Appendix
This appendix is devoted to the proofs of Lemmas 8 and 9, which we give for the reader’s convenience. The main idea is the same as in Lemma 6 or [19, Theorem 4.1].
1.1 A.1 Proof of Lemma 8
In view of Lemma 1 and Remark 6, we have for any \(m\ge 1\),
$$\begin{aligned} Y^{(m)}_t= y^{(m)}_t+\sup \limits _{t\le s\le T}L_s\big (y^{(m)}_s\big ),\quad \forall t\in [0,T], \end{aligned}$$
(22)
where \(y^{(m)}_t\) is the solution to the following quadratic BSDE
$$\begin{aligned} y^{(m)}_t=\xi +\int ^{T}_t f\left( s,Y^{(m-1)}_s,\textbf{P}_{Y^{(m-1)}_s},z^{(m)}_s\right) ds-\int ^{T}_t z^{(m)}_s dB_s. \end{aligned}$$
(23)
Applying assertion (i) of Lemma 5 and (19) yields for any \(t\in [0,T]\),
$$\begin{aligned} \exp \left\{ {\gamma }\big \vert y^{(m)}_t\big \vert \right\}&\le \textbf{E}_t\bigg [\exp \bigg \{\gamma \bigg (\vert \xi \vert +\int ^T_0 \alpha _sds+\beta (T-t)\bigg (\sup \limits _{s\in [t,T]}\big \vert Y^{(m-1)}_{s}\big \vert \nonumber \\&\quad +\sup \limits _{s\in [t,T]}\textbf{E}\left[ \big \vert Y^{(m-1)}_{s}\big \vert \right] \bigg )\bigg )\bigg \}\bigg ]. \end{aligned}$$
(24)
Using Doob’s maximal inequality, we get for each \(m\ge 1, p\ge 2 \) and \(t\in [0,T]\),
$$\begin{aligned}&\textbf{E}\bigg [\exp \bigg \{{p\gamma }\sup \limits _{s\in [t,T]}\big \vert y^{(m)}_s\big \vert \bigg \}\bigg ]\nonumber \\&\quad \le 4 \textbf{E}\bigg [\exp \bigg \{p\gamma \bigg (\vert \xi \vert +\int ^T_0 \alpha _sds+\beta (T-t)\bigg (\sup \limits _{s\in [t,T]}\vert Y^{(m-1)}_{s}\vert \nonumber \\&\qquad +\sup \limits _{s\in [t,T]}\textbf{E}\big [\big \vert Y^{(m-1)}_{s}\big \vert \big ]\bigg )\bigg )\bigg \} \bigg ]. \end{aligned}$$
(25)
Recalling (22), we have
$$\begin{aligned} \big \vert Y^{(m)}_t\big \vert \le \big \vert y^{(m)}_t\big \vert +\sup \limits _{0\le s\le T}\vert L_s(0)\vert +\kappa \sup \limits _{t\le s\le T}\textbf{E}\big [\vert y^{(m)}_s\vert \big ]. \end{aligned}$$
Set \(\widetilde{\alpha }=\sup \limits _{0\le s\le T}\vert L_s(0)\vert +\int ^T_0 \alpha _sds\). Using Jensen’s inequality, we get for any \(m\ge 1\), \(p\ge 2 \) and \(t\in [0,T]\),
$$\begin{aligned}&\textbf{E}\left[ \exp \bigg \{{p\gamma }\sup \limits _{s\in [t,T]}\big \vert Y^{(m)}_s\big \vert \bigg \}\right] \\&\quad \le e^{p\gamma \sup \limits _{0\le s\le T}\vert L_s(0)\vert }\,\textbf{E}\left[ \exp \big \{{p\gamma }\sup \limits _{s\in [t,T]}\big \vert y^{(m)}_s\big \vert \big \}\right] \textbf{E}\left[ \exp \big \{{\kappa p\gamma }\sup \limits _{s\in [t,T]}\big \vert y^{(m)}_s\big \vert \big \}\right] \\&\quad \le e^{p\gamma \sup \limits _{0\le s\le T}\vert L_s(0)\vert }\,\textbf{E}\left[ \exp \big \{(2+2\kappa ){p\gamma }\sup \limits _{s\in [t,T]}\big \vert y^{(m)}_s\big \vert \big \}\right] \\&\quad \le 4 \textbf{E}\left[ \exp \bigg \{(2+2\kappa )p\gamma \bigg (\vert \xi \vert +\widetilde{\alpha }+\beta (T-t)\sup \limits _{s\in [t,T]}\vert Y^{(m-1)}_{s}\vert \bigg )\bigg \}\right] \\&\qquad \times \textbf{E}\left[ \exp \bigg \{(2+2\kappa )p\gamma \beta (T-t)\sup \limits _{s\in [t,T]}\vert Y^{(m-1)}_{s}\vert \bigg \} \right] . \end{aligned}$$
