Proof. We need only to prove 15 . First we give an expression of \(\epsilon^m_{\tau^m_{k-1},t}\). Note that the solution \(Y_t\) to 10 satisfies \((Y,\sigma(Y)) \in \mathscr{D}_X^{2\theta}([0,1],{\mathbb{R}}^n)\) and \((\sigma(Y),((D\sigma)[\sigma])(Y))\in \mathscr{D}_X^{2\theta}([0,1],\mathcal{L}({\mathbb{R}}^d, {\mathbb{R}}^n))\). Hence we can use 3 . Then by applying the formula to \(f(Y_t)-(Y_{\tau^m_{k-1}})\) for \(f\in C^3_b({\mathbb{R}}^n,{\mathbb{R}}^L)\) successively, we can decompose \(\epsilon^m_{\tau^m_{k-1},t}\) in the following way. This calculation is possible because \(\sigma\in C^4_b, b\in C^2_b\). We need the following functions to state it: \[\begin{align}F^0(y)&=(Db)(y)[b(y)], & F^1_{\alpha}(y)&=(Db)(y)[\sigma(y)e_{\alpha}], & F^2_{\alpha}(y)&=(D\sigma(y)e_{\alpha})[b(y)], \\ \begin{align} F_{\alpha,\beta,\gamma}(y) &= D\Bigl\{(D\sigma(y)e_{\gamma})[\sigma(y)e_{\beta}]\Bigr\} [\sigma(y)e_{\alpha}], & G_{\alpha,\beta}(y) &= D\Bigl\{(D\sigma(y)e_{\beta})[\sigma(y)e_{\alpha}]\Bigr\}[b(y)]. \end{align} \end{align}\] The decomposition formula is as follows, \[\begin{align} &\epsilon^m_{\tau^m_{k-1},t} = \sum_{\alpha,\beta,\gamma} \int_{\tau^m_{k-1}}^{t}\left\{\int_{\tau^m_{k-1}}^{s}\left(\int_{\tau^m_{k-1}}^{u} F_{\alpha,\beta,\gamma}(Y_v) dB^{\alpha}_v\right)dB^{\beta}_u\right\} dB^{\gamma}_s\nonumber\\ &\quad +\sum_{\alpha,\beta,\gamma} \int_{\tau^m_{k-1}}^{t}\left\{\int_{\tau^m_{k-1}}^{s}\left(\int_{\tau^m_{k-1}}^{u} G_{\alpha,\beta}(Y_v) dv\right)dB^{\alpha}_u\right\} dB^{\beta}_s +\int_{\tau^m_{k-1}}^{t}\left(\int_{\tau^m_{k-1}}^sF^0(Y_u)du\right)ds\nonumber\\ &\quad +\sum_{\alpha}\int_{\tau^m_{k-1}}^{t}\left( \int_{\tau^m_{k-1}}^sF^1_{\alpha}(Y_u)du\right)dB^{\alpha}_s +\sum_{\alpha}\int_{\tau^m_{k-1}}^{t}\left( \int_{\tau^m_{k-1}}^sF^2_{\alpha}(Y_u)dB^{\alpha}_u\right)ds\nonumber\\ &:=I_1+\cdots+I_5. \label{explicit32form32of32epm} \end{align}\tag{17}\]

By using estimates of rough integrals, we have the following estimates: for all \(\omega\in\Omega^{(m)}_0\), it holds that \[\begin{gather} \Bigl|I_1-\sum_{\alpha,\beta,\gamma}F_{\alpha,\beta,\gamma}(Y_{\tau^m_{k-1}}) B^{\alpha,\beta,\gamma}_{\tau^m_{k-1},t}\Bigr| \le C(t-\tau^m_{k-1})^{4H^-},\\ \left|I_2\right|\le C(t-\tau^m_{k-1})^{1+2H^-}, \qquad \qquad \quad \left|I_3\right|\le C(t-\tau^m_{k-1})^2, \\ \left|I_4-F^1_{\alpha}(Y_{\tau^m_{k-1}})B^{0,\alpha}_{\tau^m_{k-1},t}\right| +\left|I_5-F^2_{\alpha}(Y_{\tau^m_{k-1}})B^{\alpha,0}_{\tau^m_{k-1},t}\right|\le C(t-\tau^m_{k-1})^{1+2H^-},\label{I432I5} \end{gather}\tag{18}\] where \(C\) depends on \(\sigma\) and \(b\) polynomially. This completes the proof. ◻

Proof. We show (1). From 20 , we have \[\begin{gather} \label{eq4382908401} Y^{\mathrm{CN},m}_t-Y^{\mathrm{CN},m}_{\tau^m_{k-1}} = \sigma(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})B_{\tau^m_{k-1},t} +b(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})(t-\tau^m_{k-1})\\ + \frac{1}{2} \left( \sigma(Y^{\mathrm{CN},m}_t)-\sigma(Y^{\mathrm{CN},m}_{\tau^m_{k-1}}) \right) B_{\tau^m_{k-1},t} + \frac{1}{2} \left( b(Y^{\mathrm{CN},m}_t)-b(Y^{\mathrm{CN},m}_{\tau^m_{k-1}}) \right) (t-\tau^m_{k-1}). \end{gather}\tag{24}\] Hence applying the Taylor formula and writing \(Y^{\mathrm{CN},m}_{\tau^m_{k-1},t}=Y^{\mathrm{CN},m}_t-Y^{\mathrm{CN},m}_{\tau^m_{k-1}}\), we have \[\begin{align} Y^{\mathrm{CN},m}_t -Y^{\mathrm{CN},m}_{\tau^m_{k-1}} -\sigma(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})B_{\tau^m_{k-1},t} -b(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})(t-\tau^m_{k-1})\\&= \frac{1}{2} \left( \int_0^1 (D\sigma)(Y^{\mathrm{CN},m}_{\tau^m_{k-1}}+\theta Y^{\mathrm{CN},m}_{\tau^m_{k-1},t})[Y^{\mathrm{CN},m}_{\tau^m_{k-1},t}]d\theta \right) B_{\tau^m_{k-1},t} \\ &\phantom{=}\qquad \qquad \qquad + \frac{1}{2} \left( \int_0^1 (Db)(Y^{\mathrm{CN},m}_{\tau^m_{k-1}}+\theta Y^{\mathrm{CN},m}_{\tau^m_{k-1},t})[Y^{\mathrm{CN},m}_{\tau^m_{k-1},t}]d\theta \right) (t-\tau^m_{k-1}) \\ &= ((D\sigma)[\sigma])(Y^{\mathrm{CN},m}_{\tau^m_{k-1}}) \left[ \frac{1}{2} B_{\tau^m_{k-1},t} \otimes B_{\tau^m_{k-1},t} \right] + \hat{\epsilon}^{\mathrm{CN},m}_{\tau^m_{k-1},t}\\ &= ((D\sigma)[\sigma])(Y^{\mathrm{CN},m}_{\tau^m_{k-1}}) {\mathbb{B}}_{\tau^m_{k-1},t} + c(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})d^{\mathrm{CN},m}_{\tau^m_{k-1},t} + \hat{\epsilon}^{\mathrm{CN},m}_{\tau^m_{k-1},t}.\end{align}\]

We show (2). From 20 , we have \[\begin{align} \max_k \sup_{\tau^m_{k-1}\leq t\leq\tau^m_k} |Y^{\mathrm{CN},m}_t-Y^{\mathrm{CN},m}_{\tau^m_{k-1}}| \le C |t-\tau^m_{k-1}|^{H^-}. \end{align}\] This estimate and 24 imply \[\begin{align} \max_k \sup_{\tau^m_{k-1}\leq t\leq\tau^m_k} |Y^{\mathrm{CN},m}_t-Y^{\mathrm{CN},m}_{\tau^m_{k-1}}-\sigma(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})B_{\tau^m_{k-1},t}| \le C |t-\tau^m_{k-1}|^{2H^-}. \end{align}\] Hence, by substituting \[\begin{align} (D\sigma)(Y^{\mathrm{CN},m}_{\tau^m_{k-1}}+\theta Y^{\mathrm{CN},m}_{\tau^m_{k-1},t})[Y^{\mathrm{CN},m}_{\tau^m_{k-1},t}]\\&= (D\sigma)(Y^{\mathrm{CN},m}_{\tau^m_{k-1}}+\theta Y^{\mathrm{CN},m}_{\tau^m_{k-1},t})[\sigma(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})B_{\tau^m_{k-1},t}] + O(|t-\tau^m_{k-1}|^{2H^-})\\ &= (D\sigma)(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})[\sigma(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})B_{\tau^m_{k-1},t}] + O(|t-\tau^m_{k-1}|^{2H^-})\end{align}\] into 23 , we can estimate the first term in 23 . Because the second term can be estimated in the same way, we arrive at (2).

We show (3). By a similar calculation to the above, we have \[\begin{gather} \Big| \hat{\epsilon}^{\mathrm{CN},m}_{\tau^m_{k-1},\tau^m_k} -\frac{1}{2}(D\sigma)(Y^{\mathrm{CN},m}_{\tau^m_{k-1}}) \left[ \frac{1}{2}(D\sigma)[\sigma](Y^{\mathrm{CN},m}_{\tau^m_{k-1}}) [(B_{\tau^m_{k-1},\tau^m_k})^{\otimes 2}] +b(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})\Delta_m\right] B_{\tau^m_{k-1},\tau^m_k}\\ -\frac{1}{4}(D^2\sigma)(Y^{\mathrm{CN},m}_{\tau^m_{k-1}}) \left[ \sigma(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})B_{\tau^m_{k-1},\tau^m_k},\sigma(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})B_{\tau^m_{k-1},\tau^m_k} \right]B_{\tau^m_{k-1},\tau^m_k}\\ -\frac{1}{2}(Db)(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})\left[\sigma(Y^{\mathrm{CN},m}_{\tau^m_{k-1}})B_{\tau^m_{k-1},\tau^m_k}\right]\Delta_m \Big| \le C\Delta_m^{4H^-}. \end{gather}\] Note that the above constants depend on \(\sigma,b\) polynomially because \(\omega\in \Omega_0^{(m)}\). The proof completed. ◻

Proof. Since \[\begin{align} d^{\mathrm{FE},m}_{\tau^m_{i-1},\tau^m_i} &= - \sum_{1\le \alpha\neq \beta\le d} B^{\alpha,\beta}_{\tau^m_{i-1},\tau^m_i}e_\alpha\otimes e_\beta - \sum_{\alpha=1}^d \frac{1}{2} \left\{(B^{\alpha}_{\tau^m_{i-1},\tau^m_i})^2-\Delta_m^{2H}\right\} e_{\alpha}\otimes e_{\alpha}, \end{align}\] all components of \(d^m_{\tau^m_{i-1},\tau^m_i}\), \(d^{m,\alpha,\beta}_{\tau^m_{i-1},\tau^m_i}=(d^m_{\tau^m_{i-1},\tau^m_i},e_\alpha\otimes e_\beta)\), are written by a linear combination of \[\begin{align} \label{eq43902498109} B^{\alpha}_{\tau^m_{i-1},\tau^m_i}B^{\beta}_{\tau^m_{i-1},\tau^m_i}, \quad B^{\alpha,\beta}_{\tau^m_{i-1},\tau^m_i},\quad (B^{\alpha}_{\tau^m_{i-1},\tau^m_i})^2-\Delta_m^{2H}, \quad \alpha\ne \beta. \end{align}\tag{35}\] Hence we may assume \(d^m_{\tau^m_{i-1},\tau^m_i}\) to be one of the above without loss of generality. These quantities are considered in several papers; for example [12], [7], [17], and [3]. In what follows, we assume \(\frac{1}{3}<H<\frac{1}{2}\). For the case \(H=\frac{1}{2}\), we can easily modify the discussion.

For \(k<l\), we have \[\begin{gather} \left| E\left[B^\alpha_{\tau^m_{k-1},\tau^m_k}B^\beta_{\tau^m_{k-1},\tau^m_k}B^\alpha_{\tau^m_{l-1},\tau^m_l}B^\beta_{\tau^m_{l-1},\tau^m_l}\right] \right| \le C \left( \frac{|k-l|^{2H-2}}{2^{2mH}} \right)^2,\\ \left| E\left[B^{\alpha,\beta}_{\tau^m_{k-1},\tau^m_k}B^{\alpha,\beta}_{\tau^m_{l-1},\tau^m_l}\right] \right| \le C \left( \frac{|k-l|^{2H-2}}{2^{2mH}} \right)^2,\\ \left| E \left[ (B^{\alpha}_{\tau^m_{k-1},\tau^m_k})^2-\Delta_m^{2H}) (B^{\alpha}_{\tau^m_{l-1},\tau^m_l})^2-\Delta_m^{2H}) \right] \right| \leq C \left( \frac{|k-l|^{2H-2}}{2^{2mH}} \right)^2. \end{gather}\] For \(k=l\), the terms above can be estimate by \(C(2^{-2mH})^2\). We refer the readers for these estimates to Lemma 3.4 in [7]. Also we can find these estimates in Lemma 7.2 (1) in [12]. These estimates imply \[\begin{align} E[|d^m_{s,t}|^{2}]&\le C \left(\frac{1}{2^m}\right)^{4H-1}(t-s)\qquad \text{for s,t\in D_m with s<t.} \end{align}\] Note that all constants \(C\) above are independent of \(m\) and \(H\). By using the hypercontractivity of the Ornstein-Uhlenbeck semigroup, we get \[\begin{align} E[|d^m_{s,t}|^{p}]&\le C_p \left(\frac{1}{2^m}\right)^{(2H-\frac{1}{2})p}(t-s)^{\frac{p}{2}} \qquad \text{for s,t\in D_m with s<t.} \label{moment32estimate32for32dm} \end{align}\tag{36}\] This estimate implies the next assertion. For \(0<\kappa<\frac{1}{2}\), set \[\begin{align} G_{m,\kappa}=(2^m)^{2H-\frac{1}{2}}\max_{s,t\in D_m, s\ne t}\frac{|d^m_{s,t}|}{|t-s|^{\frac{1}{2}-\kappa}}. \end{align}\] Then \[\begin{align} &\qquad\qquad \sup_m\|G_{m,\kappa}\|_{L^p}<\infty & &\text{for all p\ge 1}, \tag{37}\\ & |d^m_{s,t}|\le \Delta_m^{2H-\frac{1}{2}} |t-s|^{\frac{1}{2}-\kappa}G_{m,\kappa} & & \text{for all s,t\in D_m with s<t} \tag{38}. \end{align}\] This can be checked as follows. Since we see 38 from the definition of \(G_{m,\kappa}\), we show integrability 37 . Let \(\{\tilde{d}^m_t\}_{t\in [0,1]}\) be the piecewise linear extension of \(\{d^m_t\}_{t\in D_m}\). By (36 ), we have \[\begin{align} E[|\tilde{d}^m_{s,t}|^p]&\le 3^{p-1}C_p \left(\frac{1}{2^m}\right)^{(2H-\frac{1}{2})p}|t-s|^{\frac{p}{2}}. \end{align}\] By the Garsia-Rodemich-Rumsey inequality, we have for any \(p,\theta>0\) \[\begin{align} \left(\sup_{s,t, s\ne t}\frac{|\tilde{d}^m_{s,t}|}{|t-s|^{\theta}}\right)^p \le 2\int_0^1\int_0^t\frac{|\tilde{d}^m_{s,t}|^p}{|t-s|^{2+p\theta}} dsdt. \end{align}\] Combining these two inequalities and setting \(\theta=\frac{1}{2}-\kappa\), we get \[\begin{align} E[G_{m,\kappa}^p]&\le 2\cdot(2^m)^{(2H-\frac{1}{2})p} \int_0^1\int_0^t\frac{E[|\tilde{d}^m_{s,t}|^p]}{|t-s|^{2+p\theta}}dsdt \le 2\cdot 3^{p-1} C_p\int_0^1\int_0^t|t-s|^{\kappa p-2}dsdt. \end{align}\] If \(p>\kappa^{-1}\), then the right-hand side is bounded and we get \[\begin{align} E[G_{m,\kappa}^p]\le 2\cdot 3^{p-1}C_p \left(\kappa p(\kappa p-1)\right)^{-1}, \end{align}\] which proves 37 .

