Stochastic Optimal Control Problem with Obstacle Constraints in Sublinear Expectation Framework

Abstract

In this paper, we consider a stochastic optimal control problem, in which the cost function is defined through a reflected backward stochastic differential equation in sublinear expectation framework. Besides, we study the regularity of the value function and establish the dynamic programming principle. Moreover, we prove that the value function is the unique viscosity solution of the related Hamilton–Jacobi–Bellman–Isaac equation.

Appendices

Appendix A: Reflected G-BSDE

The definition of reflected G-BSDEs Given an obstacle process $\{S_t\}_{t\in [0,T]}$, a terminal value $\zeta \in L^{\beta }_G(\varOmega _T)$ with $\zeta \ge S_T$ for $\beta >2$ and generator $f(t,\omega ,y,z):[0,T]\times \varOmega \times \mathbb {R}\times \mathbb {R}^{d}\rightarrow \mathbb {R}$, a triple of processes $(Y,Z,A)\in \mathfrak {S}_G^{2}(0,T)$ is called a solution of reflected G-BSDE with data $(\zeta , f,S)$ if the following properties hold:

• (i) $Y_t=\zeta +\int _t^T f(s,Y_s,Z_s)\hbox {d}s -\int _t^T Z_s \hbox {d}B_s+(A_T-A_t)$;

• (ii) $Y_t\ge S_t$, and $\{-\int _0^t (Y_s-S_s)\hbox {d}A_s\}_{t\in [0,T]}$ is a non-increasing G-martingale,

where $\mathfrak {S}_{G}^{2}(0,T)$ is the collection of processes (Y, Z, A) such that $Y\in S_G^{2}(0,T)$, $Z\in M_G^{2}(0,T)$ and $A\in S_G^{2}(0,T)$ is a continuous non-decreasing process starting from origin.

The well-posedness of reflected G-BSDEs Consider the following assumption:

• (H1) there exists a constant $\beta >2$ such that for any y, z, $f(\cdot ,\cdot ,y,z)\in M_{G}^{\beta }(0,T)$;

• (H2) there exists a constant $L_1>0$ such that $|f(t,y,z)-f(t,y^{\prime },z^{\prime })|\le L_1(|y-y^{\prime }|+|z-z^{\prime }|);$

• (H3) there exists a constant c such that $\{S_t\}_{t\in [0,T]}\in S_G^\beta (0,T)$ and $S_t\le c$ for each $t\in [0,T]$;

• (H3’) $\{S_t\}_{t\in [0,T]}$ has the following form:

  $$\begin{aligned} S_t=S_0+\int _0^t b_s\hbox {d}s+\sum _{i,j=1}^{d}\int _0^t \gamma ^{ij}_sd\langle B^i, B^j\rangle _s+\sum _{j=1}^{d}\int _0^t \kappa _s^j \hbox {d}B_s^j, \end{aligned}$$

  where the processes $b_s,\gamma ^{ij}_s=\gamma ^{ji}_s\in M_G^\beta (0,T)$ and $\kappa _s^j\in H_G^\beta (0,T)$, $1\le i,j\le d$.

Lemma A.1

([12]) Assume that f satisfies (H1)-(H2) for some $\beta >2$ and let (H3) or (H3’) hold. Then, the reflected G-BSDE has a unique solution $(Y,Z,K)\in \mathfrak {S}_{G}^{2}(0,T)$.

Lemma A.2

([12]) Let $\zeta ^{\nu }\in L_{G}^{\beta }(\varOmega _{T})$, $\nu =1,2$ and $f^{\nu }$, $S^{\nu }$ satisfy (H1)-(H3) for some $\beta >2$. Assume that $(Y^{\nu },Z^{\nu },K^{\nu })\in \mathfrak {S}_{G}^{2}(0,T)$, $\nu =1,2$ are the solutions of the reflected G-BSDE corresponding to data ($\zeta ^{\nu }$, $f^{\nu }$, $S^{\nu }$). Set $\hat{Y}_{t}=Y_{t}^{1}-Y_{t}^{2}$, $\hat{S}_{t}=S_{t}^{1}-S_{t}^{2}$ and $\hat{\zeta }=\zeta ^1-\zeta ^2$. Then, there exists a constant $\hat{C}$ depending on T, G, $\beta$, c and $L_1$ such that