Choose a constant \(h\in (0,T]\) depending only on \(\beta \) and \(\kappa \) such that
$$\begin{aligned} (32+64\kappa )\beta h <1. \end{aligned}$$
(26)
In view of Hölder’s inequality, we obtain that for any \(p\ge 2\),
$$\begin{aligned}&\textbf{E}\left[ \exp \bigg \{{p\gamma }\sup \limits _{s\in [T-h,T]}\big \vert Y^{(m)}_s\big \vert \bigg \}\right] \nonumber \\&\quad \le 4 \textbf{E}\left[ \exp \bigg \{{(4+4\kappa ) p\gamma }(\vert \xi \vert +\widetilde{\alpha })\bigg \}\right] ^{\frac{1}{2}}\nonumber \\&\quad \times \textbf{E}\left[ \exp \bigg \{(4+4\kappa )\beta hp\gamma \sup \limits _{s\in [T-h,T]}\big \vert Y^{(m-1)}_{s}\big \vert \bigg \} \right] \nonumber \\&\quad \le 4\textbf{E}\left[ \exp \bigg \{{(8+8\kappa ) p\gamma }\vert \xi \vert \bigg \}\right] ^{\frac{1}{4}}\textbf{E}\left[ \exp \bigg \{{(8+8\kappa ) p\gamma }\widetilde{\alpha }\bigg \}\right] ^{\frac{1}{4}} \nonumber \\&\qquad \times \textbf{E}\left[ \exp \bigg \{p\gamma \sup \limits _{s\in [T-h,T]}\big \vert Y^{(m-1)}_{s}\big \vert \bigg \} \right] ^{(4+4\kappa )\beta h}. \end{aligned}$$
(27)
Define \( \rho = \frac{1}{1-(4+4\kappa )\beta h}\) and
$$\begin{aligned} \mu := {\left\{ \begin{array}{ll} \frac{T}{h}, &{}\text {if}\quad \frac{T}{h} \text { is an integer };\\ {[}\frac{T}{h}{]}+1, &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$
If \(\mu =1\), it follows from (27) that for each \(p\ge 2\) and \(m\ge 1\),
$$\begin{aligned}&\textbf{E}\bigg [\exp \bigg \{p\gamma \sup \limits _{s\in [0,T]}\big \vert Y^{(m)}_s\big \vert \bigg \} \bigg ] \\&\quad \le 4\textbf{E}\bigg [\exp \bigg \{{(8+8\kappa ) p\gamma }\vert \xi \vert \bigg \}\bigg ]^{\frac{1}{4}}\textbf{E}\bigg [\exp \bigg \{{(8+8\kappa ) p\gamma }\widetilde{\alpha }\bigg \}\bigg ]^{\frac{1}{4}}\\&\qquad \times \textbf{E}\bigg [\exp \bigg \{{p\gamma } \sup \limits _{s\in [0,T]}\vert Y^{(m-1)}_s\vert \bigg \} \bigg ]^{(4+4\kappa )\beta h}. \end{aligned}$$
Iterating the above procedure m times yields, given the definition of \(\rho \),
$$\begin{aligned}&\textbf{E}\bigg [\exp \bigg \{{p\gamma }\sup \limits _{s\in [0,T]}\big \vert Y^{(m)}_s\big \vert \bigg \} \bigg ]\nonumber \\&\quad \le 4^{\rho }\textbf{E}\bigg [\exp \bigg \{{(8+8\kappa ) p\gamma } \vert \xi \vert \bigg \}\bigg ]^{\frac{\rho }{4}}\textbf{E}\bigg [\exp \bigg \{{(8+8\kappa ) p\gamma } \widetilde{\alpha }\bigg \}\bigg ]^{\frac{\rho }{4}}, \end{aligned}$$
(28)
