Appendix
In Appendix, we give the proof of Lemma 3.9.
Proof
Proof of (i) We notice by the definitions (93) and Eqs. (52), (51) that for any partition \({\mathcal {P}}=\{0=r_0<r_1<\cdots<r_{n-1}<r_n=T\}\), \(M\in {\mathbb {N}}\), any \(k_0,{i_0}\in {\mathbb {N}},{j}\in \{1,\ldots ,d\}\),
$$\begin{aligned} D^Z_{ \big (k_0,{i_0},{j} \big )}{x}^{{\mathcal {P}},M}_{t}= & {} \int _{0}^{t}\nabla _xb \left( \tau (r),{x}^{{\mathcal {P}},M}_{\tau (r)},\rho ^{{\mathcal {P}},M}_{\tau (r)} \right) D^Z_{ \big (k_0,{i_0},{j} \big )}{x}^{{\mathcal {P}},M}_{\tau (r)}\textrm{d}r \nonumber \\{} & {} + \, \mathbb {1}_{\{0<T_{{i_0}}^{k_0}\le t\}}\mathbb {1}_{\{1\le k_0\le M\}}\xi _{{i_0}}^{k_0}\partial _{z_{{j}}}{c}\nonumber \\ {}{} & {} \left( \tau \left( {T}_{{i_0}}^{k_0} \right) ,\eta ^1_{T_{i_0}^{k_0}} \big (W_{i_0}^{k_0} \big ),{Z}_{{i_0}}^{k_0},{x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{{i_0}}^{k_0} \big )-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{{i_0}}^{k_0} \big )-} \right) \nonumber \\{} & {} + \, \sum _{k=1}^M\sum _{i=1}^{J^k_t}\nabla _x{c} \left( \tau \big ({T}_{i}^{k} \big ),\eta ^1_{T_{i}^{k}} \big (W_{i}^{k} \big ),{Z}_{i}^{k},{x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-} \right) \nonumber \\ {}{} & {} D^Z_{ \big (k_0,{i_0},{j} \big )}{x}^{{\mathcal {P}},M}_{\tau \big ({T}_{i}^{k} \big )-}, \end{aligned}$$
(140)
$$\begin{aligned} D^\Delta _j{x}_{t}^{{\mathcal {P}},M}= & {} a^M_{T}{\varvec{e_j}}+\int _{0}^{t}\nabla _xb \left( \tau (r),{x}^{{\mathcal {P}},M}_{\tau (r)},\rho ^{{\mathcal {P}},M}_{\tau (r)} \right) D^\Delta _{j}{x}^{{\mathcal {P}},M}_{\tau (r)}\textrm{d}r \nonumber \\{} & {} + \, \sum _{k=1}^M\sum _{i=1}^{J^k_t}\nabla _x{c} \left( \tau \big ({T}_{i}^{k} \big ),\eta ^1_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-} \right) \nonumber \\ {}{} & {} D^\Delta _j{x}^{{\mathcal {P}},M}_{\tau \big ({T}_{i}^{k} \big )-}, \quad \quad \end{aligned}$$
(141)
where \({\varvec{e_j}}=(0,\ldots ,0,1,0,\ldots ,0)\) with value 1 at the \(j-\)th component. And
$$\begin{aligned} D^Z_{ \big (k_0,{i_0},{j} \big )}{x}^{M}_{t}= & {} \int _{0}^{t}\nabla _xb \big (r,{x}^{M}_{r},\rho _{r} \big )D^Z_{ \big (k_0,{i_0},{j} \big )}{x}^{M}_{r}\textrm{d}r \nonumber \\{} & {} + \, \mathbb {1}_{ \big \{0<T_{{i_0}}^{k_0}\le t \big \}}\mathbb {1}_{ \big \{1\le k_0\le M \big \}}\xi _{{i_0}}^{k_0}\partial _{z_{{j}}}{c}\nonumber \\ {}{} & {} \left( {T}_{{i_0}}^{k_0},\eta ^2_{T_{i_0}^{k_0}} \big (W_{i_0}^{k_0} \big ),{Z}_{{i_0}}^{k_0},{x}^{M}_{{T}_{{i_0}}^{k_0}-},\rho _{{T}_{{i_0}}^{k_0}-} \right) \nonumber \\{} & {} + \, \sum _{k=1}^M\sum _{i=1}^{J^k_t}\nabla _x{c} \left( {T}_{i}^{k},\eta ^2_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{M}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \right) D^Z_{ \big (k_0,{i_0},{j} \big )}{x}^{M}_{{T}_{i}^{k}-}, \nonumber \\ \end{aligned}$$
(142)
$$\begin{aligned} D^\Delta _j{x}_{t}^{M}= & {} a^M_{T}{\varvec{e_j}}+\int _{0}^{t}\nabla _xb \big (r,{x}^{M}_{r},\rho _{r} \big )D^\Delta _{j}{x}^{M}_{r}\textrm{d}r\nonumber \\ {}{} & {} +\sum _{k=1}^M\sum _{i=1}^{J_t^k}\nabla _x{c} \left( {T}_{i}^{k},\eta ^2_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{M}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \right) D^\Delta _j{x}^{M}_{{T}_{i}^{k}-}. \end{aligned}$$
(143)
For \(u\in l_2\), we will use the notation \(\vert u\vert _{l_2}^2=\vert u_{({\bullet },{\circ },{\diamond })}\vert _{l_2}^2=\sum \nolimits _{k=1}^\infty \sum \nolimits _{i=1}^\infty \sum \nolimits _{j=1}^d\vert u_{(k,i,j)}\vert ^2\). We write \({\mathbb {E}}\vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{t}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{t}\vert _{l_2}^2\le C[H_1+H_2+H_3]\), with