By using 37 and 38 , we show the assertion. Let us choose \(0<\varepsilon<2H-\frac{1}{2}\) and \(0<2\kappa<\varepsilon\). Using \(\Delta_m\le t-s\), we get \[\begin{align} \text{(RHS of \eqref{estimate32of32dm322})} &=\Delta_m^{\varepsilon-\kappa} \Delta_m^{2H-\frac{1}{2}-\varepsilon+\kappa}|t-s|^{\frac{1}{2}-\kappa}G_{m,\kappa}\\ &\le \Delta_m^{\varepsilon-\kappa}|t-s|^{2H-\varepsilon}G_{m,\kappa}\\ &=\Delta_m^{\varepsilon-2\kappa}|t-s|^{2H-\varepsilon}\Delta_m^{\kappa}G_{m,\kappa} \end{align}\] Let \(G_1=\sum_{m=1}^{\infty}\Delta_m^{\kappa}G_{m,\kappa}\). This infinite series converges for \(\mu\) almost all \(\omega\). Because for all \(p\ge 1\), \[\begin{align} \|G_1\|_{L^p} \le \sum_{m=1}^{\infty} \Delta_m^{\kappa} \sup_m\|G_{m,\kappa}\|_{L^p} < \infty. \end{align}\] Combining the trivial estimate \(\Delta_m^{\kappa}G_{m,\kappa}\le G_1\), we get \[\begin{align} |d^m_{s,t}| &\le \Delta_m^{\varepsilon-2\kappa}|t-s|^{2H-\varepsilon}G_1. \end{align}\] To check the validity of the statements for the pairs \((\varepsilon_1,\lambda_1)\) and \((\varepsilon_0,2H^-)\), it suffices to set \(\varepsilon=3H^--1(<2H-\frac{1}{2})\) and \(\varepsilon=2(H-H^-)(<2H-\frac{1}{2})\) respectively and choose \(\kappa\) to be sufficiently small. This completes the proof. ◻

Proof. In what follows, we assume \(\frac{1}{3}<H<\frac{1}{2}\). In the case where \(H=\frac{1}{2}\), we can easily modify the discussion. Let \((K^m_t)\in \mathcal{K}^3_m\). First, we give estimates for variance of \(K^m_{s,t}\). We have for \(s,t\in D_m\) with \(s<t\),

Note that if the schemes are implementable Milstein or Crank-Nicolson scheme, then it is enough to consider the case \(K^m=B^{\alpha,\beta,\gamma}\) only for the proof of ([K32estimate321]) because of the identities (30 ). Therefore, in those cases, from [2], we see ([K32estimate321]) holds. In [12], the same estimates are obtained in a little bit different way. If the scheme is the first-order Euler scheme, then by the same reasoning as above, it is sufficient to estimate \(E[(\Delta_m^{2H} B^{\gamma}_{s,t})^2]\). For this, we have \[\begin{align} E[(\Delta_m^{2H} B^{\gamma}_{s,t})^2]&\le C\Delta_m^{4H}|t-s|^{2H}\\ &=C\Delta_m^{4H}\cdot|t-s|^{2H-1}|t-s|\\ &\le C\Delta_m^{4H}\Delta_m^{2H-1}|t-s|=C\Delta_m^{6H-1}|t-s|. \end{align}\] Actually we use Condition 3 only to obtain this estimate.

Now we consider ([K32estimate322]). Let \(K^m_{\tau^m_{i-1},\tau^m_i}=B^{\alpha,0}_{\tau^m_{i-1},\tau^m_i}=\int_{\tau^m_{i-1}}^{\tau^m_i}B^{\alpha}_{\tau^m_{i-1},u}du\). By using \(|E[B^{\alpha}_{\tau^m_{i-1},u}B^{\alpha}_{\tau^m_{j-1},v}]| \leq |E[B^{\alpha}_{\tau^m_{i-1},\tau^m_i}B^{\alpha}_{\tau^m_{j-1},\tau^m_j}]|\) for \(\tau^m_{i-1}\leq u\leq \tau^m_i\leq \tau^m_{j-1}\leq v\leq \tau^m_j\), we have \[\begin{align} | E [ K^m_{\tau^m_{i-1},\tau^m_i} K^m_{\tau^m_{j-1},\tau^m_j} ] | &\leq \int_{\tau^m_{i-1}}^{\tau^m_i} du \int_{\tau^m_{j-1}}^{\tau^m_j} dv |E[B^{\alpha}_{\tau^m_{i-1},\tau^m_i}B^{\alpha}_{\tau^m_{j-1},\tau^m_j}]|\\ &\leq 2^{-2m} 2^{-2Hm} |E[B^{\alpha}_{0,1}B^{\alpha}_{j-i-1,j-i}]|. \end{align}\] Noting \(E[B^{\alpha}_{0,1}B^{\alpha}_{k-1,k}]\sim -H(1-2H)k^{2H-2}\) as \(k\to\infty\), we have for \(k2^{-m}=s<t=l2^{-m}\), \[\begin{align} E[(K^m_{s,t})^2]\le C\sum_{i,j=k+1}^l 2^{-2m}2^{-2Hm}|j-i|^{2H-2}\le C(2^{-m})^{2H+1}|t-s|. \end{align}\] As for \(B^{\alpha,0}_{\tau^m_{i-1},\tau^m_i}\), we have \(B^{0,\alpha}_{\tau^m_{i-1},\tau^m_i}=B^{\alpha,0}_{\tau^m_{i-1},\tau^m_i}-\Delta_mB^{\alpha}_{\tau^m_{i-1},\tau^m_i}\). Hence, we need to estimate \(E[(\Delta_m B^{\alpha}_{s,t})^2]\). Since \(\Delta_m\le \Delta_m^{2H}\), this term is smaller than \(E[(\Delta_m B^{\alpha}_{s,t})^2]\) and we get desired estimate.

Because \(6H-1\le 2H+1\), consequently, for all cases, we have \(E[|K^m_{s,t}|^2]\le C\Delta_m^{6H-1}|t-s|\). Combining the hypercontractivity of the Ornstein-Uhlenbeck semigroup and the estimates above, for all \(p\ge 2\), we obtain \[\begin{align} E[|K^{m}_{s,t}|^{p}] \le C_p\left(2^{-m}\right)^{(3H-\frac{1}{2})p}(t-s)^{\frac{p}{2}} \quad \text{for all s,t\in D_m}. \end{align}\] From the same argument as in (38 ), for any \(\frac{1}{2}>\kappa>0\) and \(m\), there exists a positive random variable \(G'_{m,\kappa}\) satisfying \(\sup_m\|G'_{m,\kappa}\|_{L^p}<\infty\) for all \(p\ge 1\) such that \[\begin{align} |K^{m}_{s,t}|&\le \Delta_m^{3H-\frac{1}{2}}|t-s|^{\frac{1}{2}-\kappa}G'_{m,\kappa}\quad \text{for all s,t\in D_m}, \end{align}\] which implies \[\begin{align} |(2^{m})^{2H-\frac{1}{2}}K^{m}_{s,t}|&\le \Delta_m^{\frac{1}{2}-H} \Delta_m^{2H-\frac{1}{2}} |t-s|^{\frac{1}{2}-\kappa}G'_{m,\kappa}\quad \text{for all s,t\in D_m}.\label{estimate32of32294-mK94m} \end{align}\tag{39}\] Note that \(\Delta_m^{2H-\frac{1}{2}}|t-s|^{\frac{1}{2}-\kappa}\) appears in the proof of Lemma 3 (see 38 ).

Let us choose \(0<\varepsilon<2H-\frac{1}{2}\) and \(0<2\kappa<\varepsilon\). Then again using \(\Delta_m\le t-s\) and similarly to the estimate of \(d^m_{s,t}\), we get \[\begin{align} |(2^{m})^{2H-\frac{1}{2}}K^{m}_{s,t}| &\le \Delta_m^{\frac{1}{2}-H} \Delta_m^{\varepsilon-2\kappa} |t-s|^{2H-\varepsilon} \Delta_m^{\kappa} G'_{m,\kappa} \end{align}\] and set \(G_2=\sum_{m=1}^{\infty}\Delta_m^{\kappa}G'_{m,\kappa}\) which converges \(\mu\)-a.s. \(\omega\) and \(\|G_2\|_{L^p}<\infty\) for all \(p\ge 1\). Again by using the trivial estimate \(\Delta_m^{\kappa}G'_{m,\kappa}\le G_2\), we get \[\begin{align} |(2^{m})^{2H-\frac{1}{2}}K^{m}_{s,t}| &\le \Delta_m^{\frac{1}{2}-H} \Delta_m^{\varepsilon-2\kappa} |t-s|^{2H-\varepsilon} G_2. \end{align}\] Putting \(\varepsilon=3H^--1(<2H-\frac{1}{2})\), we completes the proof. ◻

Proof. Recall \(c=(D\sigma)[\sigma]\in C^3_b\). We show the case \(\frac{1}{3}<H<\frac{1}{2}\). We use the result by Liu-Tindel [7]. They considered similar problems (Proposition 4.7 and Corollary 4.9 in [7]). We can use their result to show the assertion as follows. Note that \(f_t=J^{-1}_tc(Y_t)\in \mathcal{L}({\mathbb{R}}^d\otimes{\mathbb{R}}^d,{\mathbb{R}}^n)\) and \(g_t\in \mathcal{L}({\mathbb{R}}^d,\mathcal{L}({\mathbb{R}}^d\otimes{\mathbb{R}}^d,{\mathbb{R}}^n))\) defined by \(g_tv=(-J^{-1}_tD\sigma(Y_t)v)c(Y_t)+J^{-1}_tDc(Y_t)[\sigma(Y_t)v]\) for \(v\in{\mathbb{R}}^d\) satisfy [7] because \(Y\) and \(J^{-1}\) are solutions to 10 and 12 respectively and they belong to \(L^p\) for all \(p\ge 1\). The integrability of \(J_t^{-1}\) is due to [13] (see also Remark 29). Hence from Corollary 4.9 in [7], we get \(\|(2^m)^{2H-\frac{1}{2}}I^m_{s,t}\|_{L^p} \leq C(t-s)^{\frac{1}{2}}\) for some constant \(C\). This and the Garsia-Rodemich-Rumsey inequality imply the assertion. While the above proof is based on the result by Liu-Tindel [7], we can provide another proof of the assertion under the assumption that \(\sigma, b\in C^{\infty}_b\) (see [12]).