which is uniformly bounded with respect to m thanks to assumptions (H1\(''\)) and (H2\(''\)). If \(\mu = 2\), proceeding identically as above, we have for any \(p\ge 2\),
$$\begin{aligned}&\textbf{E}\bigg [\exp \bigg \{{p\gamma }\sup \limits _{s\in [T-h,T]}\big \vert Y^{(m)}_s\big \vert \bigg \} \bigg ]\nonumber \\&\quad \le 4^{\rho }\textbf{E}\bigg [\exp \bigg \{{(8+8\kappa )p\gamma } \vert \xi \vert \bigg \}\bigg ]^{\frac{\rho }{4}}\textbf{E}\bigg [\exp \bigg \{{(8+8\kappa ) p\gamma } \widetilde{\alpha }\bigg \}\bigg ]^{\frac{\rho }{4}}. \end{aligned}$$
(29)
We then consider the following quadratic reflected BSDE over the time interval \([0,T-h]\):
$$\begin{aligned} {\left\{ \begin{array}{ll} Y_t^{(m)}=Y_{T-h}^{(m)}+\int _t^{T-h} f^{Y^{(m-1)}}(s, Z^{(m)}_s)ds-\int _t^{T-h} Z^{(m)}_sdB_s+K^{(m)}_{{T-h}}-K^{(m)}_t,\\ \textbf{E}[\ell (t,Y^{(m)}_t)]\ge 0,\quad \forall t\in [0,T-h] \,\, \text{ and } \,\, \int _0^{T-h} \textbf{E}[\ell (t,Y^{(m)}_t)] dK^{(m)}_t = 0. \end{array}\right. } \end{aligned}$$
According to the derivation of (28), we deduce that
$$\begin{aligned}&\textbf{E}\bigg [\exp \bigg \{{p\gamma }\sup \limits _{s\in [0,T-h]}\big \vert Y^{(m)}_s\big \vert \bigg \} \bigg ]\\&\quad \le 4^{\rho }\textbf{E}\bigg [\exp \bigg \{{(8+8\kappa ) p\gamma }\big \vert Y_{T-h}^{(m)}\big \vert \bigg \}\bigg ]^{\frac{\rho }{4}}\textbf{E}\bigg [\exp \bigg \{{(8+8\kappa ) p\gamma } \widetilde{\alpha }\bigg \}\bigg ]^{\frac{\rho }{4}}\\&\quad \le 4^{\rho +{\frac{\rho ^2}{4}}}\textbf{E}\bigg [\exp \bigg \{({8+8\kappa })^2 p\gamma \vert \xi \vert \bigg \}\bigg ]^{\frac{\rho ^2}{16}}\textbf{E}\bigg [\exp \bigg \{({8+8\kappa })^2 p\gamma \widetilde{\alpha }\bigg \}\bigg ]^{\frac{\rho ^2}{16}}\\&\qquad \times \textbf{E}\bigg [\exp \bigg \{{(8+8\kappa ) p\gamma } \widetilde{\alpha }\bigg \}\bigg ]^{\frac{\rho }{4}}, \end{aligned}$$
where we used (29) in the last inequality. Putting the above inequalities together and applying Hölder’s inequality again yields for any \(p\ge 2\),
$$\begin{aligned}&\textbf{E}\bigg [\exp \bigg \{{p\gamma }\sup \limits _{s\in [0,T]}\big \vert Y^{(m)}_s\big \vert \bigg \}\bigg ]\nonumber \\&\quad \le \textbf{E}\bigg [\exp \bigg \{{2p\gamma }\sup \limits _{s\in [0,T-h]}\vert Y^{(m)}_s\vert \bigg \} \bigg ]^{\frac{1}{2}} \textbf{E}\bigg [\exp \bigg \{{2p\gamma }\sup \limits _{s\in [T-h,T]}\vert Y^{(m)}_s\vert \bigg \} \bigg ]^{\frac{1}{2}}\nonumber \\&\quad \le 4^{\rho +{\frac{\rho ^2}{8}}}\textbf{E}\bigg [\exp \bigg \{(8+8\kappa )^2 2p\gamma \vert \xi \vert \bigg \}\bigg ]^{\frac{\rho }{8}+\frac{\rho ^2}{32}}\textbf{E}\bigg [\exp \bigg \{(8+8\kappa )^2 2p\gamma \widetilde{\alpha }\bigg \}\bigg ]^{\frac{\rho }{4}+\frac{\rho ^2}{32}}, \end{aligned}$$
(30)
which is also uniformly bounded with respect to m.