$$\begin{aligned} H_1= & {} {\mathbb {E}} \Bigg \vert \int _{0}^{t}\nabla _xb \left( \tau (r),{x}^{{\mathcal {P}},M}_{\tau (r)},\rho ^{{\mathcal {P}},M}_{\tau (r)} \right) D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{\tau (r)}\textrm{d}r\\ {}{} & {} -\int _{0}^{t}\nabla _xb \left( r,{x}^{M}_{r},\rho _{r} \right) D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r}\textrm{d}r \Bigg \vert _{l_2}^2, \\ H_2= & {} {\mathbb {E}} \left| \mathbb {1}_{\{0<T_{{\circ }}^{\bullet }\le t\}}\mathbb {1}_{\{1\le {\bullet }\le M\}} \left( \partial _{z_{\diamond }}{c} \big (\tau \big ({T}_{\circ }^{\bullet } \big ),\eta ^1_{T_{\circ }^{\bullet }} \big (W_{\circ }^{\bullet } \big ),{Z}_{\circ }^{\bullet },{x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{\circ }^{\bullet } \big )-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{\circ }^{\bullet } \big )-} \right) \right. \\{} & {} - \, \left. \partial _{z_{\diamond }}{c} \left( {T}_{\circ }^{\bullet },\eta ^2_{T_{\circ }^{\bullet }} \big (W_{\circ }^{\bullet } \big ),{Z}_{\circ }^{\bullet },{x}^{M}_{{T}_{\circ }^{\bullet }-},\rho _{{T}_{\circ }^{\bullet }-} \big ) \right) \right| _{l_2}^2, \\ H_3= & {} {\mathbb {E}} \left| \sum _{k=1}^M \sum _{i=1}^{J^k_t}\nabla _x{c} \left( \tau ({T}_{i}^{k} \big ),\eta ^1_{T_{i}^{k}}\big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-} \right) D^Z_{ \big ({\bullet },{\circ },{\diamond } \big )}{x}^{{\mathcal {P}},M}_{\tau \big ({T}_{i}^{k} \big )-} \right. \\{} & {} -\sum _{k=1}^M\sum _{i=1}^{J^k_t}\nabla _x{c} \left( {T}_{i}^{k},\eta ^2_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{M}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \right) D^Z_{\big ({\bullet },{\circ },{\diamond } \big )}{x}^{M}_{{T}_{i}^{k}-}\vert _{l_2}^2. \end{aligned}$$
We take a small \(\varepsilon _*>0\). We recall \(\varepsilon _M\) in (27). Firstly, using Hypothesis 2.1, we get
$$\begin{aligned} H_1\le & {} C \left[ {\mathbb {E}} \int _{0}^{t} \left| \nabla _xb \left( \tau (r),{x}^{{\mathcal {P}},M}_{\tau (r)},\rho ^{{\mathcal {P}},M}_{\tau (r)})-\nabla _xb(r,{x}^{M}_{r},\rho _{r} \right) \right| ^2 \left| D^Z_{ \left( {\bullet },{\circ },{\diamond } \right) }{x}^{M}_{r} \right| _{l_2}^2\textrm{d}r \right. \nonumber \\{} & {} \left. + \, {\mathbb {E}} \int _{0}^{t}\vert \nabla _xb \left( \tau (r),{x}^{{\mathcal {P}},M}_{\tau (r)},\rho ^{{\mathcal {P}},M}_{\tau (r)} \right) \vert ^2\vert D^Z_{ \left( {\bullet },{\circ },{\diamond } \right) }{x}^{{\mathcal {P}},M}_{\tau (r)}-D^Z_{ \left( {\bullet },{\circ },{\diamond } \right) }{x}^{M}_{r}\vert _{l_2}^2\textrm{d}r \right] \nonumber \\\le & {} C \left[ {\mathbb {E}}\int _0^t \left[ \big \vert {\mathcal {P}} \big \vert ^2+ \big \vert {x}^{{\mathcal {P}},M}_{\tau (r)}-{x}^{M}_{r} \big \vert ^{2}+ \left( W_1 \left( {\rho }^{{\mathcal {P}},M}_{\tau (r)},{\rho }_{r} \right) \right) ^2 \right] \left| D^Z_{ \left( {\bullet },{\circ },{\diamond } \right) }{x}^{M}_{r} \right| _{l_2}^2\textrm{d}r \right. \nonumber \\{} & {} + \, \left. \int _{0}^{t}{\mathbb {E}} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{\tau (r)}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r} \right| _{l_2}^2\textrm{d}r \right] . \end{aligned}$$
Then by Lemma 3.7, using Hölder inequality with conjugates \(1+\frac{\varepsilon _*}{2}\) and \(\frac{2+\varepsilon _*}{\varepsilon _*}\), by Lemma 2.6 and (50), we have
$$\begin{aligned} H_1\le & {} C \left[ \vert {\mathcal {P}}\vert ^2+\int _0^t \left( {\mathbb {E}}\vert {x}^{{\mathcal {P}},M}_{\tau (r)}-{x}^{M}_{r}\vert ^{2+\varepsilon _*} \right) ^{\frac{2}{2+\varepsilon _*}}\textrm{d}r+\int _0^t \left( W_{2+\varepsilon _*} \left( {\rho }^{{\mathcal {P}},M}_{\tau (r)},{\rho }_{r} \right) \right) ^2\textrm{d}r \right. \nonumber \\{} & {} + \left. \int _{0}^{t}{\mathbb {E}} \left| D^Z_{ ({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{\tau (r)}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r} \right| _{l_2}^2\textrm{d}r \right] \nonumber \\\le & {} C \left[ \left( \vert {\mathcal {P}}\vert +\varepsilon _M \right) ^{\frac{2}{2+\varepsilon _*}}+\int _{0}^{t}{\mathbb {E}}\vert D^Z_{ ({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{\tau (r)}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r}\vert _{l_2}^2\textrm{d}r \right] . \end{aligned}$$