$$\begin{aligned} |Y^{\nu }_t|^2&\le \hat{C}\mathbb {\hat{E}}_t\left[ 1+|\zeta ^{\nu }|^2+\int ^T_t |\lambda _s^{\nu ,0}|^2\mathrm{d}s\right] ,\\ |\hat{Y}_t|^2&\le \hat{C}\left\{ \hat{\mathbb {E}}_t\left[ |\hat{\zeta }|^2+\int _t^T|f^{1}(s,Y_{s}^{2},Z_{s} ^{2})-f^{2}(s,Y_{s}^{2},Z_{s}^{2})|^2\mathrm{d}s\right] \right. \\&\left. \quad +\hat{\mathbb {E}}_t\left[ \sup _{s\in [t,T]}|\hat{S}_s|^2\right] ^{1/2}\varPsi _{t,T}^{1/2}\right\} , \end{aligned}$$

where $\varPsi _{t,T}=\sum _{\nu =1}^2\hat{\mathbb {E}}_t[\sup _{s\in [t,T]}\hat{\mathbb {E}}_s[1+|\zeta ^\nu |^2+\int _t^T |f^{\nu }(r,0,0)|^2\mathrm{d}r]]$.

Lemma A.3

([12]) Let $\zeta ^{\nu }\in L_{G}^{\beta }(\varOmega _{T})$, $\nu =1,2$ and $f^{\nu }$, $S^{\nu }$ satisfy (H1), (H2), (H3’) for some $\beta >2$. Assume that $(Y^{\nu },Z^{\nu },K^{\nu })\in \mathfrak {S}_{G}^{2}(0,T)$, $\nu =1,2$ are the solutions of the reflected G-BSDE with data ($\zeta ^{\nu }$, $f^{\nu }$, $S^{\nu }$). Set $\bar{Y}_{t}=(Y^1_t-S_t^1)-(Y^2_t-S_t^2)$ and $\hat{S}_{t}=S_{t}^{1}-S_{t}^{2}$. Then, there exists a constant $\bar{C}$ depending on T, G, $\beta$ and $L_1$ such that

$$\begin{aligned}&|Y^{\nu }_t|^2\le \bar{C}\mathbb {\hat{E}}_t\left[ |\zeta ^{\nu }|^2+\sup \limits _{s\in [t,T]}|S^{\nu }_s|^2+\int ^T_t |\bar{\lambda }_s^{\nu ,0}|^2\mathrm{d}s\right] ,\\&|\bar{Y}_t|^2\le \bar{ C}\left\{ \hat{\mathbb {E}}_t\left[ |\zeta ^{1}-S_T^1-\zeta ^{2}+S_T^2|^2+\int _t^T(|\hat{\lambda }_s|^2+|\hat{\rho }_s|^2+|\hat{S}_{s}|^2) \mathrm{d}s\right] \right\} , \end{aligned}$$

where $\hat{\lambda }_{s}=|f^{1}(s,Y_{s}^{2},Z_{s} ^{2})-f^{2}(s,Y_{s}^{2},Z_{s}^{2})|$, $\bar{\lambda }_s^{\nu ,0}=|f^{\nu }(s,0,0)|+|b^{\nu }_s|+\sum \nolimits _{i,j=1}^{d}|\gamma _s^{\nu ,ij}|+\sum \nolimits _{j=1}^d|\kappa _s^{\nu ,j}|$ and $\hat{\rho }_s=|b^1_s-b^2_s|+\sum \nolimits _{i,j=1}^{d}|\gamma _s^{1,ij}-\gamma _s^{2,ij}|+\sum \nolimits _{j=1}^d|\kappa _s^{1,j}-\kappa _s^{2,j}|$.

Appendix B: The Complement Proofs

The following maximal inequality for G-martingale has been firstly established by Song [24].

Lemma B.1

Assume $\alpha \ge 1$ and $\delta >0$. Set

$$\begin{aligned} C_{G}=2\inf \left\{ \frac{\gamma }{\gamma -1}\left( 1+14\sum _{i=1}^{\infty }i^{- \frac{\alpha +\delta }{\gamma }}\right) :1<\gamma <\alpha +\delta , \gamma \le 2\right\} . \end{aligned}$$

Then, we have

$$\begin{aligned} \mathbb {\hat{E}}_t\left[ \sup _{s\in [t,T]}\mathbb {\hat{E}}_{s}[|\xi |^{\alpha }]\right] \le C_{G}\{(\mathbb {\hat{E}}_t[|\xi |^{\alpha +\delta }])^{\alpha /(\alpha +\delta )}+\mathbb {\hat{E}}_t[|\xi |^{\alpha +\delta }]\}. \end{aligned}$$

Proof

The proof is immediate from the definition of conditional G-expectation and Theorem 3.4 in [24]. $\square$

Now, we are going to state the proof of Lemmas 3.2 and 3.3.