Iterating the above procedure \(\mu \) times in the general case, we get
$$\begin{aligned} \sup \limits _{m\ge 0}\textbf{E}\bigg [\exp \bigg \{p\gamma \sup \limits _{s\in [0,T]}\vert Y^{(m)}_s\vert \bigg \} \bigg ]<\infty , \ \forall p\ge 1, \end{aligned}$$
which together with (25) implies that
$$\begin{aligned} \sup \limits _{m\ge 0}\textbf{E}\bigg [\exp \bigg \{p\gamma \sup \limits _{s\in [0,T]}\vert y^{(m)}_s\vert \bigg \} \bigg ]<\infty , \quad \forall p\ge 1. \end{aligned}$$
(31)
It follows from Lemma 1 and assumption (H4) that
$$\begin{aligned} \sup \limits _{m\ge 0}K^{(m)}_T\le \sup \limits _{0\le s\le T}\vert L_s(0)\vert +\sup \limits _{m\ge 0}\textbf{E}\bigg [\sup \limits _{s\in [0,T]}\left| y^{(m)}_s\right| \bigg ]<\infty . \end{aligned}$$
Finally, noting \(Z^{(m)}=z^{(m)}\) and applying [11, Corollary 4] to the quadratic BSDE (23) leads to
$$\begin{aligned} \sup \limits _{m\ge 0}\textbf{E}\left[ \bigg (\int ^T_0\left| Z^{(m)}_t\right| ^2dt\bigg )^p \right] <\infty , \ \forall p\ge 1, \end{aligned}$$
which ends the proof.
1.2 A.2 Proof of Lemma 9
Without loss of generality, assume \(f(t,y,v,\cdot )\) is concave, since the other case can be proved by a similar analysis, as discussed in Remark 7. For each fixed \(m,q\ge 1\) and \(\theta \in (0,1)\), we can define similarly \(\delta _{\theta }\ell ^{(m,q)}\), \(\delta _{\theta }\widetilde{\ell }^{(m,q)}\) and \(\delta _{\theta }\overline{\ell }^{(m,q)}\) for y, z. Then, the pair of processes \((\delta _{\theta }y^{(m,q)},\delta _{\theta }z^{(m,q)})\) satisfies the following BSDE:
$$\begin{aligned} \delta _{\theta }y^{(m,q)}_t=&-\xi +\int ^T_t\left( \delta _{\theta }f^{(m,q)}\left( s,\delta _{\theta }z^{(m,q)}_s\right) +\delta _{\theta }f_0^{(m,q)}(s)\right) ds-\int ^T_t\delta _{\theta }z^{(m,q)}_sdB_s, \end{aligned}$$
(32)
where the generator is given by
$$\begin{aligned} \delta _{\theta }f_0^{(m,q)}(t)\!&=\! \frac{1}{1-\theta }\left( f\left( t,Y^{(m+q-1)}_{t},\textbf{P}_{Y^{(m+q-1)}_{t}}, z^{(m)}_t\right) \!-\!f\left( t,Y^{(m-1)}_{t},\textbf{P}_{Y^{(m-1)}_{t}}, z^{(m)}_t\right) \right) ,\\ \delta _{\theta }f^{(m,q)}(t,z)\!&=\!\frac{1}{1-\theta }\bigg ( \theta f\left( t,Y^{(m+q-1)}_{t},\textbf{P}_{Y^{(m+q-1)}_{t}}, z^{(m+q)}_t\right) \! \\&\quad - \! f\left( t,Y^{(m+q-1)}_{t},\textbf{P}_{Y^{(m+q-1)}_{t}}, -(1-\theta )z+\theta z^{(m+q)}_t\right) \bigg ). \end{aligned}$$
From assumption \((H2'')\), we get
