kappa _1,\kappa _2]}(T), {{\bar{x}}}_{[\kappa _3,\kappa _4]}(T) \big ) \Big |^2,\\ \text{ III } =\,&\Big | \Gamma ^\alpha _{[\kappa _1, \kappa _2]}\big ( s, {{\bar{\theta }}}_{[\kappa _1,\kappa _2]}(s), \theta _{[\kappa _3,\kappa _4]}(s) \big ) -\Gamma ^\alpha _{[\kappa _1, \kappa _2]}\big ( s, {{\bar{\theta }}}_{[\kappa _1,\kappa _2]}(s), {{\bar{\theta }}}_{[\kappa _3,\kappa _4]}(s) \big ) \Big |^2. \end{aligned} \right. \end{aligned}$$

By the definition (A.8) of $(\Psi ^\alpha ,\Phi ^\alpha ,\Gamma ^\alpha )$ and the domination conditions (see Assumption (H$^p_p$)-(i)), we calculate

$$\begin{aligned} \begin{aligned} \text{ I } =\,&\Big | -(1-\alpha )\mu _p\Lambda M_p^\top M_p {{\widehat{y}}}_{[\kappa _3,\kappa _4]}(0)\\&+\alpha \Big [ \Psi _{[\kappa _1,\kappa _2]}\big ( \bar{y}_{[\kappa _1,\kappa _2]}(0), y_{[\kappa _3,\kappa _4]}(0) \big ) -\Psi _{[\kappa _1,\kappa _2]}\big ( {{\bar{y}}}_{[\kappa _1,\kappa _2]}(0), {{\bar{y}}}_{[\kappa _3,\kappa _4]}(0) \big ) \Big ] \Big |^2\\ \le \,&K \big | M_p {{\widehat{y}}}_{[\kappa _3,\kappa _4]}(0) \big |^2. \end{aligned} \end{aligned}$$

Similarly,

$$\begin{aligned} \text{ II } \le K \big | G_p \widehat{x}_{[\kappa _3,\kappa _4]}(T)\big |^2,\qquad \text{ III } \le K \big | L_p(s){{\widehat{\theta }}}_{[\kappa _3,\kappa _4]}(s) \big |^2. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \bigg [ \sup _{s\in [0,T]} \big | \widehat{x}_{[\kappa _1,\kappa _2]}(s) \big |^2 +\sup _{s\in [0,T]} \big | \widehat{y}_{[\kappa _1,\kappa _2]}(s) \big |^2 +\int _0^T \big | \widehat{z}_{[\kappa _1, \kappa _2]}(s) \big |^2\, \mathrm ds \bigg ]\\&\quad \le \ K {\mathbb {E}}\bigg \{ {{\widehat{J}}}_{[\kappa _1, \kappa _2]} +\big | M_p {{\widehat{y}}}_{[\kappa _3,\kappa _4]}(0) \big |^2 +\big | G_p {{\widehat{x}}}_{[\kappa _3,\kappa _4]}(T) \big |^2 +\int _0^T \big | L_p(s){{\widehat{\theta }}}_{[\kappa _3,\kappa _4]}(s) \big |^2\, \mathrm ds \bigg \}. \end{aligned} \end{aligned}$$

(A.10)

Secondly, being frozen $\theta _{[\kappa _1,\kappa _2]}$, it is clear that, both $(\Psi ^\alpha +\xi ,\Phi ^\alpha +\eta , \Gamma ^\alpha +\rho )_{[\kappa _3,\kappa _4]}$ and $(\Psi ^\alpha +{{\bar{\xi }}},\Phi ^\alpha +{{\bar{\eta }}}, \Gamma ^\alpha +{{\bar{\rho }}})_{[\kappa _3,\kappa _4]}$ satisfy Assumptions (H$^{p-1}_0$)—(H$^{p-1}_{p-2}$) with all Cases 1 and (H$^{p-1}_{p-1}$) with Case 2. Theorem 5.2 for $(p-1)$ works once again to yields