(144)
Secondly, using Hypothesis 2.1 and the isometry of the Poisson point measure \({\mathcal {N}}\), we get
$$\begin{aligned} H_2= & {} {\mathbb {E}}\sum _{k=1}^M\sum _{i=1}^{J^k_t}\sum _{j=1}^d \left| \partial _{z_{j}}{c} \left( \tau \big ({T}_{i}^{k} \big ),\eta ^1_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-} \right) \right. \\{} & {} - \, \left. \partial _{z_{j}}{c} \left( {T}_{i}^{k},\eta ^2_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{M}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \right) \right| ^2\nonumber \\\le & {} C{\mathbb {E}}\sum _{k=1}^M\sum _{i=1}^{J^k_t}\vert {\bar{c}}(Z^k_i)\vert ^2 \left[ \left| \tau ^{{{\mathcal {P}}}} ({T}_{i}^{k} \big )-T^k_i \right| + \left| \eta ^1_{T_{i}^{k}} \big (W_{i}^k \big )-\eta ^2_{T_{i}^{k}} \big (W_{i}^k \big ) \right| \right. \nonumber \\{} & {} + \, \left. \left| {x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}}\big ({T}_{i}^{k} \big )-}-{x}^{M}_{{T}_{i}^{k}-} \right| +W_1 \left( \rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-},\rho _{{T}_{i}^{k}-} \right) \right] ^2\nonumber \\\le & {} C{\mathbb {E}}\int _0^t\int _{[0,1]\times {\mathbb {R}}^d}\vert {\bar{c}}(z)\vert ^2 \left[ \vert \tau ^{{{\mathcal {P}}}}(r)-r\vert ^2+\vert \eta ^1_{r}(w)-\eta ^2_{r}(w)\vert ^2 \right. \nonumber \\{} & {} + \left. \vert {x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}}(r)-}-{x}^{M}_{r-}\vert ^2+ \left( W_1 \left( \rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}}(r)-},\rho _{r-} \right) \right) ^2 \right] {\mathcal {N}} (\textrm{d}w,\textrm{d}z,\textrm{d}r)\nonumber \\= & {} C{\mathbb {E}}\int _0^t\int _{[0,1]\times {\mathbb {R}}^d}\vert {\bar{c}}(z)\vert ^2 \left[ \vert \tau ^{{{\mathcal {P}}}}(r)-r\vert ^2+\vert \eta ^1_{r}(w)-\eta ^2_{r}(w)\vert ^2 \right. \nonumber \\{} & {} + \left. \vert {x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}}(r)-}-{x}^{M}_{r-}\vert ^2+ \left( W_1 \left( \rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}}(r)-},\rho _{r-} \right) \right) ^2 \right] \textrm{d}w\mu (\textrm{d}z)\textrm{d}r. \end{aligned}$$
Then by (39), (43), Lemma 2.6, (50), and Hölder inequality with conjugates \(1+\frac{\varepsilon _*}{2}\) and \(\frac{2+\varepsilon _*}{\varepsilon _*}\), we have
$$\begin{aligned} H_2\le & {} C \left[ \vert {\mathcal {P}}\vert ^2+\int _0^t \left( {\mathbb {E}} \left| {x}^{{\mathcal {P}},M}_{\tau (r)}-{x}_{r} \right| ^{2+\varepsilon _*} \right) ^{\frac{2}{2+\varepsilon _*}}\textrm{d}r \right. \nonumber \\{} & {} \left. +\int _0^t \left( {\mathbb {E}} \left| {x}^{{\mathcal {P}},M}_{\tau (r)}-{x}^{M}_{r} \right| ^{2+\varepsilon _*} \right) ^{\frac{2}{2+\varepsilon _*}}\textrm{d}r+\int _0^t \left( W_{2+\varepsilon _*} \left( \rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}}(r)},\rho _{r} \right) \right) ^2\textrm{d}r \right] \nonumber \\\le & {} C(\vert {\mathcal {P}}\vert +\varepsilon _M)^{\frac{2}{2+\varepsilon _*}}. \end{aligned}$$
(145)
Thirdly, we write \(H_3\le C[H_{3,1}+H_{3,2}]\), where
$$\begin{aligned} H_{3,1}= & {} {\mathbb {E}} \left( \sum _{k=1}^M\sum _{i=1}^{J_t^k}\vert \nabla _x{c} \left( \tau \big ({T}_{i}^{k} \big ),\eta ^1_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k}, {x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-} \right) \right. \\{} & {} - \left. \left. \nabla _x{c} \left( {T}_{i}^{k},\eta ^2_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{M}_{{T}_{i}^{k}-}, \rho _{{T}_{i}^{k}-} \right) \right| \left| D^Z_{ \left( {\bullet },{\circ },{\diamond } \right) }{x}^{M}_{{T}_{i}^{k}-} \right| _{l_2} \right) ^2, \\ H_{3,2}= & {} {\mathbb {E}} \left( \sum _{k=1}^M\sum _{i=1}^{J^k_t} \left| \nabla _x{c} \left( \tau \big ({T}_{i}^{k} \big ),\eta ^1_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}} \big ({T}_{i}^{k} \big )-} \right) \right| \right. \\{} & {} \left. \left| D^Z_{ \big ({\bullet },{\circ },{\diamond } \big )}{x}^{{\mathcal {P}},M}_{\tau \big ({T}_{i}^{k} \big )-}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{{T}_{i}^{k}-} \right| _{l_2} \right) ^2. \end{aligned}$$