$$\begin{aligned} \delta _{\theta }f_0^{(m,q)}(t)&\le \beta \left( \vert Y^{(m+q-1)}_{t}\vert +\vert \delta _{\theta }Y^{(m-1,q)}_t\vert +\textbf{E}\bigg [\left| Y^{(m+q-1)}_{t}\right| +\left| \delta _{\theta }Y^{(m-1,q)}_t\right| \bigg ]\right) ,\\ \delta _{\theta }f^{(m,q)}(t,z)&\le -f\left( t,Y^{(m+q-1)}_t,\textbf{P}_{Y^{(m+q-1)}_{t}}, -z\right) \\&\le \alpha _t+\beta \left( \left| Y^{(m+q-1)}_t\right| +\textbf{E}\bigg [\left| Y^{(m+q-1)}_{t}\right| \bigg ]\right) +\frac{\gamma }{2}\vert z\vert ^2. \end{aligned}$$
Set \(C_3:=2\sup \limits _{m}\textbf{E}\big [\sup \limits _{s\in [0,T]}\vert Y^{(m)}_{s}\vert \big ]<\infty \) (see Lemma 8) and for any \(m,q\ge 1\), denote
$$\begin{aligned} \zeta ^{(m,q)}&=\vert \xi \vert +\beta T C_3+\int ^T_0\alpha _sds+\beta T\left( \sup \limits _{s\in [0,T]}\left| Y^{(m-1)}_{s}\right| +\sup \limits _{s\in [0,T]}\left| Y^{(m+q-1)}_{s}\right| \right) ,\\ \chi ^{(m,q)}&=2\beta T C_3+\int ^T_0\alpha _sds+2 \beta T \left( \sup \limits _{s\in [0,T]}\left| Y^{(m+q-1)}_{s}\right| +\sup \limits _{s\in [0,T]}\left| Y^{(m-1)}_{s}\right| \right) . \end{aligned}$$
Applying assertion (ii) of Lemma 5 to (32) yields for any \(p\ge 1\),
$$\begin{aligned}&\exp \left\{ {p\gamma }\big (\delta _{\theta }y^{(m,q)}_t\big )^+\right\} \\&\quad \le \textbf{E}_t\exp \bigg \{p\gamma \bigg (\vert \xi \vert +\chi ^{(m,q)}+\beta (T-t)\bigg (\sup \limits _{s\in [t,T]}\vert \delta _{\theta }Y^{(m-1,q)}_{s}\vert \\&\qquad +\sup \limits _{s\in [t,T]}\textbf{E}[\vert \delta _{\theta }Y^{(m-1,q)}_{s}\vert ] \bigg )\bigg )\bigg \} \end{aligned}$$
and in a similar way, we also have
$$\begin{aligned} \exp \left\{ {p\gamma }\big (\delta _{\theta }\widetilde{y}^{(m,q)}_t\big )^+\right\}&\le \textbf{E}_t\bigg [\exp \bigg \{p\gamma \bigg (\vert \xi \vert +\chi ^{(m,q)}+\beta (T-t)\bigg (\sup \limits _{s\in [t,T]}\vert \delta _{\theta }\widetilde{Y}^{(m-1,q)}_{s}\vert \\&\quad +\sup \limits _{s\in [t,T]}\textbf{E}[\vert \delta _{\theta }\widetilde{Y}^{(m-1,q)}_{s}\vert ] \bigg )\bigg )\bigg )\bigg \}\bigg ]. \end{aligned}$$
According to the fact that
$$\begin{aligned} \big (\delta _{\theta }{y}^{(m,q)}\big )^- \le \big (\delta _{\theta }\widetilde{y}^{(m,q)}\big )^++2\vert y^{(m)}\vert \ \text {and}\ \big (\delta _{\theta }\widetilde{y}^{(m,q)}\big )^- \le \big (\delta _{\theta }{y}^{(m,q)}\big )^++2\vert y^{(m+q)}\vert , \end{aligned}$$
we derive, using Hölder’s inequality and (24), that
