Proof. By the definition of the Skorokhod problem, we have \[d\phi(s) = \mathbf{n}(s)\,d|\phi|(s), \quad \mathbf{n}(s)\in N_{\xi(s)} \quad d|\phi|\text{-a.e.}\] Since the variation measure \(d|\phi|\) is supported on the boundary, one has \(\xi(s)\in\partial D\) for \(d|\phi|\)-almost every \(s\). Hence, by 3 , we have that for every \(y \in \overline{D}\), \[\langle \xi(s)-y,\mathbf{n}(s)\rangle \leq0 \quad d|\phi|\text{-a.e.},\] which gives \(\langle \xi(s)-y,d\phi(s)\rangle = \langle \xi(s)-y,\mathbf{n}(s)\rangle \,d|\phi|(s) \leq 0\). Integrating over \([0,t]\) proves ?? . For the second assertion, write \[d\phi_i(s) = \mathbf{n}_i(s)\,d|\phi_i|(s), \quad \mathbf{n}_i(s)\in N_{\xi_i(s)} \quad d|\phi_i|\text{-a.e.}, \quad i=1,2.\] Since \(\xi_2(s) \in\overline{D}\), the normal-cone characterization implies \[\langle \xi_{1}(s) - \xi_{2}(s), \mathbf{n}_{1}(s) \rangle \leq0 \quad d|\phi_1|\text{-a.e.}\] Similarly, since \(\xi_1(s) \in \overline{D}\), \[\langle \xi_{2}(s) - \xi_{1}(s), \mathbf{n}_{2}(s) \rangle \leq0 \quad d|\phi_2|\text{-a.e.}.\] It follows that \[\begin{align} &~\big\langle \xi_1(s)-\xi_2(s), d\phi_1(s)-d\phi_2(s) \big\rangle \\=&~ \langle \xi_1(s)-\xi_2(s), \mathbf{n}_{1}(s) \rangle \,d|\phi_1|(s) - \langle \xi_1(s)-\xi_2(s),\mathbf{n}_{2}(s) \rangle \,d|\phi_2|(s) \leq 0. \end{align}\] Integrating over \([0,t]\) proves ?? . ◻

Proof. Fix \(p \in [1,p_{\ast})\). Since \(2p-1 < 2\eta\), we may choose a constant \(\theta_{p}\in(0,1)\) such that \[\label{eq:theta-choice} 2p-1 < 2\eta\big(1 - \theta_{p}\big).\tag{9}\] For arbitrary matrices \(A,B\in\mathbb{R}^{d\times m}\), Young’s inequality yields \[\label{eq:diffusion-lower-bound} \|A\|_{\mathrm{HS}}^{2} = \|A-B+B\|_{\mathrm{HS}}^{2} \leq \frac{1}{1-\theta_{p}}\|A-B\|_{\mathrm{HS}}^{2} + \frac{1}{\theta_{p}}\|B\|_{\mathrm{HS}}^{2}.\tag{10}\] Taking \(y = x_{\ast}\) in ?? and applying 10 with \(A = \sigma(x)\) and \(B = \sigma(x_{\ast})\), we obtain \[\begin{align} 2\big\langle x-x_{\ast},b(x)-b(x_{\ast}) \big\rangle + 2\eta \big(1-\theta_{p}\big)\|\sigma(x)\|_{\mathrm{HS}}^{2} \leq 2L|x-x_{\ast}|^{2} + \frac{2\eta\big(1-\theta_{p}\big)}{\theta_{p}} \|\sigma(x_{\ast})\|_{\mathrm{HS}}^{2}. \end{align}\] Moreover, Young’s inequality implies \(2\big\langle x-x_{\ast},b(x_{\ast}) \big\rangle \leq |x-x_{\ast}|^{2} + |b(x_{\ast})|^{2}\). It follows that \[2\big\langle x-x_{\ast},b(x) \big\rangle + 2\eta\big(1-\theta_{p}\big) \|\sigma(x)\|_{\mathrm{HS}}^{2} \leq C_{p}\big(1 + |x-x_{\ast}|^{2}\big)\] with \[C_{p} := 1+2L + |b(x_{\ast})|^{2}+\frac{2\eta\big(1-\theta_{p}\big)}{\theta_{p}} \|\sigma(x_{\ast})\|_{\mathrm{HS}}^{2}.\] Define \(\varepsilon_{p} := 2\eta\big(1 - \theta_{p}\big) - (2p-1)\). By 9 , one has \(\varepsilon_{p}>0\), and thus ?? . ◻

Proof. We divide the proof into five steps.

Step 1. Construction of globally Lipschitz truncated equations. Choose an integer \(n_{0}\) such that \(n_{0} > |x_{0}| \vee |x_{\ast}|\). For every integer \(n \geq n_{0}\), we define \[\Pi_{n}(x) := \begin{cases} x, & |x|\leq n, \\[2mm]\displaystyle \frac{n}{|x|}x,& |x|>n. \end{cases}\] The mapping \(\Pi_{n}\) is the metric projection of \(\mathbb{R}^{d}\) onto the closed ball \(\overline{B}_{n} := \big\{x \in \mathbb{R}^{d} : |x| \leq n\big\}\). Since \(B_n\) is closed and convex, the metric projection \(\Pi_n=P_{B_n}\) is firmly nonexpansive, and hence nonexpansive; see, e.g., [31]. Therefore, \[\label{eq:projection-nonexpansive} |\Pi_{n}(x)-\Pi_{n}(y)| \leq |x-y|, \quad x,y \in \mathbb{R}^{d}.\tag{12}\] Define the truncated coefficients \(b_{n}(x) := b\big(\Pi_{n}(x)\big)\) and \(\sigma_{n}(x) := \sigma\big(\Pi_{n}(x)\big)\) for all \(x \in \mathbb{R}^{d}\). It follows immediately that \(b_{n}(x) = b(x), \sigma_{n}(x) = \sigma(x)\) for all \(x \in \mathbb{R}^{d}\) with \(|x| \leq n\). By ?? and 12 , we obtain \[\begin{align} |b_{n}(x)-b_{n}(y)| = \left|b\big(\Pi_{n}(x)\big) - b\big(\Pi_{n}(y)\big)\right| \leq L\big(1 + 2n^{q_{b}}\big)|\Pi_{n}(x)-\Pi_{n}(y)| \leq L\big(1 + 2n^{q_{b}}\big)|x-y|, \end{align}\] and similarly \(\|\sigma_{n}(x)-\sigma_{n}(y)\|_{\mathrm{HS}} \leq L\big(1 + 2n^{q_{\sigma}}\big)|x-y|\), i.e., both \(b_{n}\) and \(\sigma_{n}\) are globally Lipschitz continuous. For each \(n\geq n_{0}\), consider the truncated RSDE \[\label{eq:truncated-reflected-sde} X^{n}(t) = x_{0} + \int_{0}^{t} b_{n}\big(X^{n}(s)\big)\,ds + \int_{0}^{t} \sigma_{n}\big(X^{n}(s)\big)\,dW(s) + K^{n}(t), \quad t \geq 0.\tag{13}\] Here, \(X^{n}\) takes values in \(\overline{D}\), and \(K^{n}\) is a continuous adapted process of bounded variation satisfying \[\label{eq:truncated-reflection-decomposition} K^{n}(t) = \int_{0}^{t} \mathbf{n}^{n}(s)\,d|K^{n}|(s), \quad \mathbf{n}^{n}(s)\in N_{X^{n}(s)} \quad d|K^{n}|\text{-a.e.},\tag{14}\] together with \[\label{eq:truncated-reflection-support} |K^{n}|_{[0,t]} = \int_{0}^{t} \mathbf{1}_{\{X^{n}(s)\in\partial D\}} \,d|K^{n}|(s).\tag{15}\] Since \(b_{n}\) and \(\sigma_{n}\) are globally Lipschitz continuous, the classical well-posedness theory for the RSDEs on convex domains yields a unique global strong solution \((X^{n},K^{n})\) to 13 – 15 ; see, e.g., [1].

Step 2. Consistency of the truncated solutions. Let \(m > n \geq n_{0}\) and define \[\tau_{n}^{n} := \inf\big\{t \geq 0: |X^{n}(t)| \geq n \big\}, \quad \tau_{n}^{m} := \inf\big\{t \geq 0: |X^{m}(t)| \geq n\big\},\] where \(\inf\varnothing := \infty\), and set \(\rho_{n}^{n,m} := \tau_{n}^{n} \wedge \tau_{n}^{m}\). For \(t < \rho_{n}^{n,m}\), both \(X^{n}(t)\) and \(X^{m}(t)\) belong to \(B(0,n)\). Hence, \[\begin{gather} b_{n}\big(X^{n}(t)\big) = b\big(X^{n}(t)\big), \quad \sigma_{n}\big(X^{n}(t)\big) = \sigma\big(X^{n}(t)\big), \\ b_{m}\big(X^{m}(t)\big) = b\big(X^{m}(t)\big), \quad \sigma_{m}\big(X^{m}(t)\big) = \sigma\big(X^{m}(t)\big). \end{gather}\] Applying Itô’s formula to \(|X^{n}\big(t\wedge\rho_{n}^{n,m}\big) - X^{m}\big(t\wedge\rho_{n}^{n,m}\big)|^{2}\) and using ?? , we obtain \[\begin{align} &~\mathbb{E}\big[\left|X^{n}\big(t\wedge\rho_{n}^{n,m}\big) - X^{m}\big(t\wedge\rho_{n}^{n,m}\big)\right|^{2}\big] \\\leq&~ \mathbb{E}\bigg[\int_{0}^{t\wedge\rho_{n}^{n,m}} \big(2\big\langle X^{n}(s)-X^{m}(s), b\big(X^{n}(s)\big)-b\big(X^{m}(s)\big) \big\rangle \\&~+ \big\|\sigma\big(X^{n}(s)\big) - \sigma\big(X^{m}(s)\big) \big\|_{\mathrm{HS}}^{2}\big)\,ds\bigg] \\\leq&~ 2L\int_{0}^{t}\mathbb{E}\big[\left| X^{n}\big(s\wedge\rho_{n}^{n,m}\big) - X^{m}\big(s\wedge\rho_{n}^{n,m}\big)\right|^{2}\big]\,ds \end{align}\] due to ?? . Gronwall’s inequality yields \[\label{eq:truncated-solutions-consistency} X^{n}(t) = X^{m}(t), \quad 0 \leq t \leq \rho_{n}^{n,m}, \quad \mathbb{P}\text{-a.s.}\tag{16}\] By the continuity of \(X^{n}\) and \(X^{m}\), 16 implies \[\tau_{n}^{n} = \tau_{n}^{m}, \quad \mathbb{P}\text{-a.s.}\] We denote their common value by \(\tau_{n}\). Subtracting the two reflected equations and using 16 , we also obtain \[\label{eq:truncated-reflections-consistency} K^{n}(t) = K^{m}(t), \quad 0\leq t\leq\tau_{n}, \quad \mathbb{P}\text{-a.s.}\tag{17}\] The consistency relations may be assumed to hold simultaneously for all integers \(m > n \geq n_{0}\) on an event of probability one. Moreover, \(\tau_{n} \leq \tau_{n+1}\). Define \(\tau_{\infty} := \lim_{n \to \infty} \tau_{n}\). For every \(t < \tau_{\infty}\), choose \(n\) sufficiently large so that \(t < \tau_{n}\), and define \[\label{eq:maximal-local-solution} X(t) := X^{n}(t), \quad K(t) := K^{n}(t).\tag{18}\] By 16 and 17 , this definition is independent of the choice of \(n\). The pair \((X,K)\) is therefore a maximal local strong solution of 4 –6 on \([0,\tau_{\infty})\).