Using Hypothesis 2.1 and (45) with
$$\begin{aligned}{\bar{\Phi }}(r,w,z,\omega ,\rho )= & {} \left| \nabla _x{c} \left( \tau (r),\eta ^1_{r}(w),z,{x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}}(r)-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}}(r)-} \right) \right. \\{} & {} - \, \left. \nabla _x{c} \left( r,\eta ^2_{r}(w),z,{x}^{M}_{r-},\rho _{r-} \right) \right| \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r-} \right| _{l_2},\end{aligned}$$
we get
$$\begin{aligned} H_{3,1}\le & {} {\mathbb {E}} \left( \int _0^t\int _{[0,1]\times {\mathbb {R}}^d} \left| \nabla _x{c} \left( \tau (r),\eta ^1_{r}(w),z,{x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}}(r)-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}}(r)-} \right) \right. \right. \\{} & {} - \, \nabla _x{c} \big (r,\eta ^2_{r}(w),z,{x}^{M}_{r-},\rho _{r-} \big ) \big \vert \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r-}\vert _{l_2}{\mathcal {N}}(\textrm{d}w,\textrm{d}z,\textrm{d}r) \big )^2\\\le & {} C{\mathbb {E}}\int _0^t\int _{[0,1]} \big [ \big \vert \tau ^{{{\mathcal {P}}}}(r)-r \big \vert ^2+\big \vert \eta ^1_{r}(w)-\eta ^2_{r}(w) \big \vert ^2\\{} & {} +\, \big \vert {x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}}(r)-}-{x}^{M}_{r-} \big \vert ^2+ \big (W_1 \big (\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}}(r)-},\rho _{r-} \big ) \big )^2 \big ] \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r-}\vert _{l_2}^2\textrm{d}w\textrm{d}r. \end{aligned}$$
Then using (39), (43), Lemma 3.7, and Hölder inequality with conjugates \(1+\frac{\varepsilon _*}{2}\) and \(\frac{2+\varepsilon _*}{\varepsilon _*}\), we have
$$\begin{aligned} H_{3,1}\le & {} C\Big [\vert {\mathcal {P}}\vert ^2+\int _0^t \left( {\mathbb {E}}\vert {x}^{{\mathcal {P}},M}_{\tau (r)-}-{x}_{r-}\vert ^{2+\varepsilon _*} \right) ^{\frac{2}{2+\varepsilon _*}}\textrm{d}r\\{} & {} \left. +\int _0^t \left( {\mathbb {E}}\vert {x}^{{\mathcal {P}},M}_{\tau (r)-}-{x}^{M}_{r-}\vert ^{2+\varepsilon _*} \right) ^{\frac{2}{2+\varepsilon _*}}\textrm{d}r+\int _0^t \left( W_{2+\varepsilon _*} \left( \rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}}(r)-},\rho _{r-} \right) \right) ^2\textrm{d}r \right] \\\le & {} C \left( \vert {\mathcal {P}}\vert +\varepsilon _M \right) ^{\frac{2}{2+\varepsilon _*}}, \end{aligned}$$
where the last inequality is a consequence of Lemma 2.6 and (50).
Using Hypothesis 2.1, (45) with
$$\begin{aligned}{} & {} {\bar{\Phi }}(r,w,z,\omega ,\rho )= \left| \nabla _x{c} \left( \tau (r),\eta ^1_{r}(w),z, {x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}}(r)-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}}(r)-} \right) \right| \\ {}{} & {} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{\tau (r)-}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r-} \right| _{l_2}, \end{aligned}$$
we have
$$\begin{aligned} H_{3,2}\le & {} {\mathbb {E}} \left( \int _0^t\int _{[0,1]\times {\mathbb {R}}^d} \left| \nabla _x{c} \left( \tau (r),\eta ^1_{r}(w),z,{x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}}(r)-}, \rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}}(r)-} \right) \right| \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{\tau (r)-} \right. \right. \\{} & {} \left. -D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r-} \right| _{l_2}{\mathcal {N}}(\textrm{d}w,\textrm{d}z,\textrm{d}r))^2\\\le & {} C\int _{0}^{t}{\mathbb {E}} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{\tau (r)-}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r-} \right| _{l_2}^2\textrm{d}r. \end{aligned}$$
Therefore,