$$\begin{aligned}&\exp \left\{ p\gamma \big \vert \delta _{\theta }{y}^{(m,q)}_t\big \vert \right\} \vee \exp \left\{ p\gamma \big \vert \delta _{\theta }\widetilde{y}^{(m,q)}_t\big \vert \right\} \\&\quad \le \exp \left\{ {p\gamma }\bigg (\bigg ( \delta _{\theta }{y}^{(m,q)}_t\bigg )^++\bigg (\delta _{\theta }\widetilde{y}^{(m,q)}_t\bigg )^++2\left| y^{(m)}_t\right| +2\left| y^{(m+q)}_t\right| \bigg )\right\} \\&\quad \le \textbf{E}_t\bigg [\exp \bigg \{p\gamma \bigg (\vert \xi \vert +\chi ^{(m,q)}+\beta (T-t)\bigg (\sup \limits _{s\in [t,T]}\delta _{\theta }\overline{Y}^{(m-1,q)}_{s}\\&\qquad +\sup \limits _{s\in [t,T]}\textbf{E}\left[ \delta _{\theta }\overline{Y}^{(m-1,q)}_{s}\right] \bigg )\bigg )\bigg \} \bigg ]^2\exp \left\{ {2p\gamma }\bigg (\left| y^{(m)}_t\right| +\left| y^{(m+q)}_t\right| \bigg )\right\} \\&\quad \le \textbf{E}_t\bigg [\exp \bigg \{p\gamma \bigg (\vert \xi \vert +\chi ^{(m,q)}+\beta (T-t)\bigg (\sup \limits _{s\in [t,T]}\delta _{\theta }\overline{Y}^{(m-1,q)}_{s}\\&\qquad +\sup \limits _{s\in [t,T]}\textbf{E}\left[ \delta _{\theta }\overline{Y}^{(m-1,q)}_{s}\right] \bigg )\bigg )\bigg \}\bigg ]^2 \textbf{E}_t\left[ \exp \bigg \{4p\gamma \zeta ^{(m,q)} \bigg \} \right] . \end{aligned}$$
In view of Doob’s maximal inequality and Hölder’s inequality, we obtain that for all \(p> 1\) and \(t\in [0,T]\)
$$\begin{aligned}&\mathbf {{E}}\left[ \exp \bigg \{p\gamma \sup \limits _{s\in [t,T]}\delta _{\theta }\overline{y}^{(m,q)}_s\bigg \}\right] \\&\quad \le 4 \textbf{E}\bigg [\exp \bigg \{8p\gamma \bigg (\vert \xi \vert +{\chi }^{(m,q)} +\beta (T-t)\bigg (\sup \limits _{s\in [t,T]}\delta _{\theta }\overline{Y}^{(m-1,q)}_{s}\\&\qquad +\sup \limits _{s\in [t,T]}\textbf{E}\left[ \delta _{\theta }\overline{Y}^{(m-1,q)}_{s}\right] \bigg ) \bigg )\bigg \}\bigg ]^{\frac{1}{2}} \textbf{E}\left[ \exp \bigg \{16p\gamma \zeta ^{(m,q)} \bigg \} \right] ^{\frac{1}{2}}\\&\quad \le 4 \textbf{E}\left[ \exp \bigg \{8p\gamma \bigg (\vert \xi \vert +{\chi }^{(m,q)} +\beta (T-t)\sup \limits _{s\in [t,T]}\delta _{\theta }\overline{Y}^{(m-1,q)}_{s} \bigg )\bigg \} \right] ^{\frac{1}{2}} \\&\qquad \times \textbf{E}\left[ \exp \bigg \{8\beta (T-t) p\gamma \sup \limits _{s\in [t,T]}\delta _{\theta }\overline{Y}^{(m-1,q)}_{s} \bigg )\bigg \} \right] ^{\frac{1}{2}} \textbf{E}\left[ \exp \bigg \{16p\gamma \zeta ^{(m,q)} \bigg \} \right] ^{\frac{1}{2}}. \end{aligned}$$
Set \( C_4:=\sup \limits _{0\le s\le T}\vert L_s(0)\vert +2\kappa \sup \limits _{m}\textbf{E}\big [\sup \limits _{s\in [0,T]}\vert y^{(m)}_{s}\vert \big ]<\infty \) (see (31)). Recalling (22) and assumption (H4),