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\bigg [ \sup _{s\in [0,T]} \big | \widehat{x}_{[\kappa _3,\kappa _4]}(s) \big |^2 +\sup _{s\in [0,T]} \big | \widehat{y}_{[\kappa _3,\kappa _4]}(s) \big |^2 +\int _0^T \big | \widehat{z}_{[\kappa _3,\kappa _4]}(s) \big |^2\, \mathrm ds \bigg ]\\&\quad \le \ K {\mathbb {E}}\bigg \{ {{\widehat{J}}}_{[\kappa _3, \kappa _4]} +\text{ IV } +\text{ V } +\int _0^T \text{ VI }\, \mathrm ds \bigg \}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} {{\widehat{J}}}_{[\kappa _3, \kappa _4]} =\,&\big | \widehat{\xi }_{[\kappa _3,\kappa _4]} \big |^2 +\big | \widehat{\eta }_{[\kappa _3,\kappa _4]} \big |^2 +\bigg ( \int _0^T \big | {{\widehat{\varphi }}}_{[\kappa _3,\kappa _4]}(s) \big |\, \mathrm ds \bigg )^2\\&+\bigg ( \int _0^T \big | {{\widehat{\psi }}}_{[\kappa _3,\kappa _4]}(s) \big |\, \mathrm ds \bigg )^2 +\int _0^T \big |{{\widehat{\gamma }}}_{[\kappa _3,\kappa _4]}(s)\big |^2\, \mathrm ds, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{aligned} \text{ IV } =\,&\Big | \Psi ^\alpha _{[\kappa _3, \kappa _4]}\big ( y_{[\kappa _1,\kappa _2]}(0), {{\bar{y}}}_{[\kappa _3,\kappa _4]}(0) \big ) -\Psi ^\alpha _{[\kappa _3, \kappa _4]}\big ( \bar{y}_{[\kappa _1,\kappa _2]}(0), {{\bar{y}}}_{[\kappa _3,\kappa _4]}(0) \big ) \Big |^2,\\ \text{ V } =\,&\Big | \Phi ^\alpha _{[\kappa _3, \kappa _4]}\big ( x_{[\kappa _1,\kappa _2]}(T), {{\bar{x}}}_{[\kappa _3,\kappa _4]}(T) \big ) -\Phi ^\alpha _{[\kappa _3, \kappa _4]}\big ( \bar{x}_{[\kappa _1,\kappa _2]}(T), {{\bar{x}}}_{[\kappa _3,\kappa _4]}(T) \big ) \Big |^2,\\ \text{ VI } =\,&\Big | \Gamma ^\alpha _{[\kappa _3, \kappa _4]}\big ( s, \theta _{[\kappa _1,\kappa _2]}(s), {{\bar{\theta }}}_{[\kappa _3,\kappa _4]}(s) \big ) -\Gamma ^\alpha _{[\kappa _3, \kappa _4]}\big ( s, \bar{\theta }_{[\kappa _1,\kappa _2]}(s), \bar{\theta }_{[\kappa _3,\kappa _4]}(s) \big ) \Big |^2. \end{aligned} \right. \end{aligned}$$

By the definition (A.8) of $(\Psi ^\alpha ,\Phi ^\alpha ,\Gamma ^\alpha )$ and the Lipschitz conditions (see Assumption (H$^p_0$)-(ii)), we calculate

$$\begin{aligned} \begin{aligned} \text{ IV } =\,&\Big | \alpha \Big [ \Psi _{[\kappa _3,\kappa _4]}\big ( y_{[\kappa _1,\kappa _2]}(0), {{\bar{y}}}_{[\kappa _3,\kappa _4]}(0) \big ) -\Psi _{[\kappa _3,\kappa _4]}\big ( {{\bar{y}}}_{[\kappa _1,\kappa _2]}(0), {{\bar{y}}}_{[\kappa _3,\kappa _4]}(0) \big ) \Big ] \Big |^2\\ \le \,&K\big | {{\widehat{y}}}_{[\kappa _1, \kappa _2]} (0) \big |^2 \le K \sup _{s\in [0,T]} \big | {{\widehat{y}}}_{[\kappa _1, \kappa _2]} (0) \big |^2. \end{aligned} \end{aligned}$$

Similarly,

$$\begin{aligned} \text{ V }\le & {} K \big | {{\widehat{x}}}_{[\kappa _1, \kappa _2]}(T) \big |^2 \le K \sup _{s\in [0,T]}\big | {{\widehat{x}}}_{[\kappa _1, \kappa _2]}(s) \big |^2, \\ \text{ VI }\le & {} K \big | {{\widehat{\theta }}}_{[\kappa _1,\kappa _2]} \big |^2 \le K \bigg [ \sup _{s\in [0,T]}\big | {{\widehat{x}}}_{[\kappa _1, \kappa _2]}(s) \big |^2 +\sup _{s\in [0,T]}\big | {{\widehat{y}}}_{[\kappa _1, \kappa _2]}(s) \big |^2 +\big | {{\widehat{z}}}_{[\kappa _1, \kappa _2]}(s) \big |^2 \bigg ]. \end{aligned}$$