Finally, we consider the case where \(H=\frac{1}{2}\). Actually, it is not difficult to check this case by using the Itô calculus. For the reader’s convenience, we include the proof. Recall that \(I^m_t\) in Condition  is defined by \(I^m_t=\sum_{i=1}^{2^mt}F_{\tau^m_{i-1}}d^m_{\tau^m_{i-1},\tau^m_i}\) \((t\in D_m)\), where \(F_{t}=J_t^{-1}c(Y_t)\). We give an estimate of \(E[|I^m_{s,t}|^{2p}]\) by applying martingale theory. Since all components of \(d^{m}_{\tau^m_{i-1},\tau^m_i}\), \(d^{m,\alpha,\beta}_{\tau^m_{i-1},\tau^m_i}=(d^m_{\tau^m_{i-1},\tau^m_i},e_\alpha\otimes e_\beta)\), are written by a linear combination of 35 , the desired estimates follow from those of \[\begin{align} \label{eq439028014214} & \sum_{i=1}^{2^mt} F^{\alpha,\beta}_{\tau^m_{i-1}} B^{\alpha}_{\tau^m_{i-1},\tau^m_i}B^{\beta}_{\tau^m_{i-1},\tau^m_i}, & & \sum_{i=1}^{2^mt} F^{\alpha,\beta}_{\tau^m_{i-1}} B^{\alpha,\beta}_{\tau^m_{i-1},\tau^m_i}, & & \sum_{i=1}^{2^mt} F^{\alpha,\alpha}_{\tau^m_{i-1}} \{ (B^{\alpha}_{\tau^m_{i-1},\tau^m_i})^2-\Delta_m \}, \end{align}\tag{40}\] where \(F^{\alpha,\beta}_t=F_t(e_\alpha\otimes e_\beta)\) and \(\alpha\neq\beta\). For \(t\in[0,1]\), let \[\tilde{I}^m_t = \sum_{i=1}^{2^{m}} F^{\alpha,\beta}_{\tau^m_{i-1}} B^{\alpha,\beta}_{\tau^m_{i-1}\wedge t,\tau^m_i\wedge t}.\] Clearly \(I^m_t=\tilde{I}^m_t\) \((t\in D_m)\) holds. Note that \[B^{\alpha}_{s,t}B^{\beta}_{s,t}=B^{\alpha,\beta}_{s,t}+B^{\beta,\alpha}_{s,t} \quad (\alpha\ne\beta), \quad (B^{\alpha}_{s,t})^2-(t-s)=\int_s^tB^{\alpha}_{s,u}dB^{\alpha}_u,\] where the integral in the second identity is the Itô integral. Therefore, for all cases in (40 ), it suffices to give the moment estimate of \[\begin{align} \tilde{I}^{m,\alpha,\beta}_t&= \int_0^tF^{m,\alpha,\beta}_udB^{\beta}_u,\qquad 1\le \alpha,\beta\le d, \end{align}\] where the integral is an Itô integral and \(F^{m,\alpha,\beta}_u=\sum_{i=1}^{2^m}F^{\alpha,\beta}_{\tau^m_{i-1}} B^{\alpha}_{\tau^m_{i-1},u}1_{[\tau^m_{i-1},\tau^m_i)}(u).\) Let \(p>1\). We have \[\begin{align} \nonumber E\left[|\tilde{I}^{m,\alpha,\beta}_{s,t}|^{2p}\right] &\le CE\left[\left(\int_s^t|F^{m,\alpha,\beta}_u|^{2}du\right)^p\right]\\ &\le C(t-s)^{p-1}E\left[\int_s^t|F^{m,\alpha,\beta}_u|^{2p}du\right] \le C'\left(\frac{t-s}{2^m}\right)^p,\label{moment32estimate32H611472} \end{align}\tag{41}\] where we have used the Burkholder-Davis-Gundy and the Hölder inequalities, and the estimate \[E[|F^{m,\alpha,\beta}_u|^{2p}]\le CE[|F^{\alpha,\beta}_{\tau^m_{i-1}}|^{2p}]E[(B^{\alpha}_{\tau^m_{i-1},u})^{2p}] \le C2^{-pm}\sup_tE[|F_t|^{2p}], \quad \tau^m_{i-1}\le u<\tau^m_i.\] By the estimate (41 ) and a similar argument to the estimate (37 ) of \(d^m_{s,t}\), we see that the assertion holds.

We conclude this proof with mentioning that, under the assumption of this lemma, Condition  holds for all \(H^-<\frac{1}{2}\) and that we can choose \(H^-\) close to \(\frac{1}{2}\). ◻

Proof of Theorem 15. First, we prove the case of the implementable Milstein, Milstein and first-order Euler schemes. Note that in these cases, \(\hat{\epsilon}^{m}\equiv 0\) holds for the approximate solution \(\hat{Y}^{m}_t\). Hence Condition  is clearly satisfied. From Lemmas 3, 4, and 5, we see that Conditions , and hold. From the definition, 32 also holds. Hence the conditions assumed in Corollary 1 are satisfied. By Corollary 1, for any \(\varepsilon<\min\{3H^--1,4H^--2H-\frac{1}{2}\}\), we have \((2^m)^{2H-\frac{1}{2}+\varepsilon}\sup_{0\leq t\leq 1}|R^m_t|\to 0\) in \(L^p\) \((p\ge 1)\) and almost surely. Since \(H^-\) can be any positive number less than \(H\) and \(3H-1\leq 2H-\frac{1}{2}\), the proof is completed.

We consider the case of the Crank-Nicolson approximate solution \(Y^{\mathrm{CN},m}_t\). We cannot directly apply Corollary 1 to the Crank-Nicolson scheme since it satisfies only Condition 9 (1-a) and (2). However we can reduce it to Corollary 1. To this end, we introduce an auxiliary approximate solution \(\hat{Y}^{m}_t\) defined via 25 , with \(\tau^m_k\) replaced by \(t(\in [\tau^m_{k-1},\tau^m_k])\) and \[\begin{align} d^m_{\tau^m_{k-1},t} &= d^{\mathrm{CN},m}_{\tau^m_{k-1},t}, & \hat{\epsilon}^{m}_{\tau^m_{k-1},t} &= \begin{cases} \hat{\epsilon}^{\mathrm{CN},m}_{\tau^m_{k-1},t}, & \omega\in\Omega_0^{(m)},\\ 0, & \omega\in (\Omega_0^{(m)})^{\complement}. \end{cases} \end{align}\] Lemmas 2 and the definition above imply that \(\hat{\epsilon}^{m}_{\tau^m_{k-1},t}\) satisfies Condition . From Lemmas 3, 4, and 5, we see that Conditions , and hold. We see that \(d^m_{\tau^m_{k-1},t}\) and \(\hat{\epsilon}^{m}_{\tau^m_{k-1},t}\) satisfy 32 . Hence we can apply Corollary 1 to \(\hat{Y}^{m}_t\) defined above. By using \(\sup_{0\le t\le 1}|J_t|\in L^p\) \((p\ge 1)\) which is due to [13], as a consequence of Corollary 1, we see that \(\sup_m\sup_{0\le t\le 1}|\hat{Y}^{m}_t|\in L^p\) \((p\ge 1)\). Note that we will give selfcontained proof of the integrability of \(J_t\) and \(J_t^{-1}\) in Remark 29 (2) and the integrability of \(\hat{Y}^{m}_t\) holds under weaker assumption as in Lemma 9 since (32 ) holds. Let \(R^m_t=\hat{Y}^{m}_t-Y_t-J_tI^m_t\) and \(R^{\mathrm{CN},m}_t=Y^{\mathrm{CN},m}_t-Y_t-J_tI^m_t\). Then using \(\hat{Y}^{m}_t=Y^{\mathrm{CN},m}_t\) \((\omega\in \Omega_0^{(m)})\) and \(Y^{\mathrm{CN},m}_t\equiv \xi\) \((\omega\in (\Omega_0^{(m)})^{\complement})\), we have \[\begin{align} R^{\mathrm{CN},m}_t &= R^m_t+Y^{\mathrm{CN},m}_t-\hat{Y}^{m}_t\\ &= R^m_t + \{Y^{\mathrm{CN},m}_t-\hat{Y}^{m}_t\} 1_{\Omega_0^{(m)}} + \{Y^{\mathrm{CN},m}_t-\hat{Y}^{m}_t\} 1_{(\Omega_0^{(m)})^\complement}\\ &= R^m_t+\{\xi-\hat{Y}^{m}_t\} 1_{(\Omega_0^{(m)})^\complement}. \end{align}\] By Corollary 1, we have \((2^m)^{2H-\frac{1}{2}+\varepsilon}\sup_{0\le t\le 1}|R^m_t|\to 0\) for all \(p\ge 1\) and almost surely. By the integrability of \(\sup_m\sup_{0\le t\le 1}|\hat{Y}^{m}_t|\) and the estimate (9 ), we have \((2^m)^{2H-\frac{1}{2}+\varepsilon}\sup_{0\le t\le 1}|(\xi-\hat{Y}^{m}_t) 1_{(\Omega_0^{(m)})^\complement}|\to 0\) in \(L^p\) and almost surely. This completes the proof. ◻

Proof. Under the assumption, \(E^{m,\rho}(Y^{m,\rho}_{\tau^m_{k-1}},\theta_{\tau^m_{k-1}}B)^{-1}\) is given by the Neumann series of \(A^{m,\rho}_{\tau^m_{k-1}} = I - E^{m,\rho}(Y^{m,\rho}_{\tau^m_{k-1}},\theta_{\tau^m_{k-1}}B)\). The estimate of the residual terms implies 56 . ◻

Proof. The second statement follows from 53 and 57 . We show 57 . Write \(k=2^mt\) and denote by \(\zeta_k\) the quantity on the right-hand side of 57 . For simplicity we write \[\begin{align} c_{i-1}d_{i-1,i}= c(Y^{m,\rho}_{\tau^m_{i-1}})d^m_{\tau^m_{i-1},\tau^m_i},\quad \epsilon_{i-1,i}= \hat{\epsilon}^m_{\tau^m_{i-1},\tau^m_i}- \epsilon^m_{\tau^m_{i-1},\tau^m_i}. \end{align}\] From 51 , we have \[\begin{align} \zeta_k-(\zeta_{k-1}+c_{k-1}d_{k-1,k}+\epsilon_{k-1,k})\\&= \sum_{i=1}^{k-1} \left\{ \tilde{J}^{m,\rho}_{\tau^m_{k-i}}(Y^{m,\rho}_{\tau^m_i},\theta_{\tau^m_i}B) - \tilde{J}^{m,\rho}_{\tau^m_{k-i-1}}(Y^{m,\rho}_{\tau^m_i},\theta_{\tau^m_i}B) \right\} (c_{i-1}d_{i-1,i}+\epsilon_{i-1,i}) \\ &= \sum_{i=1}^{k-1} \{ E^{m,\rho}(Y^{m,\rho}_{\tau^m_{k-1}},\theta_{\tau^m_{k-1}}B)-I \} \tilde{J}^{m,\rho}_{\tau^m_{k-i-1}}(Y^{m,\rho}_{\tau^m_i},\theta_{\tau^m_i}B) (c_{i-1}d_{i-1,i}+\epsilon_{i-1,i})\\ &= \{ E^{m,\rho}(Y^{m,\rho}_{\tau^m_{k-1}},\theta_{\tau^m_{k-1}}B)-I \} \sum_{i=1}^{k-1} \tilde{J}^{m,\rho}_{\tau^m_{k-i-1}}(Y^{m,\rho}_{\tau^m_i},\theta_{\tau^m_i}B) (c_{i-1}d_{i-1,i}+\epsilon_{i-1,i})\\ &= \{ E^{m,\rho}(Y^{m,\rho}_{\tau^m_{k-1}},\theta_{\tau^m_{k-1}}B)-I \} \zeta_{k-1},\end{align}\] which implies \[\begin{align} \zeta_k = E^{m,\rho}(Y^{m,\rho}_{\tau^m_{k-1}},\theta_{\tau^m_{k-1}}B)\zeta_{k-1} +c_{k-1}d_{k-1,k}+\epsilon_{k-1,k}. \end{align}\] Comparing the above with 47 , we complete the proof. ◻

Proof. Below, \(C\) is a constant depending only on \(\sigma\), \(b\), \(c\), \(C(B)\) and \(\|d^m\|_{\lambda_1}\) polynomially. By using \(C\), we determine \(\delta\) and \(C_1\) so that 60 holds. For simplicity we write \(\tau^m_i=t_i\). Let \(s=t_k, t=t_{k+l}\). By \(I_{t_k,t_{k+1}}=(1-\rho)\epsilon^m_{t_k,t_{k+1}} +\rho\hat{\epsilon}^m_{t_k,t_{k+1}}\) and the estimate of \(\hat{\epsilon}^m\), we see that 60 holds for any \(\delta\) and for the maximum of three constants \(C\) stated in 14 , 26 , and 27 . Let \(K\ge 1\). Suppose the following estimate: there exists \(M>0\) such that \[\begin{align} |I_{s,t}|&\le M|t-s|^{\lambda+H^-} \end{align}\] holds for \(\{(s,t)=(t_k, t_{k+l})~|~0\le k\le 2^m-1, l\le K, \,\, |t-s|\le \delta \}.\) Here \(M\) should be larger than the number \(C_1\) which is determined by the case \(K=1\).

We consider the case \(K+1\). We rewrite \(s=t_k\) and \(t=t_{k+K+1}\). Choose maximum \(u=t_l\) satisfying \(|u-s|\le |t-s|/2\). Then \(|t-t_{l+1}|\le |t-s|/2\) holds. Note that \(l-k\le K\) and \(K+1-(l+1)\le K\). Hence by the assumption, we have \[\begin{align} \max\{|I_{s,u}|, |I_{t_{l+1},t}|\} & \le M\left|\frac{t-s}{2}\right|^{\lambda+H^-},\tag{61}\\ \max\{|Y^{m,\rho}_u-Y^{m,\rho}_s|,|Y^{m,\rho}_{t}-Y^{m,\rho}_{t_{l+1}}|\} &\le M\left|\frac{t-s}{2}\right|^{\lambda+H^-} +C|t-s|^{H^{-}}.\tag{62} \end{align}\] Next we estimate \((\delta I)_{s,u,t}=I_{s,t}-I_{s,u}-I_{u,t}\). Denote by \((\delta I)_{s,u,t}^\sigma\), \((\delta I)_{s,u,t}^b\) and \((\delta I)_{s,u,t}^c\) the terms in \((\delta I)_{s,u,t}\) being concerned with \(\sigma\), \(b\) and \(c\), respectively. Then \[\begin{align} (\delta I)_{s,u,t}^b &= -b(Y^{m,\rho}_s)(t-s) +b(Y^{m,\rho}_s)(u-s) +b(Y^{m,\rho}_u)(t-u)\\ &= \{b(Y^{m,\rho}_u)-b(Y^{m,\rho}_s)\}(t-u),\\ (\delta I)_{s,u,t}^c &= \rho \{c(Y^{m,\rho}_u)-c(Y^{m,\rho}_s)\} d^m_{u,t} \end{align}\] and \[\begin{align} (\delta I)_{s,u,t}^\sigma &= \{\sigma(Y^{m,\rho}_u)-\sigma(Y^{m,\rho}_s)\}B_{u,t} -((D\sigma)[\sigma])(Y^{m,\rho}_s)[{\mathbb{B}}_{s,t}-{\mathbb{B}}_{s,u}-{\mathbb{B}}_{u,t}]\\ &\qquad -\big\{((D\sigma)[\sigma])(Y^{m,\rho}_s) -((D\sigma)[\sigma])(Y^{m,\rho}_u)\big\}{\mathbb{B}}_{u,t}\\ &= \big\{\sigma(Y^{m,\rho}_u)-\sigma(Y^{m,\rho}_s)-D\sigma(Y^{m,\rho}_s)[Y^{m,\rho}_u-Y^{m,\rho}_s]\big\}B_{u,t}\\ &\qquad + D\sigma(Y^{m,\rho}_s) [ I_{s,u} +((D\sigma)[\sigma])(Y^{m,\rho}_s){\mathbb{B}}_{s,u} +\rho c(Y^{m,\rho}_s)d^m_{s,u} +b(Y^{m,\rho}_s)(u-s) ] B_{u,t}\\ &\qquad - \big\{ ((D\sigma)[\sigma])(Y^{m,\rho}_s) - ((D\sigma)[\sigma])(Y^{m,\rho}_u) \big\} {\mathbb{B}}_{u,t}. \end{align}\] Here we used Chen’s identity and definition of \(I_{s,u}\). By (61 ) and (62 ), we obtain \[\begin{align} |(\delta I)_{s,u,t}| &\le C\big\{1+M\delta^{H^-}+(M\delta^{H^-})^2\big\}|t-s|^{\lambda+H^-}. \end{align}\] Similarly, we obtain \(|(\delta I)_{t_l,t_{l+1},t}| \le C|t-s|^{3H^{-}}\). By \[\begin{align} I_{s,t} &= I_{s,u}+I_{t_l,t_{l+1}}+I_{t_{l+1},t} +(\delta I)_{t_{l},t_{l+1},t} +(\delta I)_{s,u,t}, \end{align}\] we have \(|I_{s,t}|\leq f(C,M,\delta)|t-s|^{\lambda+H^-}\), where \[\begin{align} f(C,M,\delta) = 2^{1-(\lambda+H^-)}M +C\big\{1+M\delta^{H^-}+(M\delta^{H^-})^2\big\}. \end{align}\] Note that the function \(f\) and \(C\) do not depend on \(K\). Let \((M,\delta)\) be a pair such that \(f(C,M,\delta)\le M\) holds and \(M\) is greater than or equal to the maximum of three constants \(C\) stated in 14 , 26 , and 27 . Then (60 ) holds for \((C_1,\delta)=(M,\delta)\). One choice is as follows. \[\begin{align} M=\frac{3C}{1-2^{1-(\lambda+H^-)}},\quad \delta= \min\bigg\{\bigg(\frac{3C}{1-2^{1-(\lambda+H^-)}}\bigg)^{-\frac{1}{H^-}}, 1\bigg\}, \end{align}\] where \(C\) is greater than or equal to the maximum of three constants \(C\) stated in 14 , 26 , and 27 . This completes the proof. ◻