$$\begin{aligned} H_3\le & {} C[H_{3,1}+H_{3,2}]\nonumber \\ {}{} & {} \le C \left[ (\vert {\mathcal {P}}\vert +\varepsilon _M)^{\frac{2}{2+\varepsilon _*}}+\int _{0}^{t}{\mathbb {E}}\vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{\tau (r)}-D^Z_{({\bullet },{\circ },{\diamond })} {x}^{M}_{r}\vert _{l_2}^2\textrm{d}r \right] .\nonumber \\ \end{aligned}$$
(146)
Combining (144), (145) and (146),
$$\begin{aligned}{} & {} {\mathbb {E}} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{t}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{t} \right| _{l_2}^2\le C \left[ (\vert {\mathcal {P}}\vert +\varepsilon _M)^{\frac{2}{2+\varepsilon _*}} \right. \\{} & {} \quad \left. +\int _{0}^{t}{\mathbb {E}} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{\tau (r)}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r} \right| _{l_2}^2\textrm{d}r \right] . \end{aligned}$$
In a similar way, we notice, by (116), the isometry of the Poisson point measure N, and (45) with
$$\begin{aligned}{\bar{\Phi }}(r,w,z,\omega ,\rho )=\nabla _x{c} \left( \tau (r),\eta ^1_{r}(w),z,{x}^{{{\mathcal {P}}},M}_{\tau ^{{{\mathcal {P}}}}(r)-},\rho ^{{\mathcal {P}},M}_{\tau ^{{{\mathcal {P}}}}(r)-} \right) D^Z_{(k_0,{i_0},{j})}{x}^{{\mathcal {P}},M}_{\tau (r)-}\end{aligned}$$
that
$$\begin{aligned} {\mathbb {E}} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{\tau (t)}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{t} \right| _{l_2}^2\le C\vert {\mathcal {P}}\vert ,\end{aligned}$$
(147)
so
$$\begin{aligned}{} & {} {\mathbb {E}} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{t}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{t} \right| _{l_2}^2\le C \left[ (\vert {\mathcal {P}}\vert +\varepsilon _M)^{\frac{2}{2+\varepsilon _*}} \right. \\{} & {} \quad \left. +\int _{0}^{t}{\mathbb {E}} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{r}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{r} \right| _{l_2}^2\textrm{d}r \right] . \end{aligned}$$
We conclude by Gronwall lemma that \({\mathbb {E}}\vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{{\mathcal {P}},M}_{t}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M}_{t}\vert _{l_2}^2\le C(\vert {\mathcal {P}}\vert +\varepsilon _M)^{\frac{2}{2+\varepsilon _*}}\). Finally, by a similar argument, \({\mathbb {E}}\vert D^{\Delta }_{({\diamond })}{x}^{{\mathcal {P}},M}_{t}-D^{\Delta }_{({\diamond })}{x}^{M}_{t}\vert _{{\mathbb {R}}^d}^2\le C(\vert {\mathcal {P}}\vert +\varepsilon _M)^{\frac{2}{2+\varepsilon _*}}\), and we obtain what we need.
Proof of (ii) We only need to prove that for any \(M_1,M_2\in {\mathbb {N}}\) with \(\varepsilon _{M_1\wedge M_2}\le 1\) and \(\vert {\bar{c}}(z)\vert ^2\mathbb {1}_{\{\vert z\vert >M_1\wedge M_2\}}\le 1\), we have
$$\begin{aligned} \left\| Dx^{M_1}_t-Dx^{M_2}_t \right\| _{L^2(\Omega ;l^2\times {\mathbb {R}}^d)}\le (\varepsilon _{M_1}+\varepsilon _{M_2})^{\frac{1}{2+\varepsilon _*}}. \end{aligned}$$
(148)
In fact, if \((Dx^M_t)_{M\in {\mathbb {N}}}\) is a Cauchy sequence in \(L^2(\Omega ;l^2\times {\mathbb {R}}^d)\), then it has a limit Y in \(L^2(\Omega ;l^2\times {\mathbb {R}}^d)\). But when we apply Lemma 3.3(A) with \(F_M=X^M_t\) and \(F=X_t\), we know that there exists a convex combination \(\sum \limits _{M^\prime =M}^{m_{M}}\gamma _{M^\prime }^{M}\times F^{M^\prime }_t,\) with \(\gamma _{M^\prime }^{M}\ge 0,M^\prime =M, \ldots ,m_{M}\) and \( \sum \limits _{M^\prime =M}^{m_{M}}\gamma _{M^\prime }^{M}=1\), such that
$$\begin{aligned} \left\| \sum _{M^\prime =M}^{m_{M}}\gamma _{M^\prime }^{M}\times Dx^{M^\prime }_t-Dx_t\right\| _{L^2(\Omega ;l^2\times {\mathbb {R}}^d)}\rightarrow 0, \end{aligned}$$
as \(M\rightarrow \infty \). Meanwhile, we have
$$\begin{aligned}\left\| \sum _{M^\prime =M}^{m_{M}}\gamma _{M^\prime }^{M}\times DF^{M^\prime }_t-Y\right\| _{L^2(\Omega ;l^2\times {\mathbb {R}}^d)}\le \sum _{M^\prime =M}^{m_{M}}\gamma _{M^\prime }^{M}\left\| Dx^{M^\prime }_t-Y\right\| _{L^2(\Omega ;l^2\times {\mathbb {R}}^d)}\rightarrow 0.\end{aligned}$$
So \(Y=Dx_t\) and we conclude by passing to the limit \(M_2\rightarrow \infty \) in (148).