Proof

It is sufficient to prove the second inequalities in both cases, since the first ones can be proved similarly. For convenience, we omit superscripts t. We fist prove the second inequality in Lemma 3.2. Set $\hat{X}_s=X_s^{\xi ,u}-X_s^{\xi ',u'}$, $\hat{\varPhi }(X_T)=\varPhi (X^{\xi ,u}_T)-\varPhi (X^{\xi ^{\prime },u^{\prime }}_T)$ and $\hat{\lambda }_s=f(s,X^{\xi ^{\prime },u^{\prime }}_s,Y^{\xi ^{\prime },u^{\prime }}_s,Z^{\xi ^{\prime },u^{\prime }}_s,u^{\prime }_s) -f(s,X^{\xi ,u}_s,Y^{\xi ^{\prime },u^{\prime }}_s,Z^{\xi ^{\prime },u^{\prime }}_s, u_s)$. Applying Lemma A.2 and Lemma B.1 yields that

$$\begin{aligned} |\hat{Y}_t|^2 \le C_2\left\{ \hat{\mathbb {E}}_t\left[ |\hat{X}_T|^2+\int ^T_t(|\hat{X}_s|^2+|\hat{u}_s|^2)\hbox {d}s\right] + \hat{\mathbb {E}}_t\left[ \sup _{s\in [t,T]}|\hat{X}_s|^2\right] ^{1/2}\varPsi _{t,T}\right\} , \end{aligned}$$

where $C_2$ is a generic constant depending on T, G, c, L and n (may vary from line to line), and $\varPsi _{t,T}^2\le C_2(1+|\xi |^3+|\xi ^{\prime }|^3)$. From Lemma 2.1, we could get the desired result.

Then, we prove the second inequality in Lemma 3.3. Applying G-Itô’s formula (see Theorem 6.5 of Chap. III in [14]) to $l(s,X_s^{\xi ,u})$ yields that

$$\begin{aligned} l(s,X_s^{\xi ,u}) =l(t,\xi )+\int ^s_t b^{\xi ,u}_r\hbox {d}r+\int ^s_t \gamma ^{\xi ,u,ij}_rd\langle B^i, B^j\rangle _r+\int ^s_t \kappa ^{\xi ,u,j}_r\hbox {d}B^j_r, \end{aligned}$$

where $b^{\xi ,u},\gamma ^{\xi ,u}$ and $\kappa ^{\xi ,u}$ are given by

$$\begin{aligned} {b}^{\xi ,u}_s&=\partial _s l(s,X_s^{\xi ,u})+\langle \partial _x l(s,X_s^{\xi ,u}), b(s,X_s^{\xi ,u},u_s)\rangle ,\\ {\gamma }^{\xi ,u,{ij}}_s&=\langle \partial _x l(s,X_s^{\xi ,u}), h_{ij}(s,X_s^{\xi ,u},u_s)\rangle \\&\quad +\frac{1}{2}(\sigma ^{\top }(s,X_s^{\xi ,u},u_s)\partial _{xx}^2 l(s,X_s^{\xi ,u})\sigma (s,X_s^{\xi ,u},u_s))_{ij},\\ \kappa ^{\xi ,u,j}_s&=\langle \partial _x l(s,X_s^{\xi ,u}), \sigma ^{\top }_j(s,X_s^{t,\xi ,u},u_s)\rangle , \ \sigma ^{\top }_j \text { is the } j\text {-th row of } \sigma ^{\top }. \end{aligned}$$

Denote by $C_3$ a generic constant depending on T, G, n and L, which may vary from line to line. Then, recalling Lemma A.3, we deduce that

$$\begin{aligned} |\bar{Y}_t|^2\le C_3\left\{ \hat{\mathbb {E}}_t\left[ |\bar{\xi }|^2+\int _t^T(|\hat{\lambda }_s|^2+|\hat{\rho }_s|^2+|\hat{S}_{s}|^2) \hbox {d}s\right] \right\} , \end{aligned}$$