Proof. Below, \(C\) denote constants depending only on \(\sigma, b, c\), \(C(B)\) and \(\|d^m\|_{\lambda_1}\) polynomially. We have proved the case where \(s,t\) with \(t-s\le \delta\). Suppose \(t-s>\delta\). In this case, from the definition of \(I_{s,t}\) and \((\delta^{-1}|t-s|)^\lambda \geq 1\), we have \[\begin{align} |I_{s,t}| \leq |Y^{m,\rho}_{s,t}| +C|t-s|^{H-} \leq |Y^{m,\rho}_{s,t}| +C\delta^{-1}|t-s|^{\lambda+H^-}. \end{align}\] Here we wrote \(Y^{m,\rho}_{s,t}=Y^{m,\rho}_t-Y^{m,\rho}_s\). In what follows, we will give an estimates of \(|Y^{m,\rho}_{s,t}|\).

First, we consider the case \(2^{-m}\ge \delta\). For \(s=2^{-m}k<t=2^{-m}l\), we have \[\begin{align} |Y^{m,\rho}_{s,t}|&=\left|\sum_{i=k+1}^lY^{m,\rho}_{\tau^m_{i-1},\tau^m_i}\right| \le C (l-k)\Delta_m^{H^-} =C (2^m)^\lambda(l-k)^{1-(\lambda+H^-)}|t-s|^{\lambda+H^-}. \end{align}\] Noting \((2^m)^\lambda\le \delta^{-\lambda}\), we obtain \(|Y^{m,\rho}_{s,t}|\le C \delta^{-\lambda}|t-s|^{\lambda+H^-}\).

We next consider the case \(2^{-m}<\delta\). Let \(\tau^m_K=\max\{\tau^m_k~|~\tau^m_k\le \delta\}\). Then \(2^{-1}\delta\le \tau^m_K\). Let \(s_i=s+i\tau^m_K\) \((0\leq i\leq N-1)\), where \(N\) is a positive integer such that \(0\le t-s_{N-1}<\tau^m_K\). For notational simplicity, we set \(s_N=t\). Then we have \(N\le (\tau^m_K)^{-1}(t-s)+1\le 2(t-s)(\tau^m_K)^{-1} \le 4\delta^{-1}(t-s).\) By the estimate in Lemma 8, we have \[\begin{align} |Y^{m,\rho}_{s_{i-1},s_i}| \leq C \big\{ |t-s|^{\lambda+H^-}+|t-s|^{H^-}+ |t-s|^{2H^-}+|t-s|^\lambda+|t-s| \big\} \leq C |t-s|^{H^-}. \end{align}\] Hence \[\begin{align} |Y^{m,\rho}_{s,t}| \le \sum_{i=1}^N|Y^{m,\rho}_{s_i}-Y^{m,\rho}_{s_{i-1}}| \le \delta^{-1}|t-s| \cdot C |t-s|^{H^-}. \end{align}\] Since \(1>\lambda\), we obtain \(|Y^{m,\rho}_t-Y^{m,\rho}_s|\le C\delta^{-1}|t-s|^{\lambda+H^-}\). Since \(\delta^{-1}\) depends on \(\sigma\), \(b\), \(c\), \(C(B)\), \(\|d^m\|_{\lambda_1}\) polynomially, we complete the proof. ◻

Proof. Let \(I_{st}\) be the function defined in 59 . \[\begin{align} \delta \Xi(f,g,h)_{s,u,t} = \Xi(f,g,h)_{s,t}-\Xi(f,g,h)_{s,u}-\Xi(f,g,h)_{u,t}\\&= -\left\{f(Y^{m,\rho}_u)-f(Y^{m,\rho}_s) -(Df)(Y^{m,\rho}_s)[Y^{m,\rho}_u-Y^{m,\rho}_s]\right\}B_{u,t}\\ &\qquad -(Df)(Y^{m,\rho}_s) \left[ I_{s,u} +((D\sigma)[\sigma])(Y^{m,\rho}_s){\mathbb{B}}_{s,u} +\rho c(Y^{m,\rho}_{s})d^m_{s,u}+b(Y^{m,\rho}_s)(u-s) \right] B_{u,t}\\ &\qquad + \left\{ (Df)(Y^{m,\rho}_s)[\sigma(Y^{m,\rho}_s)]-(Df)(Y^{m,\rho}_u)[\sigma(Y^{m,\rho}_u)] \right\}{\mathbb{B}}_{u,t}\nonumber\\ &\qquad +\left\{g(Y^{m,\rho}_s)-g(Y^{m,\rho}_u)\right\}(t-u) +\left\{h(Y^{m,\rho}_s)-h(Y^{m,\rho}_u)\right\}d^m_{u,t}.\end{align}\] Hence \(|\delta I(f,g,h)_{s,u,t}|\le C|t-s|^{\lambda+H^-}\). By a standard argument (for example, use the sewing lemma (see [14])), we complete the proof of the lemma. ◻

Proof. These follows from the definition and the following identity. Let \(u\ge s\). \[\begin{align} Y^{m,\rho}_u(Y^{m,\rho}_{\tau},\theta_{\tau}B) &= Y^{m,\rho}_{u+\tau}(\xi,B) = Y^{m,\rho}_{u-s}(Y^{m,\rho}_{s+\tau},\theta_{s+\tau} B),\\ \tilde{J}^{m,\rho}_u(Y^{m,\rho}_\tau,\theta_\tau B) &= \tilde{J}^{m,\rho}_{u-s}(Y^{m,\rho}_{s+\tau},\theta_{s+\tau}B) \tilde{J}^{m,\rho}_s(Y^{m,\rho}_\tau,\theta_\tau B),\\ (\theta_\tau \Xi)_{u,t} &= (\theta_{s+\tau}\Xi)_{u-s,t-s} \qquad \text{for} \qquad \Xi=B,{\mathbb{B}},d^m. \end{align}\] ◻

Proof. The proof of this lemma is similar to that of Lemma 8 and is done by induction. The difference is that we do not use 14 and 27 and use 15 and 26 . Here we give a sketch of the proof. Below, \(\tau^m_i=t_i\) and \(C\) denotes a constant depending only on \(\sigma\), \(b\), \(c\), and \(H^-\) polynomially.

The first step of the induction is as follows. Note \(I_{t_k,t_{k+1}} = (1-\rho)\epsilon^m_{t_k,t_{k+1}} +\rho\hat{\epsilon}^m_{t_k,t_{k+1}}\). The estimates 15 and 26 imply \(|\epsilon^m_{t_{k-1},t_k}| + |\hat{\epsilon}^{m}_{t_{k-1},t_k}| \le C \tilde{w}(t_{k-1},t_k)^{3H^-}\) for all \(1\le k\le 2^m\) and \(\omega\in \Omega_0^{(m)}\). Hence \(|I_{t_k,t_{k+1}}|\leq C\tilde{w}(t_{k-1},t_k)^{3H^-}\). The induction works well by noting \[\begin{align} |B_{s,t}|&\le \tilde{w}(s,t)^{H^-}, & |{\mathbb{B}}_{s,t}|&\le \tilde{w}(s,t)^{2H^-}, & |d^m_{s,t}|&\le \tilde{w}(s,t)^{2H^-} \qquad \text{for all}\qquad s,t\in D_m. \end{align}\] The last estimate above follows from \(\omega\in\Omega_0^{(m,d^m)}\). For example, we need to change the sentence “maximum \(u=t_l\) satisfying \(|u-s|\le |t-s|/2\)” to “maximum \(u=t_l\) satisfying \(\tilde{w}(s,u)\le \tilde{w}(s,t)/2\)”. For this \(l\), we see \(\tilde{w}(t_{l+1},t)\le \frac{1}{2}\tilde{w}(s,t)\). We omit the details. ◻

Proof. Below, we write \(\tilde{w}_{\tau}(s,t)=\tilde{w}(s+\tau,t+\tau)\) and \(C\) is a constant depending only on \(\sigma,b,c, H^-\) which may change line by line. The proof is similar to that of Lemma 8. We take \(\delta\) smaller than \(\delta\) in Lemma 12. For simplicity we write \(t_k=\tau^m_k\). It suffices to consider the case where \(\tau\le 1-2^{-m}\). We consider the following claim depending on a positive integer \(K\).

(Claim \(K\)) (64 ) holds for all \(\tau\) and \(t_k\) satisfying \(\tau+t_k\le 1\), \(\tilde{w}_\tau(0,t_k)\le \delta\) and \(1\le k\le K\).

Since \(I_{0,t_1}= I_{0,t_1}(Y^{m,\rho}_{\tau},\theta_{\tau}B)=0\) holds for all \(\tau\), (Claim 1) holds for \(C_4=0\) and any \(\delta\). We assume (Claim \(K\)) holds and we will find the condition on \(C_4\) and \(\delta\) independent of \(K\) under which (Claim \(K+1\)) holds. Assume the case \(K\) holds for a positive constant \(C_4\) and \(\delta\). Suppose \(\tau+t_{K+1}\le 1\) and \(\tilde{w}_\tau(0,t_{K+1})\le \delta\), where \(K\ge 1\). Define \(0\le t_l<t_{K+1}\) as the maximum number such that \(\tilde{w}_{\tau}(0,t_l)\le \tilde{w}_{\tau}(0,t_{K+1})/2\). On the other hand, for \(t_{l+1}\), we have \(\tilde{w}_{\tau}(t_{l+1},t_{K+1})\le \tilde{w}_{\tau}(0,t_{K+1})/2\). We will write \(u=t_l\) and \(t=t_{K+1}\). By (Claim \(K\)), we have \[\begin{align} \tag{65} |I_{0,u}(Y^{m,\rho}_\tau,\theta_\tau B)| &\le C_4(\tilde{w}_{\tau}(0,t)/2)^{3H^{-}},\\ \tag{66} |I_{0,t-t_{l+1}}(Y^{m,\rho}_{t_{l+1}+\tau},\theta_{t_{l+1}+\tau}B)| &\le C_4(\tilde{w}_{\tau}(0,t)/2)^{3H^{-}}. \end{align}\] The estimate (65 ) implies \[\begin{align} \tag{67}|\tilde{J}^{m,\rho}_{u}(Y^{m,\rho}_\tau,\theta_\tau B)-I| &\le C_4(\tilde{w}_{\tau}(0,t)/2)^{3H^{-}} +C\tilde{w}_{\tau}(0,t)^{H^{-}} +C\tilde{w}_{\tau}(0,t)^{2H^{-}} \\ &\le \{C_4(\delta/2)^{2H^{-}}+C\} \tilde{w}_{\tau}(0,t)^{H^{-}}, \\ \tag{68} | \tilde{J}^{m,\rho}_{u}(Y^{m,\rho}_\tau,\theta_\tau B) -I -(D\sigma)(Y^{m,\rho}_{\tau})B_{\tau,u+\tau} | \le \{C_4(\delta/2)^{H^{-}}+C\} \tilde{w}_{\tau}(0,t)^{2H^{-}}. \end{align}\] For simplicity, we write \(I_{0,t}=I_{0,t}(Y^{m,\rho}_\tau,\theta_\tau B)\) and set \((\delta I)_{0,u,t}=I_{0,t}-I_{0,u}-I_{u,t}\). Hereafter we will estimate \((\delta I)_{0,u,t}\) and \(I_{u,t}\). By the results on them and the inductive assumption, we will obtain a bound of \(I_{0,t}\)