Now we prove (148). We recall the equation (142) and we write \({\mathbb {E}}\vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{t}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{t}\vert _{l_2}^2\le C[O_1+O_2+O_3]\), with
$$\begin{aligned} O_1= & {} {\mathbb {E}}\vert \int _{0}^{t}\nabla _xb \left( r,{x}^{M_1}_{r},\rho _{r} \right) D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{r}\textrm{d}r-\int _{0}^{t}\nabla _xb(r,{x}^{M_2}_{r},\rho _{r})D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r}\textrm{d}r \vert _{l_2}^2, \\ O_2= & {} {\mathbb {E}} \big \vert \mathbb {1}_{\{0<T_{{\circ }}^{\bullet }\le t\}} \big (\mathbb {1}_{\{1\le {\bullet }\le M_1\}}\partial _{z_{\diamond }}{c} \big ({T}_{\circ }^{\bullet },\eta ^2_{T_{\circ }^{\bullet }} \big (W_{\circ }^{\bullet } \big ),{Z}_{\circ }^{\bullet },{x}^{M_1}_{{T}_{\circ }^{\bullet }-},\rho _{{T}_{\circ }^{\bullet }-} \big )\\{} & {} - \, \mathbb {1}_{\{1\le {\bullet }\le M_2\}}\partial _{z_{\diamond }}{c} \big ({T}_{\circ }^{\bullet },\eta ^2_{T_{\circ }^{\bullet }} \big (W_{\circ }^{\bullet } \big ),{Z}_{\circ }^{\bullet },{x}^{M_2}_{{T}_{\circ }^{\bullet }-},\rho _{{T}_{\circ }^{\bullet }-} \big ) \big )\vert _{l_2}^2, \\ O_3= & {} {\mathbb {E}}\vert \sum _{k=1}^{M_1}\sum _{i=1}^{J^k_t}\nabla _x{c} \big ({T}_{i}^{k},\eta ^2_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{M_1}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-})D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{{T}_{i}^{k}-}\\{} & {} -\sum _{k=1}^{M_2}\sum _{i=1}^{J^k_t}\nabla _x{c} \big ({T}_{i}^{k},\eta ^2_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{M_2}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \big )D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{{T}_{i}^{k}-} \big \vert _{l_2}^2. \end{aligned}$$
Firstly, using Hypothesis 2.1, we have
$$\begin{aligned} O_1\le & {} C \Big [{\mathbb {E}} \int _{0}^{t}\vert \nabla _xb \big (r,{x}^{M_1}_{r},\rho _{r} \big )-\nabla _xb \big (r,{x}^{M_2}_{r},\rho _{r} \big ) \big \vert ^2 \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r} \big \vert _{l_2}^2\textrm{d}r\nonumber \\{} & {} + \, {\mathbb {E}} \int _{0}^{t} \big \vert \nabla _xb \big (r,{x}^{M_1}_{r},\rho _{r} \big ) \big \vert ^2 \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{r}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r}\vert _{l_2}^2\textrm{d}r \Big ]\nonumber \\\le & {} C \Big [{\mathbb {E}}\int _0^t \big \vert {x}^{M_1}_{r}-{x}^{M_2}_{r} \big \vert ^{2} \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r} \big \vert _{l_2}^2\textrm{d}r+ \int _{0}^{t}{\mathbb {E}} \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{r}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r} \big \vert _{l_2}^2\textrm{d}r \Big ]. \end{aligned}$$
Then by Lemma 3.7, Hölder inequality with conjugates \(1+\frac{\varepsilon _*}{2}\) and \(\frac{2+\varepsilon _*}{\varepsilon _*}\), and Lemma 2.6, we obtain
$$\begin{aligned} O_1\le & {} C \left[ \int _0^t \big ({\mathbb {E}} \big \vert {x}^{M_1}_{r}-{x}^{M_2}_{r} \big \vert ^{2+\varepsilon _*} \big )^{\frac{2}{2+\varepsilon _*}}\textrm{d}r+ \int _{0}^{t}{\mathbb {E}} \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{r}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r} \big \vert _{l_2}^2\textrm{d}r \right] \nonumber \\\le & {} C \left[ \big (\varepsilon _{M_1}+\varepsilon _{M_2} \big )^{\frac{2}{2+\varepsilon _*}}+\int _{0}^{t}{\mathbb {E}} \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{r}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r} \big \vert _{l_2}^2\textrm{d}r \right] . \end{aligned}$$
(149)
Secondly, using Hypothesis 2.1, the isometry of the Poisson point measure \({\mathcal {N}}\), we have