$$\begin{aligned} \delta _{\theta }\overline{Y}^{(m,q)}_t\le \delta _{\theta }\overline{y}^{(m,q)}_t+2\kappa \sup \limits _{t\le s\le T}\textbf{E}\left[ \delta _{\theta }\overline{y}^{(m,q)}_t\right] +2C_4, \end{aligned}$$
which together with Jensen’s inequality implies that for each \(p\ge 1\) and \(t\in [0,T]\)
$$\begin{aligned}&\textbf{E}\left[ \exp \bigg \{p\gamma \sup \limits _{s\in [t,T]}\delta _{\theta }\overline{Y}^{(m,q)}_s\bigg \}\right] \le e^{2p\gamma C_4}\textbf{E}\left[ \exp \bigg \{(2+4\kappa )p\gamma \sup \limits _{s\in [t,T]}\delta _{\theta }\overline{y}^{(m,q)}_s\bigg \}\right] \\&\quad \le 4 \textbf{E}\left[ \exp \bigg \{(16+32\kappa )p\gamma \bigg (\vert \xi \vert +{\chi }^{(m,q)}+C_4 +\beta (T-t)\sup \limits _{s\in [t,T]}\delta _{\theta }\overline{Y}^{(m-1,q)}_{s} \bigg )\bigg \} \right] ^{\frac{1}{2}} \\&\qquad \times \textbf{E}\left[ \exp \bigg \{(16+32\kappa )\beta (T-t) p\gamma \sup \limits _{s\in [t,T]}\delta _{\theta }\overline{Y}^{(m-1,q)}_{s} \bigg )\bigg \} \right] ^{\frac{1}{2}} \\&\qquad \times \textbf{E}\left[ \exp \bigg \{(32+64\kappa )p\gamma \zeta ^{(m,q)} \bigg \} \right] ^{\frac{1}{2}}. \end{aligned}$$
Choosing h as in (26), we have
$$\begin{aligned}&\mathbf {{E}}\left[ \exp \bigg \{p\gamma \sup \limits _{s\in [T-h,T]}\delta _{\theta }\overline{Y}^{(m,q)}_s\bigg \}\right] \nonumber \\&\quad \le 4\mathbf {{E}}\left[ \exp \bigg \{{(64+128\kappa ) p\gamma }\vert \xi \vert \bigg \} \right] ^{\frac{1}{8}}\mathbf {{E}}\left[ \exp \bigg \{(64+128\kappa ) p\gamma \big ({\chi }^{(m,q)}+C_4\big )\bigg \} \right] ^{\frac{1}{8}}\nonumber \\&\qquad \ \times \textbf{E}\left[ \exp \bigg \{(32+64\kappa ) p\gamma \zeta ^{(m,q)} \bigg \} \right] ^{\frac{1}{2}} \mathbf {{E}}\left[ \exp \bigg \{p\gamma \sup \limits _{s\in [T-h,T]}\delta _{\theta }\overline{Y}^{(m-1,q)}_{s}\bigg \} \right] ^{(16+32\kappa )\beta h}. \end{aligned}$$
(33)
Set \( \widetilde{\rho } = \frac{1}{1-(16+32\kappa )\beta h}\). If \(\mu =1\), it follows from (33) that for each \(p\ge 1\) and \(m,q\ge 1\),
$$\begin{aligned}&\textbf{E}\left[ \exp \bigg \{p\gamma \sup \limits _{s\in [0,T]}\delta _{\theta }\overline{Y}^{(m,q)}_s\bigg \}\right] \\&\quad \le 4^{{\widetilde{\rho }}}\mathbf {{E}}\left[ \exp \bigg \{(64+128\kappa ) p\gamma \vert \xi \vert \bigg \} \right] ^{\frac{\widetilde{\rho }}{8}}\sup \limits _{m,q\ge 1}\mathbf {{E}}\left[ \exp \bigg \{(64+128\kappa ) p\gamma \big ({\chi }^{(m,q)}+C_4\big )\bigg \} \right] ^{\frac{\widetilde{\rho }}{8}}\\&\qquad \times \sup \limits _{m,q\ge 1}\textbf{E}\left[ \exp \bigg \{(32+64\kappa )p\gamma \zeta ^{(m,q)} \bigg \} \right] ^{\frac{\widetilde{\rho }}{2}}\\&\qquad \times \mathbf {{E}}\left[ \exp \bigg \{p\gamma \sup \limits _{s\in [0,T]}\delta _{\theta }\overline{Y}^{(1,q)}_{s}\bigg \} \right] ^{(16\beta h+32\kappa \beta h)^{m-1}}. \end{aligned}$$