where

$$\begin{aligned}&\bar{Y}_t=(Y_t^{\xi ,u}-l(t,\xi ))-(Y_t^{\xi ',u'}-l(t,\xi ')),\ \hat{S}_s=l(s,X_s^{\xi ,u})-l(s,X_s^{\xi ',u'}),\\&\bar{\xi }=\varPhi (X^{\xi ,u}_T)-l(T,X^{\xi ,u}_T)-\varPhi (X^{\xi ^{\prime },u^{\prime }}_T)+l(T,X^{\xi ^{\prime },u^{\prime }}_T), \\&\hat{\lambda }_{s}=|f(s,X^{\xi ^{\prime },u^{\prime }}_s,Y^{\xi ^{\prime },u^{\prime }}_s,Z^{\xi ^{\prime },u^{\prime }}_s,u^{\prime }_s) -f(s,X^{\xi ,u}_s,Y^{\xi ^{\prime },u^{\prime }}_s,Z^{\xi ^{\prime },u^{\prime }}_s,u_s)|,\\&\hat{\rho }_s=|{b}^{\xi ,u}_s-b^{\xi ^{\prime },u^{\prime }}_s|+\sum \limits _{i,j=1}^{d}|\gamma _s^{\xi ,u,ij}-\gamma _s^{\xi ^{\prime },u^{\prime },ij}| +\sum \limits _{j=1}^d|\kappa _s^{\xi ,u,j}-\kappa _s^{\xi ^{\prime },u^{\prime },j}|. \end{aligned}$$

Set $\bar{\varPsi }_{t,T}=\sup \limits _{s\in [t,T]}(1+|X^{\xi ,u}_s|+|X^{\xi ^{\prime },u^{\prime }}_s| +|X^{\xi ,u}_s|^2+|X^{\xi ^{\prime },u^{\prime }}_s|^2)$ and $\hat{X}_s=X_s^{\xi ,u}-X_s^{\xi ',u'}$. Then, recalling assumptions (A1), (A2$^{\prime }$) and Lemma B.1, we derive that

$$\begin{aligned} |\bar{Y}_t|^2&\le C_3\left\{ \hat{\mathbb {E}}_t\left[ |\hat{X}_T|^2+\int ^T_t(|\hat{X}_s|^2 + |\hat{u}_s|^2)\hbox {d}s\right] +\hat{\mathbb {E}}_t[\bar{\varPsi }_{t,T}^4]^{\frac{1}{2}} \hat{\mathbb {E}}_t\left[ \int ^T_t(|\hat{X}_s|^4\right. \right. \\&\left. \left. \quad + |\hat{u}_s|^4)\hbox {d}s\right] ^{\frac{1}{2}} \right\} . \end{aligned}$$

Consequently, in spirit of Lemma 2.1, we get

$$\begin{aligned} |\bar{Y}_t|^2\le C_3(1+|\xi |^4+|\xi ^{\prime }|^4)\left( |\xi -\xi ^{\prime }|^2+\hat{\mathbb {E}}_t\left[ \int _t^T|u_s-u^{\prime }_s|^4\hbox {d}s\right] ^{\frac{1}{2}}\right) , \end{aligned}$$

which, together with $|Y^{\xi ,u}_t-Y^{\xi ^{\prime },u^{\prime }}_t|^2\le 2(|l(t,\xi )-l(t,\xi ^{\prime })|^2+|\bar{Y}_t|^2),$ implies the inequality (ii). The proof is complete. $\square$

Finally, we are ready to state the proof of Eq. (7).

Proof

For readers’ convenience, we shall give the sketch of the proof. For simplicity, we omit the superscripts (t, x).

From the proof of Lemma 4.4 in [12], it suffices to prove that

$$\begin{aligned} \lim \limits _{N\rightarrow \infty }\sup \limits _{u\in \mathcal {U}^{t}[t,T]}\mathbb {\hat{E}}\left[ \sup \limits _{s\in [t,T]}|(Y_s^{N,u}-l(s, X^{u}_s))^-|^2\right] =0. \end{aligned}$$

For simplicity, set $\varTheta ^{N,u}_r=(X_r^{u},Y_r^{N,u},Z_r^{N,u})$. Now, recalling Lemma 4.3 in [12], we derive that for each $(t,x)\in [0,T]\times \mathbb {R}^n$ and $u\in \mathcal {U}^t[t,T]$,

$$\begin{aligned} (Y_s^{N,u}-l(s, X^{u}_s))^- \le \hat{\mathbb {E}}_s\left[ |\tilde{S}_s^{N,u}|+|\int _s^T e^{N(s-r)}f(r,\varTheta _r^{N,u},u_r)\hbox {d}r|\right] , \end{aligned}$$

where $\tilde{S}_s^{N,u}=e^{N(s-T)}(\varPhi (X^{u}_T)-l(s, X^{u}_s))+\int _s^T Ne^{N(s-r)}(l(r, X^{u}_r)-l(s, X^{u}_s))\hbox {d}r$.