$$\begin{aligned} O_2\le & {} C \left[ {\mathbb {E}}\sum _{k=1}^{M_1}\sum _{i=1}^{J^k_t}\sum _{j=1}^d\vert \partial _{z_{j}}{c} \left( {T}_{i}^{k},\eta ^2_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{M_1}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \right) \right. \\{} & {} - \, \partial _{z_{j}}{c} \left( {T}_{i}^{k},\eta ^2_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{M_2}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \right) \Big \vert ^2\\{} & {} + \left. \, {\mathbb {E}}\sum _{k=M_1\wedge M_2}^{M_1\vee M_2}\sum _{i=1}^{J^k_t}\sum _{j=1}^d \Big \vert \partial _{z_{j}}{c} \Big ({T}_{i}^{k},\eta ^2_{T_{i}^{k}} \big (W_{i}^k \big ),{Z}_{i}^{k},{x}^{M_2}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \Big ) \Big \vert ^2 \right] \nonumber \\\le & {} C \left[ {\mathbb {E}}\sum _{k=1}^M\sum _{i=1}^{J^k_t} \left| {\bar{c}}(Z^k_i) \right| ^2 \left| {x}^{M_1}_{{T}_{i}^{k}-}-{x}^{M_2}_{{T}_{i}^{k}-} \right| ^2 \right. \\{} & {} + \left. \, {\mathbb {E}}\sum _{k=M_1\wedge M_2}^{M_1\vee M_2}\sum _{i=1}^{J^k_t}\vert {\bar{c}} (Z^k_i)\vert ^2 \right] \\\le & {} C \left[ {\mathbb {E}}\int _0^t\int _{[0,1]\times {\mathbb {R}}^d}\vert {\bar{c}}(z)\vert ^2\vert {x}^{M_1}_{r-}-{x}^{M_2}_{r-}\vert ^2{\mathcal {N}}(\textrm{d}w,\textrm{d}z,\textrm{d}r) \right. \\{} & {} + \left. \, {\mathbb {E}}\int _0^t\int _{[0,1]}\int _{\{\vert z\vert>M_1\wedge M_2\}}\vert {\bar{c}}(z)\vert ^2{\mathcal {N}}(\textrm{d}w,\textrm{d}z,\textrm{d}r) \right] \\= & {} C \left[ {\mathbb {E}}\int _0^t\int _{[0,1]\times {\mathbb {R}}^d}\vert {\bar{c}}(z)\vert ^2\vert {x}^{M_1}_{r-}-{x}^{M_2}_{r-}\vert ^2\textrm{d}w\mu (\textrm{d}z)\textrm{d}r \right. \\{} & {} + \left. \, {\mathbb {E}}\int _0^t\int _{[0,1]}\int _{\{\vert z\vert >M_1\wedge M_2\}}\vert {\bar{c}}(z)\vert ^2\textrm{d}w\mu (\textrm{d}z)\textrm{d}r \right] . \end{aligned}$$
Then by Hölder inequality with conjugates \(1+\frac{\varepsilon _*}{2}\) and \(\frac{2+\varepsilon _*}{\varepsilon _*}\), Hypothesis 2.1 and Lemma 2.6,
$$\begin{aligned} O_2\le & {} C \left[ \int _0^t \left( {\mathbb {E}}\vert {x}^{M_1}_{r}-{x}^{M_2}_{r}\vert ^{2+\varepsilon _*} \right) ^{\frac{2}{2+\varepsilon _*}}\textrm{d}r+\varepsilon _{M_1\wedge M_2} \right] \nonumber \\\le & {} C(\varepsilon _{M_1}+\varepsilon _{M_2})^{\frac{2}{2+\varepsilon _*}}. \end{aligned}$$
(150)
Thirdly, we write \(O_3\le C[O_{3,1}+O_{3,2}+O_{3,3}]\), where
$$\begin{aligned} O_{3,1}= & {} {\mathbb {E}} \left( \sum _{k=M_1\wedge M_2}^{M_1\vee M_2}\sum _{i=1}^{J^k_t}\vert \nabla _x{c}\left( {T}_{i}^{k},\eta ^2_{T_{i}^{k}}(W_{i}^k),{Z}_{i}^{k},{x}^{M_1\vee M_2}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \right) \vert \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1\vee M_2}_{{T}_{i}^{k}-}\vert _{l_2} \right) ^2, \\ O_{3,2}= & {} {\mathbb {E}} \left( \sum _{k=1}^{M_1\wedge M_2}\sum _{i=1}^{J_t^k}\vert \nabla _x{c} \left( {T}_{i}^{k},\eta ^2_{T_{i}^{k}}(W_{i}^k),{Z}_{i}^{k},{x}^{M_1}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \right) \right. \\{} & {} -\nabla _x{c} \left( {T}_{i}^{k},\eta ^2_{T_{i}^{k}}(W_{i}^k),{Z}_{i}^{k},{x}^{M_2}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \right) \vert \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{{T}_{i}^{k}-}\vert _{l_2})^2, \\ O_{3,3}= & {} {\mathbb {E}} \left( \sum _{k=1}^{M_1\wedge M_2}\sum _{i=1}^{J^k_t}\vert \nabla _x{c} \left( {T}_{i}^{k},\eta ^2_{T_{i}^{k}}(W_{i}^k),{Z}_{i}^{k},{x}^{M_1}_{{T}_{i}^{k}-},\rho _{{T}_{i}^{k}-} \right) \vert \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{{T}_{i}^{k}-}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{{T}_{i}^{k}-}\vert _{l_2} \right) ^2. \end{aligned}$$
Using Hypothesis 2.1, Lemma 3.7, (44) with
$$\begin{aligned}{\bar{\Phi }}(r,w,z,\omega ,\rho )= \left| \nabla _x{c} \left( r,\eta ^2_{r}(w),z,{x}^{M_1\vee M_2}_{r-},\rho _{r-} \right) \right| \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1\vee M_2}_{r-} \right| _{l_2},\end{aligned}$$
we get
$$\begin{aligned} O_{3,1}\le & {} {\mathbb {E}} \Big (\int _0^t\int _{[0,1]}\int _{\{ \vert z\vert>M_1\wedge M_2\}} \Big \vert \nabla _x{c}(r,\eta ^2_{r}(w),z,{x}^{M_1\vee M_2}_{r-},\rho _{r-} \Big ) \Big \vert \\{} & {} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1\vee M_2}_{r-} \right| _{l_2}{\mathcal {N}}(\textrm{d}w,\textrm{d}z,\textrm{d}r))^2\\\le & {} C \left[ \left( \int _{\{\vert z\vert>M_1\wedge M_2\}}{\bar{c}}(z)\mu (\textrm{d}z))^2+\int _{\{\vert z\vert >M_1\wedge M_2\}}\vert {\bar{c}}(z)\vert ^2\mu (\textrm{d}z) \right. \right] \\= & {} C\varepsilon _{M_1\wedge M_2}. \end{aligned}$$
Using Hypothesis 2.1, Lemma 3.7, (45) with