Step 3. Stopped moment estimates and nonexplosion. Fix \(1 \leq P < p_{\ast}\). Set \(Z^{n}(t) := X^{n}(t)-x_{\ast}\). Since \(b_{n}\big(X^{n}(s)\big) = b\big(X^{n}(s)\big), \sigma_{n}\big(X^{n}(s)\big) = \sigma\big(X^{n}(s)\big)\) for \(s \leq \tau_{n}\), Itô’s formula applied to \(V^{P}\big(X^{n}(t \wedge \tau_{n})\big)\) gives \[\begin{align} \label{eq:ito-stopped-exact-solution} V^{P}\big(X^{n}(t\wedge\tau_{n})\big) =&~ V^{P}(x_{0}) + P\int_{0}^{t\wedge\tau_{n}}V^{P-1}\big(X^{n}(s)\big) \big(2 \big\langle Z^{n}(s),b\big(X^{n}(s)\big) \big\rangle + \big\|\sigma\big(X^{n}(s)\big)\big\|_{\mathrm{HS}}^{2}\big) \,ds \nonumber \\&~+ 2P(P-1) \int_{0}^{t\wedge\tau_{n}} V^{P-2}\big(X^{n}(s)\big) \left|\sigma^{\top}\big(X^{n}(s)\big) Z^{n}(s) \right|^{2} \,ds \nonumber \\&~+ 2P\int_{0}^{t\wedge\tau_{n}}V^{P-1}\big(X^{n}(s)\big) \big\langle Z^{n}(s),\sigma\big(X^{n}(s)\big)\,dW(s) \big\rangle \nonumber \\&~+ 2P\int_{0}^{t\wedge\tau_{n}} V^{P-1}\big(X^{n}(s)\big) \big\langle Z^{n}(s), dK^{n}(s) \big\rangle. \end{align}\tag{19}\] By ?? with \(y = x_{\ast}\), we have \[\label{eq:exact-reflection-nonpositive} \int_{0}^{t\wedge\tau_{n}} V^{P-1}\big(X^{n}(s)\big) \big\langle Z^{n}(s),dK^{n}(s) \big\rangle \leq 0.\tag{20}\] Moreover, \[\left|\sigma^{\top}(x)(x-x_{\ast})\right|^{2} \leq |x-x_{\ast}|^{2}\|\sigma(x)\|_{\mathrm{HS}}^{2} \leq V(x) \|\sigma(x)\|_{\mathrm{HS}}^{2}.\] Therefore, the sum of the two finite-variation terms in 19 is bounded above by \[\begin{align} P\int_{0}^{t\wedge\tau_{n}}V^{P-1}\big(X^{n}(s)\big) \big(2 \big\langle Z^{n}(s),b\big(X^{n}(s)\big) \big\rangle + (2P-1)\big\|\sigma\big(X^{n}(s)\big)\big\|_{\mathrm{HS}}^{2}\big)\,ds. \end{align}\] By Lemma 2, we have \[\begin{align} 2 \langle x-x_{\ast},b(x) \rangle + (2P-1) \|\sigma(x)\|_{\mathrm{HS}}^{2} \leq C_{P}V(x) - \varepsilon_{P}\|\sigma(x)\|_{\mathrm{HS}}^{2} \leq C_{P}V(x). \end{align}\] Taking expectations in 19 , using 20 , and noting that the stopped stochastic integral has zero expectation, we obtain \[\mathbb{E}\big[V^{P}\big(X^{n}(t\wedge\tau_{n})\big)\big] \leq V^{P}(x_{0}) + C_{P} \int_{0}^{t} \mathbb{E}\big[ V^{P}\big(X^{n}(s\wedge\tau_{n})\big)\big] \,ds.\] Gronwall’s inequality yields \[\label{eq:stopped-exact-moment} \sup_{0\leq t\leq T} \mathbb{E} \big[V^{P}\big(X^{n}(t\wedge\tau_{n})\big)\big] \leq C_{P,T}V^{P}(x_{0}),\tag{21}\] where \(C_{P,T}\) is independent of \(n\). On the event \(\{\tau_{n}\leq T\}\), the continuity of \(X^{n}\) gives \(|X^{n}(\tau_{n})| = n\). Hence, \(V\big(X^{n}(\tau_{n})\big) = 1 + |X^{n}(\tau_{n})-x_{\ast}|^{2} \geq 1 + (n-|x_{\ast}|)^{2}\). It follows from 21 that \[\label{eq:truncation-exit-probability} \mathbb{P}(\tau_{n}\leq T) \leq \frac{C_{P,T}V^{P}(x_{0})}{\big(1 + (n-|x_{\ast}|)^{2}\big)^{P}}.\tag{22}\] Letting \(n \to \infty\) in 22 gives \(\mathbb{P}\big(\tau_{\infty} \leq T\big) = 0\). Since \(T > 0\) is arbitrary, we obtain \(\mathbb{P}\big( \tau_{\infty} = \infty\big) = 1\). Thus, the maximal local solution \((X,K)\) constructed in Step 2 is global.

Step 4. Uniform-in-time moment estimates. Fix \(1 \leq p < p_{\ast}\). Choose an exponent \(P\) such that \[\label{eq:intermediate-moment-index} p < P < p_{\ast}.\tag{23}\] For \(R > V(x_{0})\), define \(\vartheta_{R} := \inf\big\{t \geq 0: V\big(X(t)\big) \geq R\big\}\). Repeating the stopped Itô argument from Step 3 gives \[\label{eq:exact-solution-stopped-P-moment} \sup_{0\leq t\leq T}\mathbb{E}\big[ V^{P}\big(X(t\wedge\vartheta_{R})\big)\big] \leq C_{P,T}V^{P}(x_{0}),\tag{24}\] where the constant is independent of \(R\). Define \[M_{T} := \sup_{0\leq t\leq T} V\big(X(t)\big).\] By continuity, on the event \(\{M_{T}\geq R\}\) one has \(V\big(X(\vartheta_{R})\big) = R\). Hence, 24 implies \[\label{eq:exact-solution-tail-estimate} \mathbb{P}\big(M_{T}\geq R\big) \leq \frac{C_{P,T}V^{P}(x_{0})}{R^{P}}.\tag{25}\] Set \(A_{P,T} := C_{P,T}V^{P}(x_{0})\). Using the tail-integral representation for non-negative random variables (see, e.g., [32]) and 25 , we obtain \[\begin{align} \mathbb{E}\big[M_{T}^{p}\big] =&~ p\int_{0}^{\infty} R^{p-1}\mathbb{P}(M_{T}\geq R) \,dR \\\leq&~ p\int_{0}^{A_{P,T}^{1/P}}R^{p-1}\,dR + pA_{P,T}\int_{A_{P,T}^{1/P}}^{\infty}R^{p-P-1}\,dR \\\leq&~ C_{p,P,T}A_{P,T}^{p/P} \\\leq&~ C_{p,P,T}V^{p}(x_{0}). \end{align}\] This proves ?? . Finally, since \(|x|^{2p} \leq C_{p}\big(1 + |x-x_{\ast}|^{2p}\big) \leq C_{p}V^{p}(x)\) and \(V^{p}(x_{0}) \leq C_{p}\big(1 + |x_{0}|^{2p}\big)\), estimate ?? follows.

Step 5. Pathwise uniqueness. Let \((X,K)\) and \((Y,\widetilde{K})\) be two global strong solutions driven by the same Brownian motion and with the same initial value. For \(N \geq 1\), define \[\theta_{N} := \inf\big\{t \geq 0:|X(t)| \vee |Y(t)| \geq N\big\}.\] Applying Itô’s formula to \(|X(t\wedge\theta_{N}) - Y(t\wedge\theta_{N})|^{2}\) and using ?? , we obtain \[\begin{align} &~\mathbb{E}\big[|X(t\wedge\theta_{N})-Y(t\wedge\theta_{N})|^{2}\big] \\\leq&~ \mathbb{E}\bigg[\int_{0}^{t\wedge\theta_{N}} \big(2 \big\langle X(s)-Y(s), b\big(X(s)\big)-b\big(Y(s)\big) \big\rangle + \big\|\sigma\big(X(s)\big) - \sigma\big(Y(s)\big)\big\|_{\mathrm{HS}}^{2}\big)\,ds\bigg] \\\leq&~ 2L\int_{0}^{t}\mathbb{E}\big[ |X(s\wedge\theta_{N}) - Y(s\wedge\theta_{N})|^{2}\big]\,ds. \end{align}\] Gronwall’s inequality gives \(\mathbb{E}\big[|X(t\wedge\theta_{N}) - Y(t\wedge\theta_{N})|^{2}\big] = 0\). Letting \(N \to \infty\) and using the nonexplosion of both solutions, we conclude that \(\mathbb{P}(X(t) = Y(t), t \geq 0) = 1\). Subtracting the two reflected equations then yields \(\mathbb{P}(K(t) = \widetilde{K}(t), t \geq 0) = 1\). Thus, pathwise uniqueness holds. ◻