Then,

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\bigg [ \sup _{s\in [0,T]} \big | \widehat{x}_{[\kappa _3,\kappa _4]}(s) \big |^2 +\sup _{s\in [0,T]} \big | \widehat{y}_{[\kappa _3,\kappa _4]}(s) \big |^2 +\int _0^T \big | \widehat{z}_{[\kappa _3,\kappa _4]}(s) \big |^2\, \mathrm ds \bigg ]\\&\quad \le \ K {\mathbb {E}}\bigg \{ {{\widehat{J}}}_{[\kappa _3, \kappa _4]} +\sup _{s\in [0,T]}\big | {{\widehat{x}}}_{[\kappa _1, \kappa _2]}(s) \big |^2 +\sup _{s\in [0,T]}\big | {{\widehat{y}}}_{[\kappa _1, \kappa _2]}(s) \big |^2 + \int _0^T \big | {{\widehat{z}}}_{[\kappa _1, \kappa _2]}(s) \big |^2\, \mathrm ds \bigg \}. \end{aligned} \end{aligned}$$

(A.11)

By combining (A.10) and (A.11), we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\bigg [ \sup _{s\in [0,T]} \big | {{\widehat{x}}}(s) \big |^2 +\sup _{s\in [0,T]} \big | {{\widehat{y}}}(s) \big |^2 +\int _0^T \big | {{\widehat{z}}}(s) \big |^2\, \mathrm ds \bigg ]\\&\quad \le \ K_3 {\mathbb {E}} \bigg \{ {{\widehat{J}}} +\big | M_p \widehat{y}_{[\kappa _3,\kappa _4]}(0) \big |^2 +\big | G_p \widehat{x}_{[\kappa _3,\kappa _4]}(T) \big |^2 +\int _0^T \big | L_p(s){{\widehat{\theta }}}_{[\kappa _3,\kappa _4]}(s) \big |^2\, \mathrm ds \bigg \}, \end{aligned} \end{aligned}$$

(A.12)

where ${{\widehat{J}}}$ is defined by (3.14).

Thirdly, applying Itô’s formula to $\langle {{\widehat{x}}}(\cdot ),\ \Lambda {{\widehat{y}}}(\cdot ) \rangle = \sum _{j=\kappa _1}^{\kappa _4} \langle {{\widehat{x}}}_j(\cdot ),\ \widehat{y}_{\kappa _1+\kappa _4-j}(\cdot ) \rangle$ yields

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \big \langle \Phi ^\alpha (x(T)) -\Phi ^\alpha ({{\bar{x}}}(T)) +{{\widehat{\eta }}},\ \Lambda {{\widehat{x}}}(t) \big \rangle - \big \langle \Psi ^\alpha (y(0)) -\Psi ^\alpha ({{\bar{y}}}(0)) +\widehat{\xi },\ \Lambda {{\widehat{y}}}(0) \big \rangle \\&\quad =\ {\mathbb {E}} \int _0^T \big \langle \Gamma ^\alpha (s,\theta (s)) -\Gamma (s,{{\bar{\theta }}}(s)) +{{\widehat{\rho }}}(s),\ \Lambda {{\widehat{\theta }}}(s) \big \rangle \, \mathrm ds. \end{aligned} \end{aligned}$$

By the monotonicity conditions of $(\Psi ^\alpha ,\Phi ^\alpha ,\Gamma ^\alpha )$ (see(H$^p_p$)-(iii)-Case 1), we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \bigg \{ \big | M_p {{\widehat{y}}}_{[\kappa _3,\kappa _4]}(0) \big |^2 +\big | G_p {{\widehat{x}}}_{[\kappa _3,\kappa _4]}(T) \big |^2 +\int _0^T \big | L_p(s){{\widehat{\theta }}}_{[\kappa _3,\kappa _4]}(s) \big |^2\, \mathrm ds \bigg \}\\&\quad \le \ \frac{1}{\mu _p} {\mathbb {E}}\bigg \{ \big \langle {{\widehat{\xi }}},\ \Lambda {{\widehat{y}}}(0) \big \rangle -\big \langle {{\widehat{\eta }}},\ \Lambda {{\widehat{x}}}(T) \big \rangle +\int _0^T \big \langle {{\widehat{\rho }}}(s),\ \Lambda {{\widehat{\theta }}}(s) \big \rangle \, \mathrm ds \bigg \}. \end{aligned} \end{aligned}$$