$$\begin{aligned}{} & {} {\bar{\Phi }}(r,w,z,\omega ,\rho )=\vert \nabla _x{c} \left( r,\eta ^2_{r}(w),z,{x}^{M_1}_{r-},\rho _{r-} \right) \\{} & {} \quad -\nabla _x{c} \left( r,\eta ^2_{r}(w),z,{x}^{M_2}_{r-},\rho _{r-} \right) \big \vert \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r-} \big \vert _{l_2}, \end{aligned}$$
by Lemma 2.6, and Hölder inequality with conjugates \(1+\frac{\varepsilon _*}{2}\) and \(\frac{2+\varepsilon _*}{\varepsilon _*}\), we have
$$\begin{aligned} O_{3,2}\le & {} {\mathbb {E}} \left( \int _0^t\int _{[0,1]\times {\mathbb {R}}^d}\vert \nabla _x{c} \left( r,\eta ^2_{r}(w),z,{x}^{M_1}_{r-},\rho _{r-} \right) \right. \\{} & {} -\nabla _x{c} \big (r,\eta ^2_{r}(w),z,{x}^{M_2}_{r-},\rho _{r-} \big )\vert \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r-}\vert _{l_2}{\mathcal {N}} (\textrm{d}w,\textrm{d}z,\textrm{d}r) \big )^2\\\le & {} C{\mathbb {E}}\int _0^t\int _{[0,1]} \big \vert {x}^{M_1}_{r-}-{x}^{M_2}_{r-} \big \vert ^2 \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r-} \big \vert _{l_2}^2\textrm{d}w\textrm{d}r\\\le & {} C\int _0^t \big ({\mathbb {E}} \big \vert {x}^{M_1}_{r-}-{x}^{M_2}_{r-} \big \vert ^{2+\varepsilon _*} \big )^{\frac{2}{2+\varepsilon _*}}\textrm{d}r\\\le & {} C(\varepsilon _{M_1}+\varepsilon _{M_2})^{\frac{2}{2+\varepsilon _*}}. \end{aligned}$$
Using Hypothesis 2.1, (45) with
$$\begin{aligned}{\bar{\Phi }}(r,w,z,\omega ,\rho )= \big \vert \nabla _x{c} \big (r,\eta ^2_{r}(w),z,{x}^{M_1}_{r-},\rho _{r-} \big ) \big \vert \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{r-}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r-} \big \vert _{l_2},\end{aligned}$$
we have
$$\begin{aligned} O_{3,3}\le & {} {\mathbb {E}} \left( \int _0^t\int _{[0,1]\times {\mathbb {R}}^d} \big \vert \nabla _x{c} \big (r,\eta ^2_{r}(w),z,{x}^{M_1}_{r-},\rho _{r-} \big ) \big \vert \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{r-} \right. \\{} & {} -D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r-} \big \vert _{l_2}{\mathcal {N}}(\textrm{d}w,\textrm{d}z,\textrm{d}r) \big )^2\\\le & {} C\int _{0}^{t}{\mathbb {E}} \big \vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{r-}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r-}\big \vert _{l_2}^2\textrm{d}r. \end{aligned}$$
Therefore,
$$\begin{aligned} O_3\le & {} C \left[ O_{3,1}+O_{3,2}+O_{3,3}]\le C[(\varepsilon _{M_1}+\varepsilon _{M_2})^{\frac{2}{2+\varepsilon _*}} \right. \nonumber \\{} & {} +\, \left. \int _{0}^{t}{\mathbb {E}} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{r}\textrm{d}r-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r} \right| _{l_2}^2\textrm{d}r \right] .\quad \end{aligned}$$
(151)
Combining (149), (150) and (151),
$$\begin{aligned}{} & {} {\mathbb {E}} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{t}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{t} \right| _{l_2}^2\\{} & {} \quad \le C \left[ (\varepsilon _{M_1}+\varepsilon _{M_2})^{\frac{2}{2+\varepsilon _*}}+\int _{0}^{t}{\mathbb {E}} \left| D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{r}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{r} \right| _{l_2}^2\textrm{d}r \right] . \end{aligned}$$
So we conclude by Gronwall lemma that \({\mathbb {E}}\vert D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_1}_{t}-D^Z_{({\bullet },{\circ },{\diamond })}{x}^{M_2}_{t}\vert _{l_2}^2\le C(\varepsilon _{M_1}+\varepsilon _{M_2})^{\frac{2}{2+\varepsilon _*}}\).
Finally, we recall by (24) that \(a^M_{T}=\sqrt{T\int _{\{\vert z\vert >M\}}{\underline{c}}(z) \mu (\textrm{d}z)}\) and by Hypothesis 2.3 that \({\underline{c}}(z)\le \vert {\bar{c}}(z)\vert ^2\). We notice that
$$\begin{aligned}{\mathbb {E}} \big \vert a^{M_1}_T{\varvec{e_{\diamond }}}-a^{M_2}_T{\varvec{e_{\diamond }}} \big \vert _{{\mathbb {R}}^d}^2\le C{\mathbb {E}} \big \vert a^{M_1}_T-a^{M_2}_T \big \vert ^2\le C\int _{\{\vert z\vert >{M_1\wedge M_2}\}}{\underline{c}}(z) \mu (\textrm{d}z)\le \varepsilon _{M_1\wedge M_2}.\end{aligned}$$
Then by a similar argument as above, \({\mathbb {E}}\vert D^{\Delta }_{({\diamond })}{x}^{M_1}_{t}-D^{\Delta }_{({\diamond })}{x}^{M_2}_{t}\vert _{{\mathbb {R}}^d}^2\le C(\varepsilon _{M_1}+\varepsilon _{M_2})^{\frac{2}{2+\varepsilon _*}}\), and we obtain (148).
Proof of (iii) (iii) is an immediate consequence of (i) and (ii).
\(\square \)