In spirit of Lemma B.1 and using a similar analysis as Equation (4.3) in [12], we conclude that

$$\begin{aligned} \lim \limits _{N\rightarrow \infty }\sup \limits _{u\in \mathcal {U}^{t}[t,T]}\mathbb {\hat{E}}\left[ \sup \limits _{s\in [t,T]}\hat{\mathbb {E}}_s\left[ |\int _s^T e^{N(s-r)}f(r,\varTheta _r^{N,u},u_r)\hbox {d}r|\right] ^2\right] =0. \end{aligned}$$

(8)

Next, we shall deal with the term $\tilde{S}_s^{N,u}$. Let $C_{5}>0$ be a generic constant. It follows from Equation (4.4) in [12] that for each fixed $\varepsilon ,\delta >0$ and $\beta >2$,

$$\begin{aligned}&\lim \limits _{N\rightarrow \infty }\sup \limits _{u\in \mathcal {U}^{t}[t,T]}\hat{\mathbb {E}}\left[ \sup _{s\in [t,T-\delta ]}|\tilde{S}_s^{N,u}|^\beta \right] \\ \le&C_5\sup \limits _{u\in \mathcal {U}^{t}[t,T]}\left\{ \hat{\mathbb {E}}\left[ \sup _{s\in [t,T]}\sup _{r\in [s,s+\varepsilon ]}|l(r, X^{u}_s)-l(s, X^{u}_s)|^{\beta }\right] \right. \\&\left. +\hat{\mathbb {E}}\left[ \sup _{s\in [t,T]}\sup _{r\in [s,s+\varepsilon ]}| X^{u}_r- X^{u}_s|^{\beta }\right] \right\} . \end{aligned}$$

Noting that for each $u\in \mathcal {U}^{t}[t,T]$ and any integer $\rho >0$, we have

$$\begin{aligned}&\hat{\mathbb {E}}\left[ \sup _{s\in [t,T]}\sup _{r\in [s,s+\varepsilon ]}|l(r, X^{u}_s)-l(s, X^{u}_s)|^{\beta }\right] \\ \le&\,\hat{\mathbb {E}}\left[ \sup _{s\in [t,T]}\sup _{r\in [s,s+\varepsilon ]}|l(r, X^{u}_s)-l(s, X^{u}_s)|^{\beta }I_A\right] +\frac{C_5}{\rho }\hat{\mathbb {E}}\left[ \sup _{s\in [t,T]}|X^{u}_s|+\sup _{s\in [t,T]}|X^{u}_s|^{1+\beta }\right] \\ \le&\sup _{s\in [t,T], |x|\le \rho }\sup _{r\in [s,s+\varepsilon ]}|l(r,x)-l(s,x)|^\beta +\frac{C_5}{\rho }. \end{aligned}$$

where $A={\{\sup \limits _{s\in [t,T]}|X^{u}_s|\le \rho \}}$. Denote $t^{\rho }_i=\frac{i}{\rho }(T-t)$, $i=0, \ldots , \rho$. Then, we deduce that, for each $\varepsilon \le \frac{T-t}{\rho }$,

$$\begin{aligned} \hat{\mathbb {E}}\left[ \sup _{s\in [t,T]}\sup _{r\in [s,s+\varepsilon ]}| X^{u}_r- X^{u}_s|^{\beta }\right]&\le C_5\sum \limits _{i=0}^{\rho -1}\hat{\mathbb {E}}\left[ \sup _{s\in [t^{\rho }_i,t^{\rho }_{i+1}]}| X^{u}_{t^{\rho }_i}- X^{u}_s|^{\beta }\right] \le {C_5}{\rho ^{1-\frac{\beta }{2}}}. \end{aligned}$$

Consequently, letting $\varepsilon \rightarrow 0$ and sending $\rho \rightarrow \infty$ we obtain that

$$\begin{aligned}&\lim \limits _{N\rightarrow \infty }\sup \limits _{u\in \mathcal {U}^{t}[t,T]}\hat{\mathbb {E}}\left[ \sup _{t\in [0,T-\delta ]}|\tilde{S}_s^{N,u}|^\beta \right] =0, \end{aligned}$$

which, together with Eq. (8) and (4.5) in [12], implies that for each $t<\delta <T$

$$\begin{aligned}&\lim \limits _{N\rightarrow \infty }\sup \limits _{u\in \mathcal {U}^{t}[t,T]}\mathbb {\hat{E}}\left[ \sup \limits _{s\in [t,T]}|(Y_s^{N,u}-l(s, X^{u}_s))^-|^2\right] \\&\le \sup \limits _{u\in \mathcal {U}^{t}[t,T]}\mathbb {\hat{E}}\left[ \sup \limits _{s\in [T-\delta ,T]}|(Y_s^{1,u}-l(s, X^{u}_s))^-|^2\right] . \end{aligned}$$