Similar to (3.18), by Young’s inequality, for any $\varepsilon >0$, we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \bigg \{ \big | M_p {{\widehat{y}}}_{[\kappa _3,\kappa _4]}(0) \big |^2 +\big | G_p {{\widehat{x}}}_{[\kappa _3,\kappa _4]}(T) \big |^2 +\int _0^T \big | L_p(s){{\widehat{\theta }}}_{[\kappa _3,\kappa _4]}(s) \big |^2\, \mathrm ds \bigg \}\\&\quad \le \ \frac{1}{4\mu _p\varepsilon } {\mathbb {E}}\Big [ {{\widehat{J}}} \Big ] +\frac{2\varepsilon }{\mu _p} {\mathbb {E}}\bigg [ \sup _{s\in [0,T]} \big | {{\widehat{x}}}(s) \big |^2 +\sup _{s\in [0,T]} \big | {{\widehat{y}}}(s) \big |^2 +\int _0^T \big | {{\widehat{z}}}(s) \big |^2\, \mathrm ds \bigg ]. \end{aligned} \end{aligned}$$

(A.13)

By taking $\varepsilon =\mu _p/(4K_3)$ where $K_3$ is the constant appearing in (A.12), we obtain the a priori estimate (3.13) from (A.12) and (A.13). $\square$

Next, with the help of Lemma A.1, we establish the following continuation lemma.

Lemma A.2

Let $p\in {\mathbb {N}}$ and a set of coefficients $(\Psi ,\Phi ,\Gamma )$ are given to satisfy Assumptions (H$^p_0$)—(H$^p_p$) with all Cases 1. Then there exists an absolute constant $\delta _0>0$ such that, if for some $\alpha _0\in [0,1)$, FBSDE (A.7) is unique solvable in the space $(M^2_{\mathbb F}(0,T;{\mathbb {R}}^{n(2+d)}))^{2^{p-1}}$ for any $(\xi ,\eta ,\rho (\cdot )) \in {\mathcal {H}}[0,T;p]$, then the same result is also true for $\alpha =\alpha _0+\delta$ with $\delta \in [0,\delta _0]$ and $\alpha \le 1$.

Proof

Let $\delta _0>0$ be determined below. Let $\delta \in [0,\delta _0]$. Similar to the proof Lemma 3.7, we shall define a mapping through FBSDEs with $\alpha =\alpha _0$. In detail, for any $\theta (\cdot ) \in (M^2_{{\mathbb {F}}}(0,T;\mathbb R^{n(2+d)}))^{2^{p-1}}$, we consider an FBSDE with unknown $\Theta (\cdot ) = (X(\cdot )^\top ,Y(\cdot )^\top ,Z(\cdot )^\top )^\top$ as follows:

$$\begin{aligned} \left\{ \begin{aligned}&\mathrm dX_i(s) = \Big [ b^{\alpha _0}_i\big (s, \Theta (s)\big ) +{{\widetilde{\psi }}}_i(s) \Big ]\, \mathrm ds +\sum _{j=1}^d \Big [ \sigma _{ij}^{\alpha _0}\big (s,\Theta (s)\big ) +{{\widetilde{\gamma }}}_{ij}(s) \Big ]\, \mathrm dW_j(s),\\&\mathrm dY_i(s) = \Big [ g^{\alpha _0}_i \big ( s, \Theta (s) \big ) +{{\widetilde{\varphi }}}_i(s) \Big ]\, \mathrm ds +\sum _{j=1}^d Z_{ij}(s)\, \mathrm dW_j(s),\\&X_i(0) =\Psi ^{\alpha _0}_i \big (Y(0)\big ) +{{\widetilde{\xi }}}_i, \qquad Y_i(T) =\Phi ^{\alpha _0}_i \big (X(T)\big ) +{{\widetilde{\eta }}}_i,\\&i=1,2,\dots ,2^{p-1}, \end{aligned} \right. \end{aligned}$$

(A.14)

where

$$\begin{aligned} \left\{ \begin{aligned}&{{\widetilde{\xi }}} =\delta \bigg [ \Psi \big ( y(0)\big ) -\sum _{q=1}^p {\mathbb {M}}_q y(0) \bigg ] +\xi , \\&{{\widetilde{\eta }}} =\delta \bigg [ \Phi \big ( x(T) \big ) -\sum _{q=1}^p {\mathbb {G}}_q x(T) \bigg ] +\eta ,\\&{{\widetilde{\rho }}}(\cdot ) = \delta \bigg [ \Gamma \big (\cdot ,\theta (\cdot )\big ) -\sum _{q=1}^p {\mathbb {L}}_q(\cdot )\theta (\cdot ) \bigg ] +\rho (\cdot ). \end{aligned} \right. \end{aligned}$$