First we consider \((\delta I)_{0,u,t}\). Denote by \((\delta I)_{0,u,t}^\sigma\), \((\delta I)_{0,u,t}^b\) and \((\delta I)_{0,u,t}^c\) the terms in \((\delta I)_{0,u,t}\) being concerned with \(\sigma\), \(b\) and \(c\), respectively. Then we have \[\begin{align} (\delta I)_{0,u,t}^b &= -(Db)(Y^{m,\rho}_\tau)[I]t +(Db)(Y^{m,\rho}_\tau)[I]u\\ &\phantom{=}\qquad +(Db)(Y^{m,\rho}_u(Y^{m,\rho}_\tau,\theta_\tau B))[\tilde{J}^{m,\rho}_{u}(Y^{m,\rho}_\tau,\theta_\tau B)](t-u)\\ &= \big\{ (Db)(Y^{m,\rho}_{u+\tau})[\tilde{J}^{m,\rho}_{u}(Y^{m,\rho}_\tau,\theta_\tau B)] - (Db)(Y^{m,\rho}_\tau)[I] \big\} (t-u)\\ (\delta I)_{0,u,t}^c &= \rho \big\{ (Dc)(Y^{m,\rho}_{u+\tau})[\tilde{J}^{m,\rho}_u(Y^{m,\rho}_{\tau},\theta_{\tau}B)] - (Dc)(Y^{m,\rho}_\tau)[I] \big\} d^m_{u+\tau,t+\tau} \end{align}\] and \[\begin{align} (\delta I)_{0,u,t}^\sigma &= -(D\sigma)(Y^{m,\rho}_{\tau})[I]B_{u+\tau,t+\tau} - D((D\sigma)[\sigma])(Y^{m,\rho}_{\tau})[I] \left( {\mathbb{B}}_{\tau,\tau+t}-{\mathbb{B}}_{\tau,\tau+u} \right) \\ &\qquad\qquad +(D\sigma)(Y^{m,\rho}_{u+\tau})[\tilde{J}^{m,\rho}_{u}(Y^{m,\rho}_\tau,\theta_\tau B)] B_{u+\tau,t+\tau} \\ &\qquad\qquad + D((D\sigma)[\sigma])(Y^{m,\rho}_{u+\tau})[\tilde{J}^{m,\rho}_{u}(Y^{m,\rho}_\tau,\theta_\tau B)] {\mathbb{B}}_{u+\tau,t+\tau}. \end{align}\] Here by getting the first and third terms together, we have \[\begin{gather} (D\sigma)(Y^{m,\rho}_{u+\tau}) \Big[ \tilde{J}^{m,\rho}_{u}(Y^{m,\rho}_\tau,\theta_\tau B)-I-D\sigma(Y^{m,\rho}_\tau)[I]B_{\tau,\tau+u} \Big] B_{u+\tau,t+\tau}\\ + \Big\{ (D\sigma)(Y^{m,\rho}_{u+\tau})[I] - (D\sigma)(Y^{m,\rho}_{\tau})[I] - D(D\sigma)(Y^{m,\rho}_\tau)[\sigma(Y^{m,\rho}_{\tau})B_{\tau,u+\tau}] \Big\} B_{u+\tau,t+\tau}\\ + \uwave{ (D\sigma)(Y^{m,\rho}_{u+\tau}) \left[ D\sigma(Y^{m,\rho}_\tau)[I]B_{\tau,\tau+u} \right] B_{u+\tau,t+\tau} } + \uwave{D(D\sigma)(Y^{m,\rho}_\tau)[\sigma(Y^{m,\rho}_{\tau})B_{\tau,u+\tau}]B_{u+\tau,t+\tau}}. \end{gather}\] Because of Chen’s identity, the summation of the second and fourth terms gives \[\begin{gather} \Big\{ D((D\sigma)[\sigma])(Y^{m,\rho}_{u+\tau})[\tilde{J}^{m,\rho}_{u}(Y^{m,\rho}_\tau,\theta_\tau B)] - D((D\sigma)[\sigma])(Y^{m,\rho}_{\tau})[I] \Big\} {\mathbb{B}}_{u+\tau,t+\tau}\\ + \uwave{ ( - D((D\sigma)[\sigma])(Y^{m,\rho}_{\tau})[I] \left\{ B_{\tau,\tau+u}\otimes B_{\tau+u,\tau+t} \right\} ) }. \end{gather}\] Since the summation of terms with    vanishes due to 48 , we have \[\begin{align} (\delta I)_{0,u,t}^\sigma &= (D\sigma)(Y^{m,\rho}_{u+\tau}) \Big[\tilde{J}^{m,\rho}_{u}(Y^{m,\rho}_\tau,\theta_\tau B)-I-(D\sigma)(Y^{m,\rho}_{\tau})B_{\tau,u+\tau}\Big] B_{u+\tau,t+\tau}\\ &\qquad + \Big\{ (D\sigma)(Y^{m,\rho}_{u+\tau}) - (D\sigma)(Y^{m,\rho}_{\tau})-D(D\sigma)(Y^{m,\rho}_\tau)[\sigma(Y^{m,\rho}_{\tau})B_{\tau,u+\tau}] \Big\} B_{u+\tau,t+\tau} \\ &\qquad + \Big\{ D((D\sigma)[\sigma])(Y^{m,\rho}_{u+\tau})[\tilde{J}^{m,\rho}_{u}(Y^{m,\rho}_\tau,\theta_\tau B)] - D((D\sigma)[\sigma])(Y^{m,\rho}_{\tau})[I] \Big\} {\mathbb{B}}_{u+\tau,t+\tau}. \end{align}\] Thus, combining Lemma 12, (67 ) and (68 ) , we get \[\begin{align} |(\delta I)_{0,u,t}^\sigma| &\le C\tilde{w}_{\tau}(0,t)^{3H^{-}} + C\big\{ 1 + C_4(\delta/2)^{H^-} \big\}\tilde{w}_{\tau}(0,t)^{3H^-},\\ |(\delta I)_{0,u,t}^b| &\le C\big\{ 1 + C_4(\delta/2)^{2H^-} \big\}\tilde{w}_{\tau}(0,t)^{1+H^-},\\ |(\delta I)_{0,u,t}^c| &\le C\big\{ 1 + C_4(\delta/2)^{2H^-} \big\}\tilde{w}_{\tau}(0,t)^{3H^-}. \end{align}\] Hence, \[\begin{align} |(\delta I)_{0,u,t}| &\le C\big\{1+C_4\delta^{H^-}\big\}\tilde{w}_{\tau}(0,t)^{3H^-}. \end{align}\]

We estimate \(I_{u,t}\). We have \(I_{u,t}=I_{t_l,t} = (\delta I)_{t_l,t_{l+1},t}+ I_{t_l,t_{l+1}}+I_{t_{l+1},t}\). It is clear that \(I_{t_l,t_{l+1}}=0\). First we consider \((\delta I)_{t_l,t_{l+1},t}\). Using Lemma 11 and (67 ), we get \[\begin{align} |(\delta I)_{t_l,t_{l+1},t}| &= \big| \big\{ I_{0,t-t_l}(Y^{m,\rho}_{t_l+\tau},\theta_{t_l+\tau}B) -I_{0,t_{l+1}-t_l}(Y^{m,\rho}_{t_l+\tau}, \theta_{t_l+\tau}B)\\ &\qquad\qquad\qquad -I_{t_{l+1}-t_l,t-t_l}(Y^{m,\rho}_{t_l+\tau},\theta_{t_l+\tau}B) \big\} \cdot \tilde{J}^{m,\rho}_{t_l}(Y^{m,\rho}_\tau,\theta_\tau B) \big|\\ &\le C \big\{ 1 + C_4\delta^{H^-} \big\} \tilde{w}_{\tau+t_l}(0,t-t_l)^{3H^-} \big|\tilde{J}^{m,\rho}_{t_l}(Y^{m,\rho}_\tau,\theta_\tau B)\big|, \end{align}\] where we have used a similar estimate of \((\delta I)_{0,t_{l+1}-t_l,t-t_l}\) to \((\delta I)_{0,u,t}\) and note \(\tilde{w}_{\tau+t_l}(0,t-t_l)=\tilde{w}_{\tau}(t_l,t)\leq \tilde{w}_{\tau}(0,t)\). Next we consider \(I_{t_{l+1},t}\). Lemma 11 implies \[\begin{align} I_{t_{l+1},t} &= I_{0,t-t_{l+1}}(Y^{m,\rho}_{t_{l+1}+\tau},\theta^m_{t_{l+1}+\tau}B) \tilde{J}^{m,\rho}_{t_{l+1}}(Y^{m,\rho}_{\tau},\theta_\tau B)\\ &= I_{0,t-t_{l+1}}(Y^{m,\rho}_{t_{l+1}+\tau},\theta^m_{t_{l+1}+\tau}B) E^{m,\rho}(Y^{m,\rho}_{t_l+\tau},\theta_{t_l+\tau}B) \tilde{J}^{m,\rho}_{t_l}(Y^{m,\rho}_\tau,\theta_\tau B). \end{align}\] By 66 and the definition of \(E^{m,\rho}\) (see 50 ), we obtain \[\begin{align} |I_{t_{l+1},t}| &\le C_4 \left( \frac{1}{2} \tilde{w}_{\tau}(0,t) \right)^{3H^{-}} \big\{ 1 + C\tilde{w}_\tau(0,t)^{H^-} \big\} \big|\tilde{J}^{m,\rho}_{t_l}(Y^{m,\rho}_\tau,\theta_\tau B)\big|. \end{align}\] Hence noting \(|\tilde{J}^{m,\rho}_{t_l}(Y^{m,\rho}_\tau,\theta_\tau B)| \leq 1+C\{1+C_4\delta^{H^-}\}\), we have \[\begin{align} |I_{u,t}| &\leq \big\{ C \big\{ 1+C_4\delta^{H^-} \big\} + C_4 2^{-3H^-} \big\{ 1+C\delta^{H^-} \big\} \big\} \big\{ 1+C\{ 1 + C_4\delta^{H^-} \} \big\} \tilde{w}_\tau(0,t)^{3H^-}\\ &\leq \big\{ C_4 2^{-3H^-} + C \big\{ 1 + C_4\delta^{H^-} \big\} \big\} \big\{ 1+C\{ 1 + C_4\delta^{H^-} \} \big\} \tilde{w}_\tau(0,t)^{3H^-}\\ &\leq \big\{ C_4 2^{-3H^-} + C \big\{ 1+C_4\delta^{H^-}+(C_4\delta^{H^-})^2 \big\} \big\} \tilde{w}_\tau(0,t)^{3H^-}. \end{align}\]

Consequently, noting \(I_{0,t}=I_{0,u}+(\delta I)_{0,u,t}+I_{u,t}\), we obtain \[\begin{align} |I_{0,t}| &\le \big\{ 2C_4 2^{-3H^{-}} + C \big\{1+(C_4\delta^{H^-})+(C_4\delta^{H^-})^2\big\} \big\}\tilde{w}_{\tau}(0,t)^{3H^{-}}. \end{align}\] Hence if \(C_4\) and \(\delta\) satisfies \(C_4 2^{1-3H^{-}} + C \big\{ 1+(C_4\delta^{H^-})+(C_4\delta^{H^-})^2 \big\} \le C_4\), then (64 ) holds in the case of \(K+1\). One choice of \(C_4, \delta\) is \[\begin{align} C_4 &= \frac{3C}{1-2^{1-3H^-}}, & \delta &= \min \bigg\{ \bigg(\frac{3C}{1-2^{1-3H^-}}\bigg)^{-\frac{1}{H^-}}, 1 \bigg\}. \end{align}\]

Under this choice, we see that (64 ) holds for any \(t,\tau\in D_m\) with \(\tilde{w}(\tau,\tau+t)\le \delta\) and \(t+\tau\le 1\). This completes the proof. ◻

Proof. We show (1). We use \(\sigma^\beta_i\) to denote the dependence of \(\sigma_i\) on \(\beta\). Assume \(\beta'<\beta\). Then \(\sigma^{\beta'}_i\leq \sigma^\beta_i\) for all \(i\geq 0\), which implies \(N_{\beta'}(w)\geq N_\beta(w)\). Conversely, by setting \(\Lambda_i = \{j:\sigma^\beta_i\leq \sigma^{\beta'}_j,\sigma^{\beta'}_{j+1}\leq \sigma^\beta_{i+1}\}\) for \(0\leq i\leq N_\beta(w)-1\), we have \[\begin{align} \beta = w(\sigma^\beta_i,\sigma^\beta_{i+1}) \geq \sum_{j\in \Lambda_i} w(\sigma^{\beta'}_j,\sigma^{\beta'}_{j+1}) = \sharp \Lambda_i \beta'. \end{align}\] Since the number of \(j\) such that \(\sigma^\beta_i\in(\sigma^{\beta'}_j,\sigma^{\beta'}_{j+1})\) for some \(1\leq i\leq N_\beta(w)\) is bounded by \(N_\beta(w)\) from above and the number of \(j\) such that \((\sigma^{\beta'}_j,\sigma^{\beta'}_{j+1}]\subset(\sigma^\beta_{N_\beta(w)},1]\) is bounded by \(\beta/\beta'\), we have \(\sum_{i=0}^{N_\beta(w)-1} \sharp \Lambda_i \geq N_{\beta'}(w)-N_\beta(w)-\beta/\beta'\). Hence \(\beta N_\beta(w)\geq \beta'(N_{\beta'}(w)-N_\beta(w)-\beta/\beta')\). Hence we see the assertion for \(\beta'<\beta\). It can be generalized easily. We can show (2) easily from the definition. We prove (3). Let \(\{\tilde{\sigma}_i\}_{i=0}^{N_{\beta}(\tilde{w})}\) and \(\{\sigma_i\}_{i=0}^{N_1(w)}\) be corresponding increasing sequences. Then by the definition, we have \(w(\tilde{\sigma}_{i-1},\tilde{\sigma}_i)\ge 2\) for \(1\le i\le N_{\beta}(\tilde{w})\). This implies \(\sigma_i\le \tilde{\sigma}_i\)   \((1\le i\le N_{\beta}(\tilde{w}))\) and so the proof is finished. ◻