Proof. We construct the approximation recursively over the temporal grid. Set \(X_{h}(0) = x_{0}\) and \(K_{h}(0) = 0\). Suppose that \(X_{h}\) and \(K_{h}\) have already been uniquely constructed on \([0,t_{k}]\) for some \(k \in \{0,1,\cdots,N-1\}\). Since \(X_{h}(t_{k})\) is \(\mathcal{F}_{t_{k}}\)-measurable, the random variables \(b_{h}\big(X_{h}(t_{k})\big)\) and \(\sigma_{h}\big(X_{h}(t_{k})\big)\) are \(\mathcal{F}_{t_{k}}\)-measurable. Hence, the path \(Y_{h}^{k}\) defined by 32 is continuous and adapted on \([t_{k},t_{k+1}]\), and satisfies \(Y_{h}^{k}(t_{k}) = X_{h}(t_{k}) \in \overline{D}\). By the pathwise solvability of the Skorokhod problem on convex domains \(D\) recalled in Subsection 2.1, there exists a unique pair \(\{\big(X_{h}(t), K_{h}(t)-K_{h}(t_{k})\big)\}_{t \in [t_{k},t_{k+1}]}\) satisfying \[X_{h}(t) = Y_{h}^{k}(t) + K_{h}(t)-K_{h}(t_{k}), \quad t \in [t_{k},t_{k+1}],\] together with the corresponding normal-reflection and boundary-support conditions. Repeating this construction for \(k = 0, 1, \cdots, N-1\) yields a unique continuous adapted pair \((X_{h},K_{h})\) on \([0,T]\). By construction, this pair satisfies ?? –?? . ◻

Proof. By the definition of \(\lambda_{h}\), we have \(0 < \lambda_{h}(x) = \frac{1}{1+h^{1/2}G(x)} \leq 1\), which proves ?? . Hence, \(|b_{h}(x)| = \lambda_{h}(x)|b(x)| \leq |b(x)|\), and \(\|\sigma_{h}(x)\|_{\mathrm{HS}} = \lambda_{h}^{1/2}(x)\|\sigma(x)\|_{\mathrm{HS}} \leq \|\sigma(x)\|_{\mathrm{HS}}\). This proves ?? . Since \(|b(x)| \leq G(x)\) and \(\|\sigma(x)\|_{\mathrm{HS}}^{2} \leq G(x)\), we have \[\begin{gather} h^{1/2}|b_{h}(x)| = \frac{h^{1/2}|b(x)|}{1 + h^{1/2}G(x)} \leq \frac{h^{1/2}G(x)}{1 + h^{1/2}G(x)} \leq 1, \\ h^{1/2}\|\sigma_{h}(x)\|_{\mathrm{HS}}^{2} = h^{1/2}\lambda_{h}(x)\|\sigma(x)\|_{\mathrm{HS}}^{2} = \frac{h^{1/2}\|\sigma(x)\|_{\mathrm{HS}}^{2}}{1+h^{1/2}G(x)} \leq \frac{h^{1/2}G(x)}{1+h^{1/2}G(x)} \leq 1. \end{gather}\] Thus, ?? holds. Finally, ?? follows immediately from 7 , ?? and the definitions \(\gamma_{b} = q_{b} + 1\) and \(\gamma_{\sigma} = q_{\sigma} + 1\). ◻

Proof. By Lemma 2, we have \[\begin{align} 2 \big\langle x-x_{\ast},b(x) \big\rangle + \big(2p-1+\varepsilon_{p}\big) \|\sigma(x)\|_{\mathrm{HS}}^{2} \leq C_{p}V(x). \end{align}\] From \(b_{h}(x) = \lambda_{h}(x)b(x)\), \(\|\sigma_{h}(x)\|_{\mathrm{HS}}^{2} = \lambda_{h}(x) \|\sigma(x)\|_{\mathrm{HS}}^{2}\) and ?? , it follows that \[\begin{align} &~2\big\langle x-x_{\ast}, b_{h}(x) \big\rangle + \big(2p-1+\varepsilon_{p}\big) \|\sigma_{h}(x)\|_{\mathrm{HS}}^{2} \\=&~ \lambda_{h}(x) \big(2\big\langle x-x_{\ast}, b(x) \big\rangle + \big(2p-1+\varepsilon_{p}\big) \|\sigma(x)\|_{\mathrm{HS}}^{2}\big) \\\leq&~ C_{p}\lambda_{h}(x)V(x) \\\leq&~ C_{p}V(x). \end{align}\] Subtracting \(\varepsilon_{p}\|\sigma_{h}(x)\|_{\mathrm{HS}}^{2}\) from both sides gives the desired result. ◻

Proof. Since \(1 - \lambda_{h}(x) = \frac{h^{1/2}G(x)}{1 + h^{1/2}G(x)} \leq h^{1/2}G(x)\), we obtain \[\begin{align} |b(x)-b_{h}(x)| = \big(1-\lambda_{h}(x)\big)|b(x)| \leq h^{1/2}G(x)|b(x)|, \end{align}\] which proves ?? . Moreover, the inequality \[1-\lambda_{h}^{1/2}(x) = \frac{1-\lambda_{h}(x)}{1+\lambda_{h}^{1/2}(x)} \leq 1-\lambda_{h}(x) \leq h^{1/2}G(x)\] enables us to get \[\begin{align} \|\sigma(x)-\sigma_{h}(x)\|_{\mathrm{HS}} = \big(1 - \lambda_{h}^{1/2}(x)\big) \|\sigma(x)\|_{\mathrm{HS}} \leq h^{1/2} G(x) \|\sigma(x)\|_{\mathrm{HS}}, \end{align}\] proving ?? . By 7 , we have \[\begin{align} G(x) = |b(x)| + \|\sigma(x)\|_{\mathrm{HS}}^{2} \leq C\big(1 + |x|^{\gamma_{b}} + |x|^{2\gamma_{\sigma}}\big) \leq C\big(1 + |x|^{\widehat{\gamma}}\big). \end{align}\] It follows that \[\begin{gather} G(x)|b(x)| \leq C\big(1 + |x|^{\widehat{\gamma}}\big)\big(1 + |x|^{\gamma_{b}}\big) \leq C\big(1 + |x|^{\widehat{\gamma}+\gamma_{b}}\big), \\ G(x)\|\sigma(x)\|_{\mathrm{HS}} \leq C\big(1 + |x|^{\widehat{\gamma}}\big) \big(1 + |x|^{\gamma_{\sigma}}\big) \leq C\big(1 + |x|^{\widehat{\gamma}+\gamma_{\sigma}}\big). \end{gather}\] Combining these estimates with ?? and ?? proves ?? and ?? . ◻

Proof. For \(t \in [t_{k},t_{k+1}]\), define \(Z^{\tau}_{h,k}(t) := X_{h}(t \wedge \tau) - X_{h}(t_{k})\). Applying Itô’s formula gives \[\begin{align} \label{eq:ito-one-step-increment} |Z^{\tau}_{h,k}(t)|^{r} =&~ r \int_{t_{k}}^{t \wedge \tau} |Z_{h,k}(s)|^{r-2} \big\langle Z_{h,k}(s),b_{h}(X_{h}(t_{k})) \big\rangle \,ds \notag \\&~+ \frac{r}{2} \int_{t_{k}}^{t \wedge \tau} |Z_{h,k}(s)|^{r-2} \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2} \,ds \nonumber \\&~+ \frac{r(r-2)}{2}\int_{t_{k}}^{t \wedge \tau}|Z_{h,k}(s)|^{r-4} \left|\sigma_{h}(X_{h}(t_{k}))^{\top}Z_{h,k}(s)\right|^{2} \,ds \nonumber \\&~+ r\int_{t_{k}}^{t \wedge \tau} |Z_{h,k}(s)|^{r-2} \big\langle Z_{h,k}(s),\sigma_{h}(X_{h}(t_{k})) \,dW(s) \big\rangle \nonumber \\&~+ r\int_{t_{k}}^{t \wedge \tau}|Z_{h,k}(s)|^{r-2} \big\langle Z_{h,k}(s),dK_{h}(s) \big\rangle. \end{align}\tag{34}\] By Lemma 1 and \(X_{h}(t_{k}) \in \overline{D}\), we have \[\begin{align} r\int_{t_{k}}^{t\wedge\tau}|Z_{h,k}(s)|^{r-2} \big\langle Z_{h,k}(s),dK_{h}(s) \big\rangle \leq 0. \end{align}\] Together with \(\left|\sigma_{h}(X_{h}(t_{k}))^{\top}Z_{h,k}(s)\right|^{2} \leq \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2} |Z_{h,k}(s)|^{2}\), one gets \[\begin{align} |Z^{\tau}_{h,k}(t)|^{r} \leq{}&~ r\int_{t_{k}}^{t\wedge\tau}|Z_{h,k}(s)|^{r-1}|b_{h}(X_{h}(t_{k}))|\,ds + \frac{r(r-1)}{2} \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2} \int_{t_{k}}^{t\wedge\tau}|Z_{h,k}(s)|^{r-2} \,ds \nonumber \\&~ + r\int_{t_{k}}^{t \wedge \tau}|Z_{h,k}(s)|^{r-2} \big\langle Z_{h,k}(s),\sigma_{h}(X_{h}(t_{k}))\,dW(s) \big\rangle. \end{align}\] Setting \(\mathcal{Z}_{h,k}^{\tau} := \sup\limits_{t_{k} \leq t \leq t_{k+1}}|Z^{\tau}_{h,k}(t)|\) and taking the supremum over \(t \in [t_{k}, t_{k+1}]\), we obtain \[\begin{align} \label{eq:one-step-increment-supremum} \big(\mathcal{Z}_{h,k}^{\tau}\big)^{r} \leq&~ rh\big(\mathcal{Z}_{h,k}^{\tau}\big)^{r-1}|b_{h}(X_{h}(t_{k}))| + \frac{r(r-1)}{2}h\big(\mathcal{Z}_{h,k}^{\tau}\big)^{r-2} \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2} \nonumber \\&~+ r \sup_{t_{k} \leq t \leq t_{k+1}} \left|\int_{t_{k}}^{t \wedge \tau}|Z_{h,k}(s)|^{r-2} \big\langle Z_{h,k}(s),\sigma_{h}(X_{h}(t_{k}))\,dW(s) \big\rangle\right|. \end{align}\tag{35}\] Utilizing Young’s inequality shows that for every \(\delta > 0\), \[\begin{gather} \tag{36} h\big(\mathcal{Z}_{h,k}^{\tau}\big)^{r-1} |b_{h}(X_{h}(t_{k}))| \leq \delta \big(\mathcal{Z}_{h,k}^{\tau}\big)^{r} + C_{r,\delta} h^{r}|b_{h}(X_{h}(t_{k}))|^{r}, \\\tag{37} h \big( \mathcal{Z}_{h,k}^{\tau} \big)^{r-2} \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2} \leq \delta \big( \mathcal{Z}_{h,k}^{\tau} \big)^{r} + C_{r,\delta} h^{r/2} \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{r}. \end{gather}\] Besides, the conditional Burkholder–Davis–Gundy inequality implies \[\begin{align} \label{eq:one-step-martingale-estimate} &~\mathbb{E}_{k}\left[ \sup_{t_{k} \leq t \leq t_{k+1}} \left|\int_{t_{k}}^{t \wedge \tau} |Z_{h,k}(s)|^{r-2} \big\langle Z_{h,k}(s),\sigma_{h}(X_{h}(t_{k}))\,dW(s) \big\rangle \right| \right] \nonumber \\\leq&~ C_{r}\mathbb{E}_{k}\left[ \left(\int_{t_{k}}^{\tau} |Z_{h,k}(s)|^{2r-2} \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2} \,ds \right)^{1/2} \right] \nonumber \\\leq&~ C_{r}\mathbb{E}_{k}\left[ \big(\mathcal{Z}_{h,k}^{\tau}\big)^{r-1} h^{1/2} \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}} \right] \nonumber \\\leq&~ \delta \mathbb{E}_{k}\left[ \big(\mathcal{Z}_{h,k}^{\tau}\big)^{r} \right] + C_{r,\delta} h^{r/2} \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{r}, \end{align}\tag{38}\] where we have used the fact that \(\sigma_{h}(X_{h}(t_{k}))\) is \(\mathcal{F}_{t_{k}}\)-measurable. Taking conditional expectations in 35 , and then using 36 , 37 and 38 , we derive that \[\begin{align} \mathbb{E}_{k} \left[ \big( \mathcal{Z}_{h,k}^{\tau} \big)^{r} \right] \leq \frac{r(r+3)}{2}\delta \mathbb{E}_{k} \left[ \big(\mathcal{Z}_{h,k}^{\tau}\big)^{r} \right] + C_{r,\delta}\left(h^{r}|b_{h}(X_{h}(t_{k}))|^{r} + h^{r/2}\|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{r}\right). \end{align}\] Since \(r \geq 2\) is fixed and \(\delta > 0\) is arbitrary, one can choose \(\delta = \delta_{r} > 0\) sufficiently small (e.g., \(\delta_{r} = \frac{1}{r(r+3)+1})\) such that \(r(r+3)\delta < 1\). In this way, the first term on the right-hand side can be absorbed into the left-hand side, thus yielding ?? .