The result from Lemma 8 insures that for any \(\theta \in (0,1)\)
$$\begin{aligned} \lim _{m\rightarrow \infty }\sup \limits _{q\ge 1}\mathbf {{E}}\left[ \exp \bigg \{p\gamma \sup \limits _{s\in [0,T]}\delta _{\theta }\overline{Y}^{(1,q)}_{s}\bigg \} \right] ^{(16\beta h+32\kappa \beta h)^{m-1}}=1, \end{aligned}$$
which implies that
$$\begin{aligned}&\sup \limits _{\theta \in (0,1)}\lim _{m\rightarrow \infty }\sup \limits _{q\ge 1}\textbf{E}\left[ \exp \big \{p\gamma \sup \limits _{s\in [0,T]}\delta _{\theta }\overline{Y}^{(m,q)}_s\big \}\right] \\&\quad \le 4^{\widetilde{\rho }}\sup \limits _{m,q\ge 1}\mathbf {{E}}\left[ \exp \bigg \{(64+128\kappa ) p\gamma \big ({\chi }^{(m,q)}+C_4\big )\bigg \}\right] ^{\frac{\widetilde{\rho }}{8}}\\&\qquad \times \mathbf {{E}}\left[ \exp \bigg \{(64+128\kappa ) p\gamma \vert \xi \vert \bigg \} \right] ^{\frac{\widetilde{\rho }}{8}}\sup \limits _{m,q\ge 1}\textbf{E}\left[ \exp \bigg \{(32+64\kappa )p\gamma \zeta ^{(m,q)} \bigg \} \right] ^{\frac{\widetilde{\rho }}{2}}<\infty . \end{aligned}$$
If \(\mu =2\), in view of the derivation of (30), we conclude that for any \(p\ge 1\),
$$\begin{aligned}&\textbf{E}\left[ \exp \big \{{p\gamma }\sup \limits _{s\in [0,T]}\delta _{\theta }\overline{Y}^{(m,q)}_s\big \} \right] \\&\quad \le 4^{\widetilde{\rho }+{\frac{\widetilde{\rho }^2}{16}}}\textbf{E}\left[ \exp \bigg \{(64+128\kappa )^2 2p\gamma \vert \xi \vert \bigg \} \right] ^{{\frac{\widetilde{\rho }}{16}+\frac{\widetilde{\rho }^2}{128}}}\\&\qquad \times \sup \limits _{m,q\ge 1}\mathbf {{E}}\left[ \exp \bigg \{(64+128\kappa )^2 2p\gamma \big ({\chi }^{(m,q)}+C_4\big )\bigg \} \right] ^{{\frac{\widetilde{\rho }}{8}+\frac{\widetilde{\rho }^2}{128}}}\\&\qquad \times \sup \limits _{m,q\ge 1}\textbf{E}\left[ \exp \bigg \{(32+64\kappa )(64+128\kappa )2p\gamma \zeta ^{(m,q)} \bigg \} \right] ^{{\frac{\widetilde{\rho }}{2}+\frac{\widetilde{\rho }^2}{32}}}\\&\qquad \times \mathbf {{E}}\left[ \exp \bigg \{(64+128\kappa )2p\gamma \sup \limits _{s\in [0,T]}\delta _{\theta }\overline{Y}^{(1,q)}_{s}\bigg \} \right] ^{({\frac{1}{2}+\frac{\widetilde{\rho }}{16})}(16\beta h+32\kappa \beta h)^{m-1}}, \end{aligned}$$
which also implies the desired assertion when \(\mu =2\). Iterating the above procedure \(\mu \) times in the general case, we complete the proof.