Proof. Let \(\delta\) and \(C_4\) be numbers given in Lemma 13. Let us take \(m\) satisfying \(2^{-m}\le \delta\). Let \(0<\varepsilon\le\delta\). By Lemma 13, for \(t,\tau\) satisfying \(\tilde{w}(\tau,\tau+t)\le\varepsilon\) and \(\tau+t\le 1\), we have \[\begin{align} |\tilde{J}^{m,\rho}_{t}(Y^{m,\rho}_{\tau},\theta_{\tau}B)-I|&\le C_4\varepsilon^{3H^-}+C(\varepsilon^{H^-}+\varepsilon^{2H^-}+\varepsilon), \end{align}\] where \(C\) is a constant depending only on \(\sigma, b,c\). Hence, for sufficiently small \(\varepsilon\) which depends only on \(C_4, C\), that is, depends only on \(\sigma,b,c\), it holds that for any \(t,\tau\in D_m\) with \(t+\tau\le 1\) and \(\tilde{w}(\tau,t+\tau)\le \varepsilon\), \(\tilde{J}^{m,\rho}_t(Y^{m,\rho}_{\tau},\theta_{\tau}B)\) are invertible and \[\begin{align} \max\big\{|\tilde{J}^{m,\rho}_{t}(Y^{m,\rho}_{\tau},\theta_{\tau}B)|, |\tilde{J}^{m,\rho}_{t}(Y^{m,\rho}_{\tau},\theta_{\tau}B)^{-1}|\big\} &\le 2.\label{small32tJmr} \end{align}\tag{69}\]

By the definition of \(w\), we see that there exists a constant \(C_{H^-}(\ge 1)\) such that for any \(0\le s<u<t\le 1\) \[\begin{align} w(s,t)\le C_{H^-}\left(w(s,u)+w(u,t)\right). \end{align}\] For \(\omega\in \Omega_0^{(m)}\), \(w\left(u,(u+2^{-m})\wedge 1\right)\le 2^{-m}\) holds for any \(0\le u\le 1\). Therefore, we get \[\begin{align} w\left(s,(u+2^{-m})\wedge 1\right)\le C_{H^-}\left(w(s,u)+2^{-m}\right), \quad 0\le s\le u\le 1. \end{align}\] By using this, we get \[\begin{align} \tilde{w}\left(s,(u+2^{-m})\wedge 1\right) \le C_{H^-}\left(\tilde{w}(s,u)+2^{1-m}\right), \quad 0\le s\le u\le 1. \end{align}\] Let us take a positive number \(\beta\) and \(m\) such that \[\begin{align} C_{H^-}\left(\beta+2^{1-m}\right)\le \varepsilon. \end{align}\] Note that \(\beta\) and \(m\) depends on \(C_{H^-}\) and \(\varepsilon\). Let \(\{\tilde{\sigma}_i\}_{i=0}^{N_{\beta}(\tilde{w})}\) be the increasing sequence defined by \(\tilde{w}\) and \(\beta\). Let \(\hat{\sigma}_i=\inf\{t\in D_m~|~t\ge \tilde{\sigma}_i\}\) \((0\le i\le N_{\beta}(\tilde{w}))\). Also set \(\hat{\sigma}_{N_{\beta}(\tilde{w})+1}=1\). Then we have for all \(0\le i\le N_{\beta}(\tilde{w})\) \[\begin{align} \tilde{w}(\hat{\sigma}_{i},\hat{\sigma}_{i+1}) \le \tilde{w}\left(\tilde{\sigma}_{i}, (\tilde{\sigma}_{i+1}+2^{-m})\wedge 1\right) \le C_{H^-}(\tilde{w}(\tilde{\sigma}_{i},\tilde{\sigma}_{i+1})+2^{1-m}) \le \varepsilon.\label{sigmai32and32sigmai431} \end{align}\tag{70}\] Take \(t(\ne 0)\in D_m\) and choose \(j\) so that \(\hat{\sigma}_{j-1}<t\le \hat{\sigma}_j\) \((1\le j\le N_{\beta}(\tilde{w})+1)\). We have \[\begin{align} \tilde{J}^{m,\rho}_t(\xi,B)= \tilde{J}^{m,\rho}_{t-\hat{\sigma}_{j-1}}(Y^{m,\rho}_{\hat{\sigma}_{j-1}}, \theta_{\hat{\sigma}_{i-1}}B)\cdots \tilde{J}^{m,\rho}_{\hat{\sigma}_2-\hat{\sigma}_1}(Y^{m,\rho}_{\hat{\sigma}_1}, \theta_{\hat{\sigma}_1}B) \tilde{J}^{m,\rho}_{\hat{\sigma}_1}(\xi,B). \label{product32of32tJmr1} \end{align}\tag{71}\] By (69 ), (70 ) and (71 ), We obtain \[\begin{align} \max_{t\in D_m}\big\{ |\tilde{J}^{m,\rho}_t(\xi,B)|, |\tilde{J}^{m,\rho}_t(\xi,B)^{-1}| \big\}\le 2^{N_{\beta}(\tilde{w})+1}, \end{align}\] which completes the proof. ◻

Proof. We already proved that there exists \(\tilde{N}(B)\) such that \(|\tilde{J}^{m,\rho}_t|\le \tilde{N}(B)\) for all sufficiently large \(m\) and \(t\in D_m\). Noting this boundedness, we obtain desired result by the same proofs as in Lemmas 8 and 9. ◻

Proof. (1) Set \(A^{m,\rho}_{\tau^m_{i-1},\tau^m_i} = I-E^{m,\rho}(Y^{m,\rho}_{\tau^m_{i-1}},\theta_{\tau^m_{i-1}}B)\). By the equation 54 , we have \[\begin{gather} (\tilde{J}^{m,\rho}_{\tau^m_i})^{-1}-(\tilde{J}^{m,\rho}_{\tau^m_{i-1}})^{-1} = (\tilde{J}^{m,\rho}_{\tau^m_{i-1}})^{-1} \left(E^{m,\rho}(Y^{m,\rho}_{\tau^m_{i-1}},\theta_{\tau^m_{i-1}}B)^{-1}-I\right)\\ = (\tilde{J}^{m,\rho}_{\tau^m_{i-1}})^{-1} \left((I-A^{m,\rho}_{\tau^m_{i-1},\tau^m_i})^{-1}-I\right) = (\tilde{J}^{m,\rho}_{\tau^m_{i-1}})^{-1} \bigg[ A^{m,\rho}_{\tau^m_{i-1},\tau^m_i} + \sum_{l=2}^{\infty} \left\{ A^{m,\rho}_{\tau^m_{i-1},\tau^m_i} \right\}^l \bigg]. \end{gather}\] By the geometric property \(B^{\alpha,\beta}_{s,t}=B^{\alpha}_{s,t}B^{\beta}_{s,t}-B^{\beta,\alpha}_{s,t}\), we have \[\begin{align} & (D\sigma)(Y^{m,\rho}_s)[(D\sigma)(Y^{m,\rho}_s)B_{s,t}] B_{s,t}- (D\sigma)(Y^{m,\rho}_s)[(D\sigma)(Y^{m,\rho}_s)]{\mathbb{B}}_{s,t} \nonumber\\ &\quad= (D\sigma)(Y^{m,\rho}_s)[(D\sigma)(Y^{m,\rho}_s)e_{\alpha}] e_{\beta}B^{\alpha}_{s,t}B^{\beta}_{s,t}- (D\sigma)(Y^{m,\rho}_s)[(D\sigma)(Y^{m,\rho}_s)e_{\alpha}]e_{\beta} {\mathbb{B}}^{\alpha,\beta}_{s,t}\nonumber\\ &\quad=(D\sigma)(Y^{m,\rho}_s)[(D\sigma)(Y^{m,\rho}_s)e_{\alpha}]e_{\beta} {\mathbb{B}}^{\beta,\alpha}_{s,t}. \end{align}\] Using this and by the assumption of (55 ) and Lemma 15, we obtain the desired estimate.

(2) We have proved that \((\tilde{J}^{m,\rho}_t)^{-1}\) satisfies a similar equation to \(Y^{m,\rho}_t\) and the norm can be estimated as in Lemma 15. Hence, we can complete the proof in the same way as in Lemma 9. ◻

Proof. We already proved Lemma 16 and Lemma 17. Hence the proof is similar to that of Lemma 10. ◻

Proof. (1) Note that the processes \((\tilde{J}^{m,\rho})^{-1}\) and \(c(Y^{m,\rho})\) appeared in 58 admit the uniform Hölder estimates and that \(d^m\) and \(\hat{\epsilon}^{m}-\epsilon^m\) are \(\{a_m\}\)-order nice discrete processes (see Remark 21). Hence the assertion follows from Remark 22. (2) follows from (1) and Proposition 26. We prove (3). By (2), there exists \(X\in \cap_{p\ge 1}L^p(\Omega)\) such that \(\max_t|\hat{Y}^{m}_t-Y_t|\le a_mX\) on \(\Omega_0^{(m,d^m)}\). Also we have for any \(R>0\), there exists \(C_R>0\) such that \(\mu\Big((\Omega_0^{(m,d^m)})^\complement\Big)\le C_R2^{-mR}\). Using these estimates and the Schwarz inequality, we have \[\begin{align} &\|(2^m)^{\min\{3H^--1,\varepsilon_1\}-\kappa}\max_t|\hat{Y}^{m}_t-Y_t|\|_{L^p}^p\\ &\le E\left[(2^m)^{-\kappa p} X^p ; \Omega_0^{(m,d^m)}\right]+ E\left[(2^m)^{(\min\{3H^--1,\varepsilon_1\}-\kappa)p} \max_t|\hat{Y}^{m}_t-Y_t|^p ;(\Omega_0^{(m,d^m)})^\complement\right]\\ &\le 2^{-mp\kappa}\|X\|_{L^p}^p+ (2^m)^{(\min\{3H^--1,\varepsilon_1\}-\kappa)p-R/2}C_R^{\frac{1}{2}}E[\max_t|\hat{Y}^{m}_t-Y_t|^{2p}]^{\frac{1}{2}}. \end{align}\] Combining this estimate and Lemma 9, we complete the proof. ◻

Proof. (1) This follows from Lemma 16 and Remark 29.

(2) Similarly to \(\epsilon^m_t\) and \(\hat{\epsilon}^{m}_t\) (see 2 ), we set \(\epsilon(J)^m_t=\sum_{i=1}^{2^mt} \epsilon(J)_{\tau^m_{i-1},\tau^m_i}\) \((t\in D_m)\). From assertion (1), \(\epsilon(J)^m\) is a \(\{\Delta_m^{3H^--1}\}\)-order nice discrete process. Hence, using the estimate of \(\tilde{J}^m\) and Remark 22, we see assertion (2).

(3) From the definition of \(\tilde{J}^m\) and 76 , we have \[\begin{align} J_t &= \tilde{J}^m_t+\tilde{J}^m_t\sum_{i=1}^{2^mt} \left(\tilde{J}^m_{\tau^m_i}\right)^{-1} \epsilon(J)_{\tau^m_{i-1},\tau^m_i} = \tilde{J}^m_t-\tilde{J}^m_t\delta^{m}(J)_t \end{align}\] Hence \(\tilde{J}^m_t-J_t = J_t \delta^{m}(J)_t + (\tilde{J}^m_t-J_t) \delta^{m}(J)_t\), which implies 77 . Noting \(\tilde{J}^m_t-J_t=\tilde{J}^m_t\delta^m(J)_t\), we get (78 ).

(4) Note that \[\begin{align} J_{t}^{-1}-(\tilde{J}^m_{t})^{-1} &= - (\tilde{J}^m_{t})^{-1} \big(J_{t}-\tilde{J}^m_{t}\big) J_{t}^{-1} \\ &= - J_t^{-1} \big(J_t-\tilde{J}^m_t\big) J_t^{-1} + \big(J_{t}^{-1}-(\tilde{J}^m_{t})^{-1}\big) \big(J_t-\tilde{J}^m_t\big) J_t^{-1}. \end{align}\] Iterating this \(L\) times and using the first identity above, we get \[\begin{align} J_t^{-1}-(\tilde{J}^m_t)^{-1} &= - J_t^{-1} \sum_{l=1}^L \big[ \big(J_t-\tilde{J}^m_t\big) J_t^{-1} \big]^l + (J_t^{-1}-(\tilde{J}^m_t)^{-1}) \big[ \big(J_t-\tilde{J}^m_t\big) J_t^{-1} \big]^L\\ &= - J_t^{-1} \sum_{l=1}^L \big[ \big(J_t-\tilde{J}^m_t\big) J_t^{-1} \big]^l - (\tilde{J}^m_{t})^{-1} \big[ \big(J_t-\tilde{J}^m_t\big) J_t^{-1} \big]^{L+1}. \end{align}\] and \((\tilde{J}^m_{t})^{-1} \big[ \big(J_t-\tilde{J}^m_t\big) J_t^{-1} \big]^{L+1} = O(\Delta_m^{(3H^--1)(L+1)})\). Thus \[\begin{align} J_t^{-1}-(\tilde{J}^m_t)^{-1}\\&= - J_t^{-1} \sum_{l=1}^L \left[ \left( - J_t \sum_{r=1}^R (\delta^{m}(J)_t)^r + O(\Delta_m^{(3H^--1)(R+1)}) \right) J_t^{-1} \right]^l + O(\Delta_m^{(3H^--1)(L+1)}) \\ &= - J_t^{-1} \sum_{l=1}^L \left[ \left(- J_t\sum_{r=1}^R (\delta^{m}(J)_t)^r\right) J_t^{-1} \right]^l +O(\Delta_m^{(3H^--1)(L+1)}) +LO(\Delta_m^{(3H^--1)(R+1)}).\end{align}\] Since we have \[\begin{align} \left[ \left(- J_t \sum_{r=1}^R (\delta^{m}(J)_t)^r\right) J_t^{-1} \right]^l = J_t \left( - \sum_{r=1}^R (\delta^{m}(J)_t)^r \right)^l J_t^{-1}, \end{align}\] we arrive at the conclusion. ◻