Applying ?? with \(\tau=t_{k+1}\) gives \[\begin{align} \mathbb{E}_{k}\left[\sup_{t_{k}\leq t\leq t_{k+1}} |X_{h}(t)-X_{h}(t_{k})|^{r} \right] \leq C_{r}\left(h^{r}|b_{h}(X_{h}(t_{k}))|^{r} + h^{r/2}\|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{r}\right). \end{align}\] Owing to ?? , we have \(h^{r}|b_{h}(X_{h}(t_{k}))|^{r} \leq h^{r/2}\) and \(h^{r/2}\|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{r} \leq h^{r/4}\). By \(h \in (0,1]\), \(h^{r/2} \leq h^{r/4}\) and hence \[\mathbb{E}_{k}\left[\sup_{t_{k} \leq t \leq t_{k+1}} |X_{h}(t)-X_{h}(t_{k})|^{r}\right] \leq C_{r}h^{r/4}.\] Taking expectations yields ?? . ◻

Proof. Recalling \(Z_{h,k}(s) = X_{h}(s) - X_{h}(t_{k}), s \in [t_{k},t_{k+1}]\) and setting \(\widehat{Z}_{h}(s) := X_{h}(s) - x_{\ast}\) and \(z_{h,k} := X_{h}(t_{k}) - x_{\ast}\) yield \(\widehat{Z}_{h}(s) = z_{h,k} + Z_{h,k}(s)\). Applying Itô’s formula to \(V^{P}\big(X_{h}(t)\big)\) on \([t_{k},t]\) with \(t \in [t_{k},t_{k+1}]\) gives \[\begin{align} \label{eq:ito-numerical-one-step-lyapunov} V^{P}\big(X_{h}(t)\big) \nonumber =&~ V_{h,k}^{P} + P\int_{t_{k}}^{t}V^{P-1}\big(X_{h}(s)\big) \big(2 \big\langle \widehat{Z}_{h}(s),b_{h}(X_{h}(t_{k})) \big\rangle + \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2} \big)\,ds \nonumber \\&~ + 2P(P-1)\int_{t_{k}}^{t}V^{P-2}\big(X_{h}(s)\big) \big|\sigma_{h}(X_{h}(t_{k}))^{\top}\widehat{Z}_{h}(s)\big|^{2}\,ds \nonumber \\&~ + 2P\int_{t_{k}}^{t}V^{P-1}\big(X_{h}(s)\big) \big\langle \widehat{Z}_{h}(s), \sigma_{h}(X_{h}(t_{k}))\,dW(s) \big\rangle \nonumber \\&~ + 2P\int_{t_{k}}^{t} V^{P-1}\big(X_{h}(s)\big) \big\langle \widehat{Z}_{h}(s),dK_{h}(s) \big\rangle. \end{align}\tag{39}\] Owing to ?? with \(y = x_{\ast}\) and \[\big|\sigma_{h}(X_{h}(t_{k}))^{\top} \widehat{Z}_{h}(s)\big|^{2} \leq \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2}|\widehat{Z}_{h}(s)|^{2} \leq \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2}V\big(X_{h}(s)\big),\] we take conditional expectations with \(t = t_{k+1}\) in 39 to obtain \[\begin{align} \mathbb{E}_{k}\left[V_{h,k+1}^{P}\right] \notag \leq V_{h,k}^{P} + P\mathbb{E}_{k}\int_{t_{k}}^{t_{k+1}}V^{P-1}\big(X_{h}(s)\big) \big(2\big\langle \widehat{Z}_{h}(s), b_{h}(X_{h}(t_{k})) \big\rangle + (2P-1)\|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2}\big)\,ds. \end{align}\] By \(\widehat{Z}_{h}(s) = z_{h,k} + Z_{h,k}(s)\), we have \(2\big\langle \widehat{Z}_{h}(s),b_{h}(X_{h}(t_{k})) \big\rangle = 2 \big\langle z_{h,k},b_{h}(X_{h}(t_{k})) \big\rangle + 2 \big\langle Z_{h,k}(s),b_{h}(X_{h}(t_{k})) \big\rangle\). Besides, Lemma 4 gives \[\begin{align} 2\big\langle z_{h,k},b_{h}(X_{h}(t_{k}))\big\rangle + (2P-1)\|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2} \leq C_{P}V_{h,k} - \varepsilon_{P}\|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2}. \end{align}\] It follows that \[\begin{align} \label{eq:one-step-lyapunov-IJD} \mathbb{E}_{k}\left[V_{h,k+1}^{P}\right] \leq V_{h,k}^{P} + C_{P}I_{h,k} - P\varepsilon_{P}D_{h,k} + 2PJ_{h,k} \end{align}\tag{40}\] with \[\begin{gather} I_{h,k} := \mathbb{E}_{k}\left[\int_{t_{k}}^{t_{k+1}} V_{h,k} V^{P-1}\big(X_{h}(s)\big) \,ds\right], \\ D_{h,k} := \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2}\, \mathbb{E}_{k} \left[\int_{t_{k}}^{t_{k+1}} V^{P-1}\big(X_{h}(s)\big) \,ds\right], \\ J_{h,k} := |b_{h}(X_{h}(t_{k}))|\,\mathbb{E}_{k} \left[\int_{t_{k}}^{t_{k+1}} V^{P-1}\big(X_{h}(s)\big) |Z_{h,k}(s)|\,ds\right]. \end{gather}\]

We estimate the three terms in 40 . Using \(V\big(X_{h}(s)\big) \leq 2\big(V_{h,k} + |Z_{h,k}(s)|^{2}\big)\) and Young’s inequality yields \[V_{h,k}V^{P-1}\big(X_{h}(s)\big) \leq C_{P}\big(V_{h,k}^{P} + |Z_{h,k}(s)|^{2P}\big).\] Together with Lemma 6, \(h|b_{h}(X_{h}(t_{k}))| \leq h^{1/2}\), \(h^{1/2}\|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}} \leq h^{1/4}\) and \(V_{h,k}^{P} \geq 1\), we deduce that \[\begin{align} \label{eq:Ihk-final-bound} I_{h,k} \leq&~ C_{P}hV_{h,k}^{P} + C_{P}h\mathbb{E}_{k} \left[\sup_{t_{k}\leq s\leq t_{k+1}} |Z_{h,k}(s)|^{2P}\right] \notag \\\leq&~ C_{P}h V_{h,k}^{P} + C_{P}h\Big((h|b_{h}(X_{h}(t_{k}))|)^{2P} + \big(h^{1/2}\|\sigma_{h}(X_{h}(t_{k})) \|_{\mathrm{HS}}\big)^{2P}\Big) \notag \\\leq&~ C_{P}h V_{h,k}^{P}. \end{align}\tag{41}\]