By the definition of $Y^{1,u}$, we have

$$\begin{aligned} Y_s^{1,u}=\mathbb {\hat{E}}_s\left[ \varPhi (X_T^{u})+\int _s^T f(r,\varTheta _r^{1,u},u_r)\hbox {d}r+\int _s^T (Y_r^{1,u}-l(r,X_r^{u}))^-\hbox {d}r\right] . \end{aligned}$$

It is easy to check that for some $\beta >2$,

$$\begin{aligned} \lim _{\delta \rightarrow 0}\sup _{u\in \mathcal {U}^{t}[t,T]}\mathbb {\hat{E}}\left[ \sup \limits _{s\in [T-\delta ,T]}\int _s^T\{ |f(r,\varTheta _r^{1,u},u_r)|^\beta +|Y_r^{1,u}-l(r,X_r^{u})|^\beta \}\hbox {d}r\right] =0. \end{aligned}$$

By Lemma B.1, it follows that

$$\begin{aligned} \lim _{\delta \rightarrow 0}\sup _{u\in \mathcal {U}^{t}[t,T]}\mathbb {\hat{E}}\left[ \sup \limits _{s\in [T-\delta ,T]}\mathbb {\hat{E}}_s\left[ \int _s^T |f(r,\varTheta _r^{1,u},u_r)|+|Y_r^{1,u}-l(r,X_r^{u})|\hbox {d}r\right] ^2\right] =0. \end{aligned}$$

Note that $(Y_T^{1,u}-l(T, X^{u}_T))^- =0$. Then, it holds that

$$\begin{aligned} (Y_s^{1,u}-l(s, X^{u}_s))^-\le |Y_s^{1,u}-\varPhi (X_T^{u})|+|l(s, X^{u}_s)-l(T, X^{u}_T)|. \end{aligned}$$

Thus, by a similar analysis as the above we derive that for each integer $\rho >0$,

$$\begin{aligned} \mathbb {\hat{E}}[\sup \limits _{s\in [T-\delta ,T]}|(Y_s^{1,u}-l(s, X^{u}_s))^-|^2]&\le C_5\left\{ \mathbb {\hat{E}}\left[ \sup \limits _{s\in [T-\delta ,T]}|\mathbb {\hat{E}}_s[\varPhi (X_T^{u})]-\varPhi (X_T^{u})|^2\right] \right. \\&\quad +\sup _{s\in [T-\delta ,T], |x|\le \rho }|l(s,x)-l(T,x)|^2\\&\quad \left. +\frac{C_5}{\rho }+m(\delta )\right\} , \end{aligned}$$

where $m(\cdot )$ is a nonnegative continuous function satisfying $\lim _{\delta \rightarrow 0}m(\delta )=0$.

On the other hand, it follows from Lemmas 2.1 and B.1 that

$$\begin{aligned} \lim _{\delta \rightarrow 0}\sup _{u\in \mathcal {U}^{t}[t,T]}\mathbb {\hat{E}} \left[ \sup \limits _{s\in [T-\delta ,T]}|\mathbb {\hat{E}}_s[\varPhi (X_T^{u})]-\varPhi (X_T^{u})|^2\right] =0. \end{aligned}$$

Therefore, by the above analysis, it holds that for each integer $\rho >0$

$$\begin{aligned}&\lim \limits _{N\rightarrow \infty }\sup \limits _{u\in \mathcal {U}^{t}[t,T]}\mathbb {\hat{E}}\left[ \sup \limits _{s\in [t,T]}|(Y_s^{N,u}-l(s, X^{u}_s))^-|^2\right] \\&\ \ \le C_5\left( \sup _{s\in [T-\delta ,T], |x|\le \rho }|l(s,x)-l(T,x)|^2+\frac{1}{\rho }+m(\delta )\right) . \end{aligned}$$

Sending $\delta \rightarrow 0$ and then $\rho \rightarrow \infty$ yields the desired result. $\square$