(A.15)

By our assumption, FBSDE (A.14) is unique solvable in $(M^2_{{\mathbb {F}}}(0,T;{\mathbb {R}}^{n(2+d)}))^{2^{p-1}}$. Therefore, we have established a mapping $\mathscr {T}_{\alpha _0+\delta }$ by

$$\begin{aligned} \Theta (\cdot ) ={\mathscr {T}}_{\alpha _0+\delta }\big (\theta (\cdot )\big ) : (M^2_{{\mathbb {F}}}(0,T;{\mathbb {R}}^{n(2+d)}))^{2^{p-1}} \rightarrow (M^2_{{\mathbb {F}}}(0,T;{\mathbb {R}}^{n(2+d)}))^{2^{p-1}}. \end{aligned}$$

By carefully observing the definition (A.8) of the coefficients $(\Psi ^{\alpha _0+\delta },\Phi ^{\alpha _0+\delta },\Gamma ^{\alpha _0+\delta })$, it is clear that the fixed point of the mapping $\mathscr {T}_{\alpha _0+\delta }$ is equivalent to the solution of FBSDE (A.7) with $\alpha =\alpha _0+\delta$. Next, we shall prove that the mapping ${\mathscr {T}}_{\alpha _0+\delta }$ admits a unique fixed point by Banach’s Contraction Mapping Theorem.

For any given $\theta (\cdot )$, ${{\bar{\theta }}}(\cdot ) \in (M^2_{\mathbb F}(0,T;{\mathbb {R}}^{n(2+d)}))^{2^{p-1}}$, denote $\Theta (\cdot ) ={\mathscr {T}}_{\alpha _0+\delta }(\theta (\cdot ))$ and ${{\bar{\Theta }}}(\cdot ) ={\mathscr {T}}_{\alpha _0+\delta }({{\bar{\theta }}}(\cdot ))$. Moreover, we denote ${{\widehat{\theta }}}(\cdot ) = \theta (\cdot ) -{{\bar{\theta }}}(\cdot )$, ${{\widehat{\Theta }}}(\cdot ) = \Theta (\cdot ) -{{\bar{\Theta }}}(\cdot )$, and so on. We apply Lemma A.1 to get

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\bigg [ \sup _{s\in [0,T]} \big |{{\widehat{X}}}(s)\big |^2 +\sup _{s\in [0,T]} \big |{{\widehat{Y}}}(s)\big |^2 + \int _0^T \big | {{\widehat{Z}}}(s) \big |\, \mathrm ds \bigg ]\\&\quad \le \ K\delta ^2 {\mathbb {E}}\Bigg \{ \bigg | \Psi \big ( y(0) \big ) -\Psi \big ( {{\bar{y}}}(0) \big ) -\sum _{q=1}^p {\mathbb {M}}_q {{\widehat{y}}}(0) \bigg |^2\\&\qquad +\bigg | \Phi \big ( x(T) \big ) -\Phi \big ( {{\bar{x}}}(T) \big ) -\sum _{q=1}^p {\mathbb {G}}_q {{\widehat{x}}}(T) \bigg |^2\\&\qquad +\int _0^T \bigg | \Gamma \big ( s,\theta (s) \big ) -\Gamma \big ( s,{{\bar{\theta }}}(s) \big ) -\sum _{q=1}^p {\mathbb {L}}_q(s) {{\widehat{\theta }}}(s) \bigg |^2\, \mathrm ds \Bigg \}\\&\quad \le \ K_4\delta ^2 {\mathbb {E}} \bigg \{ \sup _{s\in [0,T]} \big |{{\widehat{x}}}(s)\big |^2 +\sup _{s\in [0,T]} \big |\widehat{y}(s)\big |^2 + \int _0^T \big | {{\widehat{z}}}(s) \big |\, \mathrm ds \bigg \}. \end{aligned} \end{aligned}$$

Select $\delta _0 =1/(2\sqrt{K_4})$. Then, for any $\delta \in [0,\delta _0]$, the above inequality implies that $\mathscr {T}_{\alpha _0+\delta }$ is a contraction. The proof is finished. $\square$

Similar to the proof of Theorem 3.5, with the help of Lemmas A.1 and A.2, we can easily prove Theorem 5.2 under Assumptions (H$^p_0$)–(H$^p_p$) with all Cases 1.