Concerning \(J_{h,k}\), the inequality \(V^{P-1}\big(X_{h}(s)\big) \leq C_{P} \big(V_{h,k}^{P-1} + |Z_{h,k}(s)|^{2P-2}\big)\) implies \[\label{eq:Jhk-splitting} J_{h,k} \leq C_{P}\big(J_{h,k}^{(1)}+J_{h,k}^{(2)}\big)\tag{42}\] with \[\begin{gather} J_{h,k}^{(1)} := hV_{h,k}^{P-1}|b_{h}(X_{h}(t_{k}))| \mathbb{E}_{k}\left[\sup_{t_{k}\leq s\leq t_{k+1}}|Z_{h,k}(s)|\right], \\ J_{h,k}^{(2)} := h|b_{h}(X_{h}(t_{k}))|\mathbb{E}_{k} \left[ \sup_{t_{k}\leq s\leq t_{k+1}}|Z_{h,k}(s)|^{2P-1}\right]. \end{gather}\] Making use of Lemma 6, Young’s inequality, \(h|b_{h}(X_{h}(t_{k}))|^{2}\leq1\) and \(V_{h,k} \geq 1\) shows that for every \(\delta > 0\), \[\begin{align} \label{eq:J1-with-Vk-diffusion} J_{h,k}^{(1)} \leq&~ Ch^{2}V_{h,k}^{P-1}|b_{h}(X_{h}(t_{k}))|^{2} + C h^{3/2} V_{h,k}^{P-1} |b_{h}(X_{h}(t_{k}))| \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}} \notag \\\leq&~ Ch^{2}V_{h,k}^{P-1}|b_{h}(X_{h}(t_{k}))|^{2} + C V_{h,k}^{P-1} \big(C_{\delta}h^{2}|b_{h}(X_{h}(t_{k}))|^{2} + \delta h \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2}\big) \notag \\\leq&~ C_{\delta}h V_{h,k}^{P} + C \delta h V_{h,k}^{P-1} \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2}. \end{align}\tag{43}\] We now compare the last term in 43 with \(D_{h,k}\). From \(V_{h,k}^{P-1} \leq C_{P}\big(V^{P-1}\big(X_{h}(s)\big) + |Z_{h,k}(s)|^{2P-2}\big)\), Lemma 6, \(h^{1/2} \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2} \leq 1\) and \(h|b_{h}(X_{h}(t_{k}))| \leq h^{1/2}\), it follows that \[\begin{align} \label{eq:Vk-diffusion-final-comparison} &~h V_{h,k}^{P-1} \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2} = \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2}\, \mathbb{E}_{k}\bigg[\int_{t_{k}}^{t_{k+1}} V_{h,k}^{P-1} \,ds\bigg] \notag \\\leq&~ C_{P}D_{h,k} + C_{P} h \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2}\, \mathbb{E}_{k} \left[\sup_{t_{k} \leq s\leq t_{k+1}} |Z_{h,k}(s)|^{2P-2} \right] \notag \\\leq&~ C_{P}D_{h,k} + C_{P} h \|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{2} \Big((h|b_{h}(X_{h}(t_{k}))|)^{2P-2} \notag \\&~+ \big(h^{1/2}\|\sigma_{h}(X_{h}(t_{k})) \|_{\mathrm{HS}}\big)^{2P-2}\Big) \notag \\\leq&~ C_{P}D_{h,k} + C_{P}h. \end{align}\tag{44}\] Combining 43 and 44 and using \(V_{h,k}^{P} \geq 1\) result in \[\label{eq:J1-final} J_{h,k}^{(1)} \leq C_{P,\delta} h V_{h,k}^{P} + C_{P}\delta D_{h,k}.\tag{45}\] For \(J_{h,k}^{(2)}\), Lemma 6, \(h|b_{h}(X_{h}(t_{k}))| \leq h^{1/2}\) and \(h^{1/2}\|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}} \leq h^{1/4}\) yield \[\begin{align} \label{eq:J2-final} J_{h,k}^{(2)} \leq C_{P}h|b_{h}(X_{h}(t_{k}))|\big((h|b_{h}(X_{h}(t_{k}))|)^{2P-1} + \big(h^{1/2}\|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}\big)^{2P-1}\big) \leq C_{P}h. \end{align}\tag{46}\] It follows from 42 , 45 and 46 that \[\label{eq:Jhk-final-bound} J_{h,k} \leq C_{P,\delta} h V_{h,k}^{P} + C_{P}\delta D_{h,k}.\tag{47}\]

Substituting 41 and 47 into 40 , we obtain \[\begin{align} \mathbb{E}_{k}\left[V_{h,k+1}^{P}\right] \leq \big(1+C_{P,\delta}h\big)V_{h,k}^{P} - \big(P\varepsilon_{P}-C_{P}\delta\big)D_{h,k}. \end{align}\] Choosing \(\delta > 0\) sufficiently small so that \(P\varepsilon_{P} - C_{P}\delta > 0\) proves the desired result. ◻

Proof. We first observe that \(\nu_{R}\) is a stopping time with respect to the discrete filtration \(\big(\mathcal{F}_{t_{k}}\big)_{k=0}^{N}\). Indeed, for every \(k\in\{0,1,\cdots,N\}\), we have \[\big\{\nu_{R} \leq k\big\} = \bigcup_{j=0}^{k}\big\{V_{h,j}\geq R\big\}.\] Since \(X_{h}(t_{j})\) is \(\mathcal{F}_{t_{j}}\)-measurable and \(\mathcal{F}_{t_{j}} \subset \mathcal{F}_{t_{k}}\) for \(0 \leq j \leq k\), it follows that \(\big\{\nu_{R} \leq k\big\} \in \mathcal{F}_{t_{k}}\).

For \(k\in\{0,1,\cdots,N-1\}\), define \(A_{k} := \big\{k < \nu_{R}\big\} \in \mathcal{F}_{t_{k}}\). On the event \(A_{k}\), the stopping index has not yet been reached. Since \(\nu_{R}\) is integer-valued, \(k<\nu_{R}\) implies \(\nu_{R} \geq k+1\), and hence \[k\wedge\nu_{R} = k, \quad (k+1)\wedge\nu_{R} = k+1 \quad \text{on }A_{k}.\] On the complementary event \(A_{k}^{c}=\{k\geq\nu_{R}\}\), the stopping has already occurred, and therefore \[k\wedge\nu_{R} = (k+1)\wedge\nu_{R} = \nu_{R} \quad \text{on }A_{k}^{c}.\] It follows that \[\begin{align} V_{h,(k+1)\wedge\nu_{R}}^{P} ={}&~ \mathbf{1}_{A_{k}} V_{h,k+1}^{P} + \mathbf{1}_{A_{k}^{c}} V_{h,k\wedge\nu_{R}}^{P}. \label{eq:stopped-variable-decomposition} \end{align}\tag{48}\] Since \[V_{h,k\wedge\nu_{R}}^{P} = \sum_{j=0}^{k-1} V_{h,j}^{P} \mathbf{1}_{\{\nu_{R}=j\}} + V_{h,k}^{P} \mathbf{1}_{\{\nu_{R}\geq k\}},\] and every term on the right-hand side is \(\mathcal{F}_{t_{k}}\)-measurable, we see that \(V_{h,k\wedge\nu_{R}}^{P}\) is \(\mathcal{F}_{t_{k}}\)-measurable. Taking conditional expectations in 48 and applying \(A_{k} \in \mathcal{F}_{t_{k}}\) gives \[\begin{align} \mathbb{E}_{k}\left[V_{h,(k+1)\wedge\nu_{R}}^{P}\right] = \mathbf{1}_{A_{k}}\mathbb{E}_{k}\left[V_{h,k+1}^{P}\right] + \mathbf{1}_{A_{k}^{c}}V_{h,k\wedge\nu_{R}}^{P} \leq \mathbf{1}_{A_{k}}\big(1+C_{P}h\big)V_{h,k}^{P} + \mathbf{1}_{A_{k}^{c}}V_{h,k\wedge\nu_{R}}^{P}, \end{align}\] where Lemma 7 has been used. Together with \(V_{h,k\wedge\nu_{R}}^{P} = V_{h,k}^{P}\) on \(A_{k}\), we have \[\begin{align} \mathbb{E}_{k}\left[V_{h,(k+1)\wedge\nu_{R}}^{P}\right] \leq&~ \big(1+C_{P}h\big) \big(\mathbf{1}_{A_{k}}V_{h,k}^{P} + \mathbf{1}_{A_{k}^{c}}V_{h,k\wedge\nu_{R}}^{P}\big) \\=&~ \big(1+C_{P}h\big) \big(\mathbf{1}_{A_{k}}V_{h,k\wedge\nu_{R}}^{P} + \mathbf{1}_{A_{k}^{c}}V_{h,k\wedge\nu_{R}}^{P}\big) \\=&~ \big(1+C_{P}h\big) V_{h,k\wedge\nu_{R}}^{P}. \end{align}\] Taking expectations and using the tower property of conditional expectation yields \[\label{eq:unconditional-stopped-recursion} \mathbb{E}\left[V_{h,(k+1)\wedge\nu_{R}}^{P}\right] \leq \big(1 + C_{P}h\big) \mathbb{E}\left[V_{h,k\wedge\nu_{R}}^{P}\right].\tag{49}\] Since \(R > V(x_{0}) = V_{h,0}\), one has \(\nu_{R} \geq 1\), and hence \(V_{h,0\wedge\nu_{R}}^{P} = V_{h,0}^{P} = V^{P}(x_{0})\). Iterating 49 , we obtain that for every \(k \in \{0,1,\cdots,N\}\), \[\begin{align} \mathbb{E}\left[V_{h,k\wedge\nu_{R}}^{P}\right] \leq \big(1+C_{P}h\big)^{k}V^{P}(x_{0}) \leq \exp\big(C_{P}kh\big)V^{P}(x_{0}) \leq \exp\big(C_{P}T\big)V^{P}(x_{0}). \end{align}\] The constant \(C_{P,T} := \exp\big(C_{P}T\big)\) is independent of both \(h\) and \(R\), and thus proving ?? . ◻

Proof. Define the maximal Lyapunov value over the temporal grid by \(M_{h,N} := \max_{0\leq k\leq N}V_{h,k}\). We first derive a tail estimate for \(M_{h,N}\). Fix \(R > V(x_{0})\) and recall the stopping index \[\nu_{R} := \inf\big\{k \in \{0,1,\cdots,N\}: V_{h,k} \geq R \big\}\] with the convention \(\inf\varnothing := N+1\). On the event \(\big\{M_{h,N} \geq R\big\}\), there exists at least one index \(k \in \{0,1,\cdots,N\}\) such that \(V_{h,k} \geq R\), which implies \(\nu_{R} \leq N\), \(V_{h,N\wedge\nu_{R}} = V_{h,\nu_{R}} \geq R\) and thus \[R^{P}\mathbf{1}_{\{M_{h,N}\geq R\}} \leq R^{P}\mathbf{1}_{\{\nu_{R}\leq N\}} \leq V_{h,N \wedge \nu_{R}}^{P}\mathbf{1}_{\{\nu_{R}\leq N\}} \leq V_{h,N \wedge \nu_{R}}^{P}.\] Taking expectations and applying Proposition 3 with \(k=N\) gives \[\begin{align} \label{eq:maximum-tail-large-R} \mathbb{P}\big(M_{h,N} \geq R\big) \leq \frac{\mathbb{E}\left[V_{h,N \wedge \nu_{R}}^{P}\right]}{R^{P}} \leq \frac{C_{P,T}V^{P}(x_{0})}{R^{P}}. \end{align}\tag{50}\] Since the constant in Proposition 3 has been chosen as \(C_{P,T} = \exp\big(C_{P}T\big) \geq 1\), one has \[\frac{C_{P,T}V^{P}(x_{0})}{R^{P}} \geq 1, \quad 0<R\leq V(x_{0}),\] which means that one can use the trivial estimate \(\mathbb{P}\big(M_{h,N}\geq R\big) \leq 1\) for \(R \in (0,V(x_{0})]\). Hence, we obtain that for every \(R>0\), \[\label{eq:grid-maximum-tail} \mathbb{P}\big(M_{h,N}\geq R\big) \leq \min\left\{1,\frac{C_{P,T}V^{P}(x_{0})}{R^{P}}\right\}.\tag{51}\]