Proof. From 54 , we have \[\begin{align} N^{m,\rho}_{\tau^m_j} = N^{m,\rho}_{\tau^m_{j-1}} + (-\tilde{J}^{m,\rho}_{\tau^m_j})^{-1} \left\{\partial_{\rho}E^{m,\rho}(Y^{m,\rho}_{\tau^m_{j-1}},\theta^m_{\tau^m_{j-1}}B)\right\} \tilde{J}^{m,\rho}_{\tau^m_{j-1}}. \end{align}\] Using \((\tilde{J}^{m,\rho}_{\tau^m_j})^{-1} = (\tilde{J}^{m,\rho}_{\tau^m_{j-1}})^{-1} \{ I -(D\sigma)(Y^{m,\rho}_{\tau^m_{j-1}})B_{\tau^m_{j-1},\tau^m_j} +O(\Delta_m^{2H^-}) \}\) due to Lemma 6 and the expression of \(\partial_\rho E^{m,\rho}(Y^{m,\rho}_{\tau^m_{j-1}},\theta^m_{\tau^m_{j-1}}B)\), we have \[\begin{align} N^{m,\rho}_{\tau^m_j}-N^{m,\rho}_{\tau^m_{j-1}} &= (-\tilde{J}^{m,\rho}_{\tau^m_{j-1}})^{-1} \Big\{ (D^2\sigma)(Y^{m,\rho}_{\tau^m_{j-1}})[Z^{m,\rho}_{\tau^m_{j-1}},\tilde{J}^{m,\rho}_{\tau^m_{j-1}}]B_{\tau^m_{j-1},\tau^m_j} \nonumber\\ &\qquad\qquad\qquad\qquad - (D\sigma)(Y^{m,\rho}_{\tau^m_{j-1}}) \left[ (D^2\sigma)(Y^{m,\rho}_{\tau^m_{j-1}})[Z^{m,\rho}_{\tau^m_{j-1}},\tilde{J}^{m,\rho}_{\tau^m_{j-1}}]B_{\tau^m_{j-1},\tau^m_j} \right] B_{\tau^m_{j-1},\tau^m_j} \nonumber \\ &\qquad\qquad\qquad\qquad + D^2((D\sigma)[\sigma])(Y^{m,\rho}_{\tau^m_{j-1}})[Z^{m,\rho}_{\tau^m_{j-1}},\tilde{J}^{m,\rho}_{\tau^m_{j-1}}]{\mathbb{B}}_{\tau^m_{j-1},\tau^m_j} \Big\} \nonumber \\ &\phantom{=}\quad + (-\tilde{J}^{m,\rho}_{\tau^m_{j-1}})^{-1} \Big[ (Dc)(Y^{m,\rho}_{\tau^m_{j-1}})[\tilde{J}^{m,\rho}_{\tau^m_{j-1}}] + \rho(D^2c)(Y^{m,\rho}_{\tau^m_{j-1}})[Z^{m,\rho}_{\tau^m_{j-1}},\tilde{J}^{m,\rho}_{\tau^m_{j-1}}] \Big]d^m_{\tau^m_{j-1},\tau^m_j} \nonumber \\ &\phantom{=}\quad + (-\tilde{J}^{m,\rho}_{\tau^m_{j-1}})^{-1} \Big[ (D^2b)(Y^{m,\rho}_{\tau^m_{j-1}})[Z^{m,\rho}_{\tau^m_{j-1}},\tilde{J}^{m,\rho}_{\tau^m_{j-1}}]\Delta_m \Big] + O(\Delta_m^{3H^-}). \label{eq4349023211} \end{align}\tag{79}\] Next we take the sum over \(0\leq j\leq 2^mt\). Applying \(B^\alpha_{s,t}B^\beta_{s,t}-B^{\alpha,\beta}_{s,t}=B^{\beta,\alpha}_{s,t}\) and substituting \(Z^{m,\rho}_{\tau^m_{j-1}}=\tilde{J}^{m,\rho}_{\tau^m_{j-1}}\tilde{Z}^{m,\rho}_{\tau^m_{j-1}}=\sum_{\nu=1}^n\tilde{Z}^{m,\rho,\nu}_{\tau^m_{j-1}} \tilde{J}^{m,\rho}_{\tau^m_{j-1}}f_\nu\), we see that the summation of the first term in 79 gives \[\begin{align} \sum_{\nu=1}^n \sum_{j=1}^{2^mt} \tilde{Z}^{m,\rho,\nu}_{\tau^m_{j-1}} I^{m,\rho}(\varphi_\nu)_{\tau^m_j,\tau^m_{j-1}} = \sum_{\nu=1}^n \tilde{Z}^{m,\rho,\nu}_t I^{m,\rho}(\varphi_\nu)_t - \sum_{\nu=1}^n \sum_{j=1}^{2^mt} \tilde{Z}^{m,\rho,\nu}_{\tau^m_j,\tau^m_{j-1}} I^{m,\rho}(\varphi_\nu)_{\tau^m_j}. \end{align}\] The summations of the second and third terms in 79 give \(I_2(N^{m,\rho})\) and \(I_1(N^{m,\rho})\), respectively. The summation of the fourth term \(O(\Delta_m^{3H^-})\) in 79 is \(I_3(N^{m,\rho})\), which is an \(\{a_m\}\)-order nice discrete process. This completes the proof of (1).

We show assertion (2). Recall that the discrete Hölder norm \(\|I^{m,\rho}(\varphi_\nu)\|_{H^-}\) can be estimated by a constant which depends on \(\sigma, b, c\), \(C(B)\) and \(\tilde{N}(B)\) polynomially (see Lemma ) and that \(\tilde{Z}^{m,\rho,\nu}\) is an \(\{a_m\}\)-order nice discrete process (see Theorem 28). Thus, the discrete version of the estimate of Young integrals (Remark 22) implies that \(I_0(N^{m,\rho})\) is an \(\{a_m\}\)-order nice discrete process. Noting that we have good estimates of \(H^-\)-Hölder norm of \(Y^{m,\rho}\), \(\tilde{J}^{m,\rho}\), \((-\tilde{J}^{m,\rho})^{-1}\) (Lemma 9, Lemma 16, Lemma 17) and that \(Z^{m,\rho}\) is an \(\{a_m\}\)-order nice discrete process (Theorem 28), we see that \(I_1(N^{m,\rho})\) is an \(\{a_m\}\)-order nice discrete process. Since \(d^m\) is an \(\{a_m\}\)-order nice discrete process, \(I_2(N^{m,\rho})\) is as well. As for \(I_3^{m,\rho}\), we already proved the assertion. Here we used Lemmas 16, 17, and 18 and Theorem 28. Since \(\sup_{\rho}\|\tilde{Z}^{m,\rho}\|_{H^-}=O(a_m)\) and other terms are \(\{a_m\}\)-order nice discrete processes, we have \(\sup_{\rho}\|N^{m,\rho}\|_{H^-}=O(a_m)\) which completes the proof of assertion (2). ◻

Proof. Note that \[\begin{align} \tilde{J}^{m,\rho}_t-\tilde{J}^m_t &= \int_0^\rho \partial_{\rho_1}J^{m,\rho_1}_t d\rho_1 = \int_0^\rho (-J^{m,\rho_1}_t)N^{m,\rho_1}_t d\rho_1,\\ (\tilde{J}^{m,\rho}_t)^{-1}-(\tilde{J}^m_t)^{-1} &= \int_0^{\rho} \partial_{\rho_1} (J^{m,\rho_1}_t)^{-1} d\rho_1 = \int_0^{\rho} N^{m,\rho_1}_t (J^{m,\rho_1}_t)^{-1} d\rho_1. \end{align}\] From Lemmas 15 and 22, we see that \(\sup_{t,\rho}|\tilde{J}^{m,\rho}_t-\tilde{J}^m_t|=O(a_m)\) and \(\sup_{t,\rho}|(\tilde{J}^{m,\rho}_t)^{-1}-(\tilde{J}^m_t)^{-1}|=O(a_m)\). This and Remark 30 yield the assertion. ◻

Proof. Set \(F^{m,\rho}_t=(\tilde{J}^{m,\rho}_{t+\Delta_m})^{-1}(c(Y^{m,\rho}_t)-c(Y_t))\). We have \[\begin{align} c(Y^{m,\rho}_t)-c(Y_t)&= \int_0^{\rho}(Dc)(Y^{m,\rho_1}_t)[Z^{m,\rho_1}_t]d\rho_1 = \int_0^{\rho}(Dc)(Y^{m,\rho_1}_t)[\tilde{J}^{m,\rho_1}_t \tilde{Z}^{m,\rho_1}_t]d\rho_1 \end{align}\] and we obtain Hölder estimate of the discrete process \(\| F^{m,\rho}\|_{H^-} \le C \sup_{\rho}\|\tilde{Z}^{m,\rho}\|_{H^-}\). Here, \(C\) depends on the Hölder norms of \(Y^{m,\rho}\) and \(\tilde{J}^{m,\rho}\). By combining the estimate \(\|d^m\|_{\lambda_1}\le 2^{-m\varepsilon_1}G_1\leq a_m G_1\) (\(\omega\in \Omega_0\)) and Remark 22, we complete the proof. ◻

Proof. This follows from Lemma 17 and Remark 29. We used \(\lambda_1>H^-\). ◻

Proof. From 56 , Condition 9 (1) and Lemma 1 (1), we have \[\begin{align} (\tilde{J}^{m,\rho}_{\tau^m_i})^{-1}\big(\hat{\epsilon}^{m}_{\tau^m_{i-1},\tau^m_i}-\epsilon^m_{\tau^m_{i-1},\tau^m_i}\big) = (\tilde{J}^{m,\rho}_{\tau^m_{i-1}})^{-1}\big(\hat{\epsilon}^{m}_{\tau^m_{i-1},\tau^m_i}-\epsilon^m_{\tau^m_{i-1},\tau^m_i}\big) + O(\Delta_m^{4H^-}). \end{align}\] Combining this identity with Condition 9 (2) and Lemma 1 (2) yields the desired estimate. ◻

Proof. By the assumption on the Hölder norm of \(F^m\) and Condition 11 and using Remark 22, we have \(\|(2^m)^{2H-\frac{1}{2}}I^m(F^m)\|_{\lambda_2} \le \Delta_m^{\varepsilon_2} C G_2,\) which implies the assertion. ◻

Proof. We use the decompositions in Lemmas 25 and 26. First, we consider the sum of \(O(\Delta_m^{4H^-})\). Let \(s=\tau^m_k<\tau^m_l=t\). We have \[\begin{align} \left| (2^m)^{2H-\frac{1}{2}} \sum_{i=k}^{l-1} O(\Delta_m^{4H^-}) \right| \le (2^m)^{2H-\frac{1}{2}}(l-k) C\Delta_m^{4H^-} =\Delta_m^{4H^--2H-\frac{1}{2}}C(t-s). \end{align}\] where \(C\) depends on \(\tilde{C}(B)\) and \(\tilde{N}(B)\) polynomially. This term can be estimated as in the assertion. As for sum process \(K^m_{s,t}=\Delta_m B^{\alpha}_{s,t}\) which defined by the term \(\Delta_m B^{\alpha}_{\tau^m_{i-1},\tau^m_i}\) in Lemma 26, we have similar estimate to the elements in \(\mathcal{K}^3_m\). See the proof of Lemma 4. Note that we use Condition 3 only in that proof. The remaining main terms can be handled by Lemma 27 and Condition 11. This completes the proof. ◻

Proof. Let \(R\) be a positive integer. From Remark 30, we have \[\begin{align} (\tilde{J}^m_{\tau^m_i})^{-1}-J_{\tau^m_i}^{-1} &= K^{2,m,R}_{\tau^m_i}J_{\tau^m_i}^{-1} +L^{2,m,R}_{\tau^m_i} \nonumber \\ &= K^{2,m,R}_{\tau^m_i}J^{-1}_{\tau^m_{i-1}} +K^{2,m,R}_{\tau^m_i}J^{-1}_{\tau^m_{i-1},\tau^m_i} +L^{2,m,R}_{\tau^m_i}. \label{eq48930184231} \end{align}\tag{81}\] where \(K^{2,m,R}\) is an \(\{a_m\}\)-order nice discrete processes and \(L^{2,m,R}\) is a small discrete process. Hence \[\begin{align} S^{m,\rho,3}_t &= \sum_{i=1}^{2^m t} \left( K^{2,m,R}_{\tau^m_i}J^{-1}_{\tau^m_{i-1}} +K^{2,m,R}_{\tau^m_i}J^{-1}_{\tau^m_{i-1},\tau^m_i} +L^{2,m,R}_{\tau^m_i} \right) c(Y_{\tau^m_{i-1}}) d^m_{\tau^m_{i-1},\tau^m_i}\\ &= S^{m,\rho,3,1}_t +S^{m,\rho,3,2}_t +S^{m,\rho,3,3}_t, \end{align}\] Then with the help of the summation by parts formula 45 , we have \[\begin{align} S^{m,\rho,3,1}_t = \sum_{i=1}^{2^m t} K^{2,m,R}_{\tau^m_i} I^m_{\tau^m_{i-1},\tau^m_i} = K^{2,m,R}_t I^m_t - \sum_{i=1}^{2^m t} K^{2,m,R}_{\tau^m_{i-1},\tau^m_i} I^m_{\tau^m_{i-1}}. \end{align}\] Recalling that \((2^m)^{2H-\frac{1}{2}}I^m|_{D_m}\) is discrete \(H^-\)-Hölder continuous and using Remark 30, using \(X_m\) defined by 80 , we have \[\begin{align} \|(2^m)^{2H-\frac{1}{2}}S^{m,\rho,3,1}\|_{H^-}\\&\leq 2 \|K^{2,m,R}\|_{H^-} \|(2^m)^{2H-\frac{1}{2}}I^m|_{D_m}\|_{H^-} + \left\| \sum_{i=1}^{2^m \cdot} K^{2,m,R}_{\tau^m_{i-1},\tau^m_i} (2^m)^{2H-\frac{1}{2}}I^m_{\tau^m_{i-1}} \right\|_{H^-}\\ &\leq C \{ a_m \cdot X_m + a_m \cdot X_m \}.\end{align}\] In a similar way to Lemma 28, using Lemma 25, we have \[\begin{align} \|(2^m)^{2H-\frac{1}{2}} S^{m,\rho,3,2} \|_{\lambda_2} \le a_m C\{ G_2+1 \}. \end{align}\] The term \(\|(2^m)^{2H-\frac{1}{2}}S^{m,\rho,3,3}\|_{H^-}\) becomes small for large \(R\). The proof is completed. ◻