Setting \(A := C_{P,T}^{1/P}V(x_{0}) \geq V(x_{0})\) and using the tail-integral representation for nonnegative random variables (see, e.g., [32]), we obtain \[\begin{align} \mathbb{E}\left[M_{h,N}^{p}\right] =&~ p\int_{0}^{\infty}R^{p-1}\mathbb{P} \big(M_{h,N} \geq R\big)\,dR \\=&~ p\int_{0}^{A}R^{p-1}\mathbb{P} \big(M_{h,N} \geq R\big)\,dR + p\int_{A}^{\infty}R^{p-1} \mathbb{P}\big(M_{h,N} \geq R\big)\,dR \\\leq{}&~ p\int_{0}^{A}R^{p-1}\,dR + pC_{P,T}V^{P}(x_{0}) \int_{A}^{\infty}R^{p-P-1}\,dR. \end{align}\] Since \(p < P\), the second improper integral is finite and satisfies \[\int_{A}^{\infty} R^{p-P-1} \,dR = \frac{A^{p-P}}{P-p}.\] It follows that \[\begin{align} \mathbb{E}\left[M_{h,N}^{p}\right] \leq A^{p} + \frac{pA^{p-P}}{P-p}C_{P,T}V^{P}(x_{0}) \leq \left(1+\frac{p}{P-p}\right) C_{P,T}^{p/P}V^{p}(x_{0}) = \frac{P}{P-p}C_{P,T}^{p/P}V^{p}(x_{0}), \end{align}\] which yields ?? . Finally, ?? follows from \(|x|^{2p} \leq C_{p}V^{p}(x)\) and \(V^{p}(x_{0}) \leq C_{p}(1 + |x_{0}|^{2p})\). ◻

Proof. Applying Lemma 6, ?? , \(\gamma_{b} \leq \gamma\) and \(\gamma_{\sigma} \leq \gamma\) leads to \[\begin{align} \mathbb{E}\left[\sup_{t_{k}\leq t\leq t_{k+1}} |X_{h}(t)-X_{h}(t_{k})|^{r}\right] \leq&~ C_{r}h^{r}\mathbb{E}\left[|b_{h}(X_{h}(t_{k}))|^{r}\right] + C_{r}h^{r/2}\mathbb{E} \left[\|\sigma_{h}(X_{h}(t_{k}))\|_{\mathrm{HS}}^{r}\right]. \\\leq&~ C_{r}h^{r} \big(1 + \mathbb{E}\left[|X_{h}(t_{k})|^{r\gamma_{b}}\right]\big) + C_{r}h^{r/2}\big(1 + \mathbb{E}\left[|X_{h}(t_{k})|^{r\gamma_{\sigma}}\right]\big) \\\leq&~ C_{r}(h^{r}+h^{r/2}) \big(1 + \mathbb{E}\left[|X_{h}(t_{k})|^{r\gamma}\right]\big). \end{align}\] Together with \(|x|^{r\gamma} \leq C_{r,\gamma}V^{r\gamma/2}(x)\) for all \(x \in \mathbb{R}^{d}\), the condition ?? allows us to apply Proposition 4 with the moment exponent \(r\gamma/2\) to obtain \[\mathbb{E}\left[\sup_{t_{k}\leq t\leq t_{k+1}} |X_{h}(t)-X_{h}(t_{k})|^{r}\right] \leq C_{r,P,T}\big(h^{r} + h^{r/2}\big).\] Since \(h \in (0,1]\) and \(r \geq 2\), we have \(h^{r} \leq h^{r/2}\). Therefore, ?? follows. ◻

Proof. For \(k = 0,1,\cdots,N-1\), we define \(\mathcal{I}_{h,k} := \sup_{t_{k}\leq t\leq t_{k+1}}|X_{h}(t)-X_{h}(t_{k})|\). It follows that for every \(t \in [t_{k},t_{k+1}]\), \[\begin{align} V\big(X_{h}(t)\big) \leq 1 + 2|X_{h}(t_{k})-x_{\ast}|^{2} + 2|X_{h}(t)-X_{h}(t_{k})|^{2} \leq 2V_{h,k} + 2\mathcal{I}_{h,k}^{2}, \end{align}\] which in combination with Proposition 4 and ?? implies \[\begin{align} \label{eq:path-maximum-grid-increment} \mathbb{E}\left[\sup_{0\leq t\leq T}V^{p}\big(X_{h}(t)\big)\right] \leq&~ C_{p}\mathbb{E}\left[\max_{0\leq k\leq N}V_{h,k}^{p}\right] + C_{p}\mathbb{E}\left[\max_{0\leq k\leq N-1}\mathcal{I}_{h,k}^{2p}\right] \notag \\\leq&~ C_{p,P,T}V^{p}(x_{0}) + C_{p}\mathbb{E}\left[\max_{0\leq k\leq N-1}\mathcal{I}_{h,k}^{2p}\right]. \end{align}\tag{52}\] It remains to estimate the maximal excursion between grid points. By Hölder’s inequality, one has \[\mathbb{E}\left[\max_{0\leq k\leq N-1}\mathcal{I}_{h,k}^{2p}\right] \leq \left(\mathbb{E}\left[\max_{0\leq k\leq N-1} \mathcal{I}_{h,k}^{2P}\right]\right)^{p/P}.\] Since the maximum of nonnegative numbers is bounded by their sum, ?? gives \[\begin{align} \mathbb{E}\left[\max_{0\leq k\leq N-1}\mathcal{I}_{h,k}^{2P}\right] \leq \sum_{k=0}^{N-1}\mathbb{E}\left[\mathcal{I}_{h,k}^{2P}\right] \leq C_{P}Nh^{P/2} = C_{P}Th^{P/2-1}. \end{align}\] From \(P > 2\) and \(h \in (0,1]\), \(h^{P/2-1} \leq 1\), it follows that \[\label{eq:max-increment-uniform} \mathbb{E}\left[\max_{0\leq k\leq N-1} \mathcal{I}_{h,k}^{2p}\right] \leq C_{p,P,T}.\tag{53}\] Combining 52 and 53 , we obtain \[\mathbb{E}\left[\sup_{0\leq t\leq T} V^{p}\big(X_{h}(t)\big)\right] \leq C_{p,P,T}\big(1+V^{p}(x_{0})\big).\] Observing \(V(x_{0}) \geq 1\), the constant term may be absorbed into \(V^{p}(x_{0})\), proving ?? . Finally, ?? follows from \(|x|^{2p} \leq C_{p}V^{p}(x)\) and \(V^{p}(x_{0}) \leq C_{p}\big(1 + |x_{0}|^{2p}\big)\). ◻

Proof. We first estimate the freezing defects. By ?? , we have \[\begin{align} \left|R_{b,h}^{\mathrm{fr}}(t)\right| \leq L\big(1 + |X_{h}(t)|^{q_{b}} + |\overline{X}_{h}(t)|^{q_{b}}\big) |\Delta_{h}X(t)|. \end{align}\] Suppose first that \(q_{b}>0\), and define the conjugate exponents \(a_{b} := \frac{q_{b}+\gamma}{q_{b}}, a_{b}' := \frac{q_{b}+\gamma}{\gamma}\) satisfying \(\frac{1}{a_{b}} + \frac{1}{a_{b}'} = 1\). Applying Hölder’s inequality gives \[\begin{align} \label{eq:drift-freezing-holder} \mathbb{E}\left[|R_{b,h}^{\mathrm{fr}}(t)|^{2p}\right] \leq C_{p}\left(\mathbb{E}\big[1 + |X_{h}(t)|^{2p(q_{b}+\gamma)} + |\overline{X}_{h}(t)|^{2p(q_{b}+\gamma)}\big]\right)^{1/a_{b}} \left(\mathbb{E}|\Delta_{h}X(t)|^{2p(q_{b}+\gamma)/\gamma}\right)^{1/a_{b}'}. \end{align}\tag{56}\] Since \(p(q_{b}+\gamma) < P\), Proposition 6 implies that the first factor on the right-hand side of 56 is uniformly bounded. Noting that \[\frac{\gamma}{2}\frac{2p(q_{b}+\gamma)}{\gamma} = p(q_{b}+\gamma) < P,\] Proposition 5 yields \(\mathbb{E}\big[|\Delta_{h}X(t)|^{2p(q_{b}+\gamma)/\gamma}\big] \leq C_{p,P,T}h^{p(q_{b}+\gamma)/\gamma}\) and consequently \[\begin{align} \left(\mathbb{E}\left[|\Delta_{h}X(t)|^{\frac{2p(q_{b}+\gamma)}{\gamma}}\right]\right)^{1/a_{b}'} \leq C_{p,P,T}h^{\frac{2p(q_{b}+\gamma)}{\gamma}/(2a_{b}')} = C_{p,P,T}h^{p}. \end{align}\] Substituting this estimate into 56 gives \[\label{eq:drift-freezing-pointwise-moment} \mathbb{E}\left[|R_{b,h}^{\mathrm{fr}}(t)|^{2p}\right] \leq C_{p,P,T}h^{p}.\tag{57}\] If \(q_{b}=0\), ?? gives \(\left|R_{b,h}^{\mathrm{fr}}(t)\right| \leq C|\Delta_{h}X(t)|\). Since \(p\gamma < P\), Proposition 5 with \(r=2p\) again gives 57 . The diffusion freezing defect is treated in the same way. Indeed, by ?? , one gets \[\begin{align} \left\|R_{\sigma,h}^{\mathrm{fr}}(t)\right\|_{\mathrm{HS}} \leq L\big(1 + |X_{h}(t)|^{q_{\sigma}} + |\overline{X}_{h}(t)|^{q_{\sigma}}\big)|\Delta_{h}X(t)|. \end{align}\] Since ?? also implies \(p(q_{\sigma}+\gamma) < P\), repeating the proceeding arguments gives \[\label{eq:diffusion-freezing-pointwise-moment} \mathbb{E}\left[\| R_{\sigma,h}^{\mathrm{fr}}(t) \|_{\mathrm{HS}}^{2p} \right] \leq C_{p,P,T}h^{p}.\tag{58}\]

We next estimate the taming defects. Owing to ?? and ?? , one gets \[\begin{gather} \left|R_{b,h}^{\mathrm{tm}}(t)\right| = \left|b\big(\overline{X}_{h}(t)\big) - b_{h}\big(\overline{X}_{h}(t)\big)\right| \leq Ch^{1/2}\left(1 + |\overline{X}_{h}(t)|^{\widehat{\gamma}+\gamma_{b}}\right), \\ \left\|R_{\sigma,h}^{\mathrm{tm}}(t)\right\|_{\mathrm{HS}} = \left\|\sigma\big(\overline{X}_{h}(t)\big) - \sigma_{h}\big(\overline{X}_{h}(t)\big)\right\|_{\mathrm{HS}} \leq Ch^{1/2}\left( 1 + |\overline{X}_{h}(t)|^{\widehat{\gamma}+\gamma_{\sigma}}\right). \end{gather}\] Since \(\widehat{\gamma}+\gamma_{b} \leq \widehat{\gamma}+\gamma = \Lambda(q_{b},q_{\sigma})\) and \(\widehat{\gamma}+\gamma_{\sigma} \leq \widehat{\gamma} + \gamma = \Lambda(q_{b},q_{\sigma})\), ?? and Proposition 6 imply that \[\begin{align} \label{eq:drift-taming-defect-before-moment} \mathbb{E}\big[|R_{b,h}^{\mathrm{tm}}(t)|^{2p}\big] + \mathbb{E}\big[\|R_{\sigma,h}^{\mathrm{tm}} (t)\|_{\mathrm{HS}}^{2p}\big] \leq C_{p,P,T}h^{p}. \end{align}\tag{59}\]

Finally, by means of the decompositions 54 and 55 , one has \[\begin{gather} |R_{b,h}(t)|^{2p} \leq C_{p}\big(\left|R_{b,h}^{\mathrm{fr}}(t)\right|^{2p} + \left|R_{b,h}^{\mathrm{tm}}(t)\right|^{2p}\big), \\ \|R_{\sigma,h}(t)\|_{\mathrm{HS}}^{2p} \leq C_{p}\big(\left\|R_{\sigma,h}^{\mathrm{fr}}(t)\right\|_{\mathrm{HS}}^{2p} + \left\|R_{\sigma,h}^{\mathrm{tm}}(t)\right\|_{\mathrm{HS}}^{2p}\big). \end{gather}\] Together with 57 , 58 and 59 , we obtain the desired result. ◻

Proof. Let \(e_{h}(t) := X(t)-X_{h}(t), t \in [0,T]\) and define the principal coefficient differences \[\Delta b_{h}(t) := b\big(X(t)\big) - b\big(X_{h}(t)\big), \quad \Delta\sigma_{h}(t) := \sigma\big(X(t)\big) - \sigma\big(X_{h}(t)\big).\] It follows from 54 and 55 that \[b\big(X(t)\big) - b_{h}\big(\overline{X}_{h}(t)\big) = \Delta b_{h}(t) + R_{b,h}(t), \quad \sigma\big(X(t)\big) - \sigma_{h}\big(\overline{X}_{h}(t)\big) = \Delta\sigma_{h}(t) + R_{\sigma,h}(t),\] which shows that the error process satisfies \[\begin{align} e_{h}(t) = \int_{0}^{t} \big(\Delta b_{h}(s) + R_{b,h}(s)\big) \,ds + \int_{0}^{t} \big(\Delta\sigma_{h}(s) + R_{\sigma,h}(s)\big)\,dW(s) + K(t)-K_{h}(t). \end{align}\] Noting that there exists an exponent \(P\) satisfying \(p\Lambda(q_{b},q_{\sigma}) < P < p_{\ast}\), we may choose an exponent \(q\) such that \[\label{eq:auxiliary-error-index} p < q < \frac{P}{\Lambda(q_{b},q_{\sigma})} < p_{\ast}\tag{60}\] due to \(\Lambda(q_{b},q_{\sigma}) \geq 1\). Since \(q < p_{\ast} = \eta+\frac{1}{2}\), one has \(2q-1 < 2\eta\). We may therefore choose a constant \(\delta_{q}>0\) sufficiently small such that \[\label{eq:delta-q-choice} (2q-1)(1+\delta_{q}) < 2\eta.\tag{61}\] We next apply Itô’s formula to \(|e_{h}(t)|^{2q}\). A standard localization argument is understood in the calculations below. We obtain \[\begin{align} \label{eq:ito-state-error} |e_{h}(t)|^{2q} =&~ 2q\int_{0}^{t}|e_{h}(s)|^{2q-2}\big\langle e_{h}(s),\Delta b_{h}(s) + R_{b,h}(s)\big\rangle\,ds \nonumber \\&~ + q\int_{0}^{t}|e_{h}(s)|^{2q-2} \big\|\Delta\sigma_{h}(s) + R_{\sigma,h}(s)\big\|_{\mathrm{HS}}^{2}\,ds \nonumber \\&~ + 2q(q-1)\int_{0}^{t}|e_{h}(s)|^{2q-4} \big|\big(\Delta\sigma_{h}(s) + R_{\sigma,h}(s)\big)^{\top}e_{h}(s)\big|^{2}\,ds \nonumber \\&~ + 2q\int_{0}^{t}|e_{h}(s)|^{2q-2} \big\langle e_{h}(s),dK(s)-dK_{h}(s) \big\rangle + M_{q,h}(t), \end{align}\tag{62}\] where \[\begin{align} M_{q,h}(t) := 2q \int_{0}^{t} |e_{h}(s)|^{2q-2} \big\langle e_{h}(s), \big(\Delta\sigma_{h}(s) + R_{\sigma,h}(s)\big) \,dW(s)\big\rangle \end{align}\] is a continuous local martingale.

We first examine the reflection term. By 5 and ?? , we have \[\begin{align} \big\langle e_{h}(s),dK(s)-dK_{h}(s) \big\rangle = \big\langle X(s)-X_{h}(s),\mathbf{n}_{X}(s) \big\rangle \,d|K|(s) - \big\langle X(s)-X_{h}(s),\mathbf{n}_{h}(s) \big\rangle \,d|K_{h}|(s). \end{align}\] Owing to \(X_{h}(s) \in \overline{D}\) and \(X(s) \in \overline{D}\), the normal-cone characterization 3 gives \[\begin{gather} \big\langle X(s)-X_{h}(s),\mathbf{n}_{X}(s) \big\rangle \leq 0 \quad d|K|\text{-a.e.}, \\ \big\langle X(s)-X_{h}(s),\mathbf{n}_{h}(s) \big\rangle \geq 0 \quad d|K_{h}|\text{-a.e.}, \end{gather}\] and hence \[\label{eq:weighted-reflection-error-negative} |e_{h}(s)|^{2q-2} \big\langle e_{h}(s), dK(s)-dK_{h}(s) \big\rangle \leq 0.\tag{63}\] For the quadratic diffusion terms, we use \(|A^{\top}x|^{2} \leq \|A\|_{\mathrm{HS}}^{2}|x|^{2}\) to obtain \[\begin{align} \label{eq:quadratic-diffusion-error-bound} &~q|e_{h}(s)|^{2q-2}\big\|\Delta\sigma_{h}(s) + R_{\sigma,h}(s)\big\|_{\mathrm{HS}}^{2} \notag \\&~+ 2q(q-1)|e_{h}(s)|^{2q-4} \big|\big(\Delta\sigma_{h}(s) + R_{\sigma,h}(s)\big)^{\top}e_{h}(s)\big|^{2} \notag \\\leq&~ q(2q-1)|e_{h}(s)|^{2q-2} \big\|\Delta\sigma_{h}(s) + R_{\sigma,h}(s)\big\|_{\mathrm{HS}}^{2} \notag \\\leq&~ q(2q-1)|e_{h}(s)|^{2q-2} \bigg((1+\delta_{q})\|\Delta\sigma_{h}(s)\|_{\mathrm{HS}}^{2} + \left( 1+\frac{1}{\delta_{q}} \right) \|R_{\sigma,h}(s)\|_{\mathrm{HS}}^{2}\bigg), \end{align}\tag{64}\] where Young’s inequality has been used. As 61 ensures \(\frac{(2q-1)(1+\delta_{q})}{2} < \eta\), the coupled monotonicity condition ?? yields \[\begin{align} \label{eq:principal-coupled-error-bound} &~2q|e_{h}(s)|^{2q-2} \big\langle e_{h}(s), \Delta b_{h}(s)\big\rangle + q(2q-1)(1+\delta_{q})|e_{h}(s)|^{2q-2} \|\Delta\sigma_{h}(s)\|_{\mathrm{HS}}^{2} \notag \\=&~ 2q |e_{h}(s)|^{2q-2}\Big(\big\langle e_{h}(s), \Delta b_{h}(s) \big\rangle + \frac{(2q-1)(1+\delta_{q})}{2} \|\Delta\sigma_{h}(s)\|_{\mathrm{HS}}^{2}\Big) \notag \\\leq&~ 2qL|e_{h}|^{2q}. \end{align}\tag{65}\] Besides, utilizing Young’s inequality again results in \[\begin{align} \tag{66} 2q|e_{h}(s)|^{2q-2}\big\langle e_{h}(s),R_{b,h}(s)\big\rangle \leq&~ C_{q}|e_{h}(s)|^{2q} + C_{q}|R_{b,h}(s)|^{2q}, \\\tag{67} |e_{h}(s)|^{2q-2}\|R_{\sigma,h}(s)\|_{\mathrm{HS}}^{2} \leq&~ C_{q}|e_{h}(s)|^{2q} + C_{q}\|R_{\sigma,h}(s)\|_{\mathrm{HS}}^{2q}. \end{align}\] Substituting 63 , 64 , 65 , 66 and 67 into 62 , we obtain \[\begin{align} \label{eq:error-stochastic-gronwall-form} |e_{h}(t)|^{2q} \leq C_{q}\int_{0}^{t}|e_{h}(s)|^{2q}\,ds + C_{q}\int_{0}^{t}\big(|R_{b,h}(s)|^{2q} + \|R_{\sigma,h}(s)\|_{\mathrm{HS}}^{2q}\big)\,ds + M_{q,h}(t). \end{align}\tag{68}\]

Define \[\mathcal{R}_{q,h}(t) := C_{q}\int_{0}^{t}\big(|R_{b,h}(s)|^{2q} + \|R_{\sigma,h}(s)\|_{\mathrm{HS}}^{2q}\big)\,ds.\] Then \(\mathcal{R}_{q,h}\) is nonnegative, continuous, adapted, and nondecreasing. Moreover, 68 takes the form \[|e_{h}(t)|^{2q} \leq \mathcal{R}_{q,h}(t) + C_{q}\int_{0}^{t}|e_{h}(s)|^{2q}\,ds + M_{q,h}(t).\] Let \(\theta := \frac{p}{q} \in (0,1)\) due to 60 . Applying the stochastic Gronwall inequality (see, e.g., [34]) yields \[\begin{align} \mathbb{E}\left[\sup_{0\leq t\leq T} |e_{h}(t)|^{2p}\right] = \mathbb{E}\left[\sup_{0\leq t\leq T} \big(|e_{h}(t)|^{2q}\big)^{p/q}\right] \leq C_{p,q,T}\mathbb{E}\left[ \mathcal{R}_{q,h}^{p/q}(T)\right] \leq C_{p,q,T}\left(\mathbb{E}\left[ \mathcal{R}_{q,h}(T)\right]\right)^{p/q}, \end{align}\] where Hölder’s inequality and \(q/p > 1\) have been used in the last inequality. By \(q\Lambda(q_{b},q_{\sigma}) < P\) in 60 , Lemma 8 can be applied with \(q\) in place of \(p\). Therefore, \[\begin{align} \mathbb{E}\left[\sup_{0\leq t\leq T} |e_{h}(t)|^{2p}\right] \leq C_{p,q,T}\left(C_{q} \mathbb{E}\int_{0}^{T}\Big[|R_{b,h}(s)|^{2q} + \|R_{\sigma,h}(s)\|_{\mathrm{HS}}^{2q}\Big]\,ds\right)^{p/q} \leq C_{p,q,P,T}h^{p}. \end{align}\] Since the auxiliary exponent \(q\) depends only on \(p\), \(P\) and the growth indices, it may be absorbed into the constant. Thus, we obtain ?? . ◻

Proof. Recall the state error \(e_{h}(t) = X(t)-X_{h}(t)\), and the principal coefficient differences \[\Delta b_{h}(t) = b\big(X(t)\big) - b\big(X_{h}(t)\big), \quad \Delta\sigma_{h}(t) = \sigma\big(X(t)\big) - \sigma\big(X_{h}(t)\big).\] Subtracting ?? from 4 , we obtain \[\begin{align} \label{eq:reflection-error-identity} K(t)-K_{h}(t) = e_{h}(t) - \int_{0}^{t}\big(\Delta b_{h}(s) + R_{b,h}(s)\big)\,ds - \int_{0}^{t}\big(\Delta\sigma_{h}(s) + R_{\sigma,h}(s)\big)\,dW(s). \end{align}\tag{70}\] The strict condition ?? implies \(P-p\overline{q} > p\Lambda(q_{b},q_{\sigma}) > 0\). We may therefore choose an exponent \(\widetilde{p}\) such that \[\label{eq:auxiliary-reflection-error-index} \frac{pP}{P-p\overline{q}} < \widetilde{p} < \frac{P}{\Lambda(q_{b},q_{\sigma})}.\tag{71}\] Since \(\frac{pP}{P-p\overline{q}} \geq p\), we have \(\widetilde{p} > p\). Moreover, 71 gives \(\widetilde{p}\Lambda(q_{b},q_{\sigma}) < P\) and \[\label{eq:polynomial-error-holder-condition} \frac{p\overline{q}\widetilde{p}}{\widetilde{p}-p} < P.\tag{72}\] Applying Theorem 7 with \(\widetilde{p}\) in place of \(p\), we obtain \[\label{eq:higher-state-error-for-reflection} \mathbb{E}\left[\sup_{0\leq t\leq T} |e_{h}(t)|^{2\widetilde{p}}\right] \leq C_{\widetilde{p},P,T}h^{\widetilde{p}}.\tag{73}\] We first estimate the principal drift difference. By ?? , one has \[\begin{align} \label{eq:principal-drift-polynomial-bound} |\Delta b_{h}(t)| \leq L\big(1 + |X(t)|^{q_{b}} + |X_{h}(t)|^{q_{b}}\big)|e_{h}(t)|. \end{align}\tag{74}\] Define the conjugate exponents \(a := \frac{\widetilde{p}}{\widetilde{p}-p} > 1, a' := \frac{\widetilde{p}}{p} > 1\) satisfying \(\frac{1}{a} + \frac{1}{a'} = 1\). Raising 74 to the power \(2p\) and applying Hölder’s inequality yields \[\begin{align} \label{eq:principal-drift-holder} \mathbb{E}\big[|\Delta b_{h}(t)|^{2p}\big] \leq C_{p}\left(\mathbb{E}\big[1 + |X(t)|^{2p q_{b}a} + |X_{h}(t)|^{2p q_{b}a}\big]\right)^{1/a} \left(\mathbb{E}\big[|e_{h}(t)|^{2pa'}\big]\right)^{1/a'}. \end{align}\tag{75}\] By 72 and \(q_{b}\leq\overline{q}\), we get \(pq_{b}a = \frac{pq_{b}\widetilde{p}}{\widetilde{p}-p} < P\). Theorem 1 and Proposition 6 therefore imply that the first factor on the right-hand side of 75 is uniformly bounded. Since \(2pa' = 2\widetilde{p}\), estimate 73 gives \[\begin{align} \left(\mathbb{E}\big[|e_{h}(t)|^{2pa'}\big]\right)^{1/a'} \leq \left(\mathbb{E}\left[\sup_{0\leq s\leq T} |e_{h}(s)|^{2\widetilde{p}}\right]\right)^{p/\widetilde{p}} \leq C_{p,P,T}h^{p}. \end{align}\] It follows that \[\label{eq:principal-drift-error-rate} \sup_{0\leq t\leq T}\mathbb{E} \big[|\Delta b_{h}(t)|^{2p}\big] \leq C_{p,P,T}h^{p}.\tag{76}\] The same argument, using ?? , gives \[\label{eq:principal-diffusion-error-rate} \sup_{0\leq t\leq T}\mathbb{E} \big[\|\Delta\sigma_{h}(t)\|_{\mathrm{HS}}^{2p}\big] \leq C_{p,P,T}h^{p}.\tag{77}\] Indeed, the required state-moment exponent satisfies \(\frac{pq_{\sigma}\widetilde{p}}{\widetilde{p}-p} \leq \frac{p\overline{q}\widetilde{p}}{\widetilde{p}-p} < P\). Taking the supremum over \(t \in [0,T]\) in 70 , and using the elementary inequality \(|x_{1}+\cdots+x_{5}|^{2p} \leq C_{p} \sum_{i=1}^{5} |x_{i}|^{2p}\) for \(x_{i} \in \mathbb{R}, i = 1,2,\cdots, 5\), we obtain \[\begin{align} \label{eq:reflection-error-five-terms} \mathbb{E}\left[\sup_{0\leq t\leq T}|K(t)-K_{h}(t)|^{2p}\right] \notag \leq&~ C_{p}\mathbb{E}\left[\sup_{0\leq t\leq T}|e_{h}(t)|^{2p}\right] + C_{p}\mathbb{E}\left[\left(\int_{0}^{T} |\Delta b_{h}(s)|\,ds\right)^{2p}\right] \nonumber \\&~+ C_{p}\mathbb{E}\left[\left(\int_{0}^{T} |R_{b,h}(s)|\,ds\right)^{2p}\right] \nonumber + C_{p}\mathbb{E}\left[\sup_{0\leq t\leq T} \left|\int_{0}^{t}\Delta\sigma_{h}(s)\,dW(s) \right|^{2p}\right] \nonumber \\&~+ C_{p}\mathbb{E}\left[\sup_{0\leq t\leq T} \left|\int_{0}^{t}R_{\sigma,h}(s)\,dW(s)\right|^{2p}\right]. \end{align}\tag{78}\] Theorem 7 gives \[\label{eq:reflection-error-state-term} \mathbb{E}\left[\sup_{0\leq t\leq T} |e_{h}(t)|^{2p}\right] \leq C_{p,P,T}h^{p}.\tag{79}\] By Hölder’s inequality and 76 , we have \[\begin{align} \label{eq:reflection-error-principal-drift-integral} \mathbb{E}\left[\left(\int_{0}^{T}|\Delta b_{h}(s)|\,ds\right)^{2p}\right] \leq T^{2p-1}\int_{0}^{T}\mathbb{E}\big[|\Delta b_{h}(s)|^{2p}\big]\,ds \leq C_{p,P,T}h^{p}. \end{align}\tag{80}\] Similarly, Lemma 8 gives \[\label{eq:reflection-error-drift-defect-integral} \mathbb{E}\left[\left(\int_{0}^{T}|R_{b,h}(s)|\,ds\right)^{2p}\right] \leq C_{p,P,T}h^{p}.\tag{81}\] By the Burkholder–Davis–Gundy inequality, 77 , and Hölder’s inequality, one gets \[\begin{align} \label{eq:reflection-error-principal-diffusion} \mathbb{E}\left[\sup_{0\leq t\leq T}\left| \int_{0}^{t}\Delta\sigma_{h}(s)\,dW(s)\right|^{2p}\right] \leq&~ C_{p}\mathbb{E}\left[\left(\int_{0}^{T} \|\Delta\sigma_{h}(s)\|_{\mathrm{HS}}^{2}\,ds\right)^{p}\right] \notag \\\leq{}&~ C_{p,T}\int_{0}^{T}\mathbb{E}\big[ \|\Delta\sigma_{h}(s)\|_{\mathrm{HS}}^{2p}\big]\,ds \nonumber \\\leq&~ C_{p,P,T}h^{p}. \end{align}\tag{82}\] The same argument, together with Lemma 8, yields \[\label{eq:reflection-error-diffusion-defect} \mathbb{E}\left[\sup_{0\leq t\leq T} \left|\int_{0}^{t}R_{\sigma,h}(s)\,dW(s)\right|^{2p}\right] \leq C_{p,P,T}h^{p}.\tag{83}\]

Substituting 79 , 80 , 81 , 82 and 83 into 78 proves ?? . ◻