Proof. Noting \(\partial_{\rho}(\tilde{J}^{m,\rho}_t)^{-1}= -(\tilde{J}^{m,\rho}_t)^{-1}\partial_{\rho}\tilde{J}^{m,\rho}_t (\tilde{J}^{m,\rho}_t)^{-1}=N^{m,\rho}_t(\tilde{J}^{m,\rho}_t)^{-1}\), we have \[\begin{align} (\tilde{J}^{m,\rho}_t)^{-1}-(\tilde{J}^m_t)^{-1} = \int_{0<\rho_1<\rho} d\rho_1\, N^{m,\rho_1}_t (\tilde{J}^{m,\rho_1}_t)^{-1}\\&= \int_{0<\rho_1<\rho} d\rho_1\, N^{m,\rho_1}_t (\tilde{J}^m_t)^{-1} + \int_{0<\rho_1<\rho} d\rho_1 N^{m,\rho_1}_t \left\{ (\tilde{J}^{m,\rho_1}_t)^{-1}-(\tilde{J}^m_t)^{-1} \right\}\\ &= \int_{0<\rho_1<\rho} d\rho_1\, N^{m,\rho_1}_t (\tilde{J}^m_t)^{-1} + \int_{0<\rho_1<\rho} d\rho_1 N^{m,\rho_1}_t \int_{0<\rho_2<\rho_1} d\rho_2\, N^{m,\rho_2}_t (\tilde{J}^{m,\rho_2}_t)^{-1}.\end{align}\] Iterating this calculation, we are done. ◻

Proof. We use the same notation as in Lemmas 22 and 30. Set \[\begin{align} \tilde{N}^{m,\rho_1,\ldots,\rho_l}_t &= \prod_{r=1}^l \left\{ N^{m,\rho_r}_t - \sum_{\nu=1}^n \tilde{Z}^{m,\rho_r,\nu}_t I^{m,\rho_r}(\varphi_\nu)_t \right\} = \prod_{r=1}^l \sum_{\lambda=0}^3 I_\lambda(N^{m,\rho_r})_t,\\ R^{m,\rho_1,\ldots,\rho_l}_t &= N^{m,\rho_1}_t\cdots N^{m,\rho_l}_t - \tilde{N}^{m,\rho_1,\ldots,\rho_l}_t. \end{align}\] Note that the product \(\prod_{r=1}^l\) in the above equation should be taken according to the order. Then we have \(S^{m,\rho,2}_t=S^{m,\rho,2,1}_t+S^{m,\rho,2,2}_t+S^{m,\rho,2,3}_t\), where \[\begin{align} S^{m,\rho,2,1}_t &= \sum_{l=1}^{L-1} \int_{0<\rho_l<\cdots<\rho_1<\rho} d\rho_1 \cdots d\rho_l\, \sum_{i=1}^{2^mt} \tilde{N}^{m,\rho_1,\ldots,\rho_l}_{\tau^m_i} (\tilde{J}^m_{\tau^m_i})^{-1} c(Y_{\tau^m_{i-1}})d^m_{\tau^m_{i-1},\tau^m_i},\\ S^{m,\rho,2,2}_t &= \sum_{l=1}^{L-1} \int_{0<\rho_l<\cdots<\rho_1<\rho} d\rho_1 \cdots d\rho_l\, \sum_{i=1}^{2^mt} R^{m,\rho_1,\ldots,\rho_l}_{\tau^m_i} (\tilde{J}^m_{\tau^m_i})^{-1} c(Y_{\tau^m_{i-1}})d^m_{\tau^m_{i-1},\tau^m_i},\\ S^{m,\rho,2,3}_t &= \int_{0<\rho_{L}<\cdots<\rho_1<\rho} d\rho_1 \cdots d\rho_L\, \sum_{i=1}^{2^mt} N^{m,\rho_1}_{\tau^m_i} \cdots N^{m,\rho_L}_{\tau^m_i} (\tilde{J}^{m,\rho_L}_{\tau^m_i})^{-1} c(Y_{\tau^m_{i-1}})d^m_{\tau^m_{i-1},\tau^m_i}. \end{align}\] We estimate the terms above.

By the definition, all terms in the expansion of \(R^{m,\rho_1,\ldots,\rho_l}_t\) are given by the product of \(l\) terms from \(N^{m,\rho_r}_t\) and \(\tilde{Z}^{m,\rho_r,\nu}_t I^{m,\rho_r}(\varphi_\nu)_t\) (\(1\leq r\leq l\), \(1\leq \nu\leq n\)) and each term contains at least one \(\tilde{Z}^{m,\rho_r,\nu}_t I^{m,\rho_r}(\varphi_\nu)_t\). Thus, using Remark 22, Lemmas 18 and 22, we have \[\begin{align} \bigg\| (2^m)^{2H-\frac{1}{2}} \sum_{i=1}^{2^m \cdot} R^{m,\rho_1,\ldots,\rho_l}_{\tau^m_i} (\tilde{J}^m_{\tau^m_i})^{-1} c(Y_{\tau^m_{i-1}}) d^m_{\tau^m_{i-1},\tau^m_i} \Big\|_{\lambda_1}\\&\leq C \| (2^m)^{2H-\frac{1}{2}} R^{m,\rho_1,\ldots,\rho_l} \|_{H^-} \bigg\| \sum_{i=1}^{2^m \cdot} (\tilde{J}^m_{\tau^m_i})^{-1} c(Y_{\tau^m_{i-1}}) d^m_{\tau^m_{i-1},\tau^m_i} \bigg\|_{\lambda_1}\\ &\leq C \|(2^m)^{2H-\frac{1}{2}}\tilde{Z}^{m,\rho}\|_{H^-} \cdot a_m C G_1,\end{align}\] from which we obtain an estimate of \(S^{m,\rho,2,2}\). We next consider \(S^{m,\rho,2,1}\). Noting 81 , we have \[\begin{align} \sum_{i=1}^{2^mt} \tilde{N}^{m,\rho_1,\ldots,\rho_l}_{\tau^m_i} (\tilde{J}^m_{\tau^m_i})^{-1} c(Y_{\tau^m_{i-1}})d^m_{\tau^m_{i-1},\tau^m_i} = \sum_{i=1}^{2^mt} \tilde{N}^{m,\rho_1,\ldots,\rho_l}_{\tau^m_i} (I+K^{2,m,R}_{\tau^m_i}) J_{\tau^m_{i-1}}^{-1} c(Y_{\tau^m_{i-1}})d^m_{\tau^m_{i-1},\tau^m_i}\\&+ \sum_{i=1}^{2^mt} \tilde{N}^{m,\rho_1,\ldots,\rho_l}_{\tau^m_i} (I+K^{2,m,R}_{\tau^m_i}) J_{\tau^m_{i-1},\tau^m_i}^{-1} c(Y_{\tau^m_{i-1}}) d^m_{\tau^m_{i-1},\tau^m_i}\\ &+ \sum_{i=1}^{2^mt} \tilde{N}^{m,\rho_1,\ldots,\rho_l}_{\tau^m_i} L^{2,m,R}_{\tau^m_i} c(Y_{\tau^m_{i-1}}) d^m_{\tau^m_{i-1},\tau^m_i}.\end{align}\] All terms can be treated in the similar way as Lemma 29 because \(\tilde{N}^{m,\rho_1,\ldots,\rho_l}\) is an \(\{a_m\}\)-order nice discrete process independent of \(\rho_1,\ldots,\rho_l\) (see Lemma 22).

Finally, we consider \(S^{m,\rho,2,3}\). Noting that \[\begin{align} \sup_{\rho_1,\ldots,\rho_L}\|N^{m,\rho_1}\cdots N^{m,\rho_L}\|_{H^-} =O(a_m^L), \end{align}\] we see that this term is small for large \(L\). This completes the proof. ◻

Proof of Lemma . We write \(f_m = \sup_{\rho}\|(2^m)^{2H-\frac{1}{2}}\tilde{Z}^{m,\rho}\|_{H^-}1_{\Omega_0^{(m,d^m)}}\). From the lemmas above, there exist random variables \(\{\varGamma_m\}\) and \(\varGamma\) defined on \(\Omega_0\) which satisfy \(\sup_m\|\varGamma_m\|_{L^p}<\infty\) for all \(p\ge 1\) and \(\varGamma\in \cap_{p\ge 1}L^p(\Omega_0)\) such that \(f_m\le \varGamma_m+a_m \varGamma f_m\). Recalling \(\tilde{Z}^{m,\rho}\) is an \(\{a_m\}\)-order nice discrete process independent of \(\rho\) (Theorem 28), there exists \(\varGamma'\) such that \(f_m\leq (2^m)^{2H-\frac{1}{2}}\varGamma'\) and \(\varGamma'\in \cap_{p\ge 1}L^p(\Omega_0)\). By using this inequality \(L\)-times and Theorem 28, we get \[\begin{align} f_m &\le \left\{ \sum_{l=0}^{L-1} (a_m \varGamma)^l \right\} \varGamma_m + (a_m \varGamma)^{L} f_m \le \left\{ \sum_{l=0}^{L-1} (a_m \varGamma)^l \right\} \varGamma_m + (2^m)^{2H-\frac{1}{2}}(a_m \varGamma)^{L} \varGamma'. \end{align}\] By taking \(L\) to be sufficiently large, we arrive at the conclusion. ◻

Proof. Write \(f_m=\sup_{\rho}\|(2^m)^{2H-\frac{1}{2}}\tilde{Z}^{m,\rho}\|_{H^-}1_{\Omega_0^{(m,d^m)}}\). Lemmas 24, 31 imply \[\begin{align} \|(2^m)^{2H-\frac{1}{2}+\varepsilon}S^{m,\rho,1}\|_{H^-}1_{\Omega_0^{(m,d^m)}} &\leq (2^m)^\varepsilon \cdot a_m CG_1 f_m,\\ \|(2^m)^{2H-\frac{1}{2}+\varepsilon_1}S^{m,\rho,2}\|_{H^-}1_{\Omega_0^{(m,d^m)}} &\leq (2^m)^\varepsilon \cdot a_m C \big\{ G_1f_m +X_m +G_2 +1 \big\}. \end{align}\] Lemmas 28 and 29 gives similar estimates for \(\|(2^m)^{2H-\frac{1}{2}+\varepsilon}S^{m,\rho,r}\|_{H^-}1_{\Omega_0^{(m,d^m)}}\) for \(r=3,4,5\). Combining these estimates and Lemma 23, the proof is finished. ◻

Proof of Theorem 12. Recall that \(R^m_t\) \((t\in D_m)\) is defined by 31 . We will first consider \(R^m_t1_{\Omega_0^{(m,d^m)}}\), then \(R^m_t1_{(\Omega_0^{(m,d^m)})^\complement}\). Proposition 26 implies \[\begin{align} R^m_t1_{\Omega_0^{(m,d^m)}} = (\hat{Y}^{m}_t-Y_t-J_tI^m_t)1_{\Omega_0^{(m,d^m)}} = \int_0^1 \{ \tilde{J}^{m,\rho}_t \tilde{Z}^{m,\rho}_t - J_tI^m_t \} 1_{\Omega_0^{(m,d^m)}}\, d\rho. \end{align}\] The integrand scaled by \((2^m)^{2H-\frac{1}{2}+\varepsilon}\) is decomposed into \[\begin{gather} (2^m)^{2H-\frac{1}{2}+\varepsilon} \{ \tilde{J}^{m,\rho}_t \tilde{Z}^{m,\rho}_t - J_tI^m_t \} 1_{\Omega_0^{(m,d^m)}} = \tilde{J}^{m,\rho}_t \cdot (2^m)^{2H-\frac{1}{2}+\varepsilon} \big( \tilde{Z}^{m,\rho}_t-I^m_t \big) 1_{\Omega_0^{(m,d^m)}}\\ + (2^m)^{\varepsilon} \big(\tilde{J}^{m,\rho}_t-J_t\big) 1_{\Omega_0^{(m,d^m)}} \cdot (2^m)^{2H-\frac{1}{2}} I^m_t. \end{gather}\] Hence we have \[\begin{gather} (2^m)^{2H-\frac{1}{2}+\varepsilon} \max_{t\in D_m}|R^m_t1_{\Omega_0^{(m,d^m)}}|\\ \leq \big( \max_t | \tilde{J}^{m,\rho}_t | \big) \big( \sup_{\rho} \|(2^m)^{2H-\frac{1}{2}+\varepsilon}(\tilde{Z}^{m,\rho}-I^m)\|_{H^-} \big) 1_{\Omega^{(m,d^m)}_0}\\ + (2^m)^{\varepsilon} \big( \sup_{t,\rho}|\tilde{J}^{m,\rho}_t-J_t| \big) 1_{\Omega^{(m,d^m)}_0} \cdot \|(2^m)^{2H-\frac{1}{2}}I^m|_{D_m}\|_{H^-}. \end{gather}\] Here, \(I^m|_{D_m}\) denote the discrete process defined as the restriction of \(I^m\) on \(D_m\). The first term in the right-hand side converges to \(0\) due to Lemmas 20 and 32. The second term converges to \(0\) follows from Theorem 31 and Condition 10. From this we have \((2^m)^{2H-\frac{1}{2}+\varepsilon} \max_{t\in D_m}|R^m_t1_{\Omega_0^{(m,d^m)}}|\) converges to \(0\) in \(L^p\).

Next we consider \(R^m_t1_{(\Omega_0^{(m,d^m)})^\complement}\). Noting \[\begin{align} (2^m)^{2H-\frac{1}{2}+\varepsilon} R^m_t1_{(\Omega_0^{(m,d^m)})^\complement} &= \big(\hat{Y}^{m}_t-Y_t\big) \cdot (2^m)^{2H-\frac{1}{2}+\varepsilon} 1_{(\Omega_0^{(m,d^m)})^\complement}\\ &\phantom{\leq}\qquad - J_t \cdot (2^m)^{2H-\frac{1}{2}} I^m_t \cdot (2^m)^{\varepsilon} 1_{(\Omega_0^{(m,d^m)})^\complement}, \end{align}\] we have \[\begin{gather} (2^m)^{2H-\frac{1}{2}+\varepsilon} \max_{t\in D_m} |R^m_t1_{(\Omega_0^{(m,d^m)})^\complement}| \leq \big( \max_t | \hat{Y}^{m}_t-Y_t | \big) \cdot (2^m)^{2H-\frac{1}{2}+\varepsilon} 1_{(\Omega_0^{(m,d^m)})^\complement}\\ + \big( \max_t | J_t | \big) \|(2^m)^{2H-\frac{1}{2}}I^m|_{D_m}\|_{H^-} \cdot (2^m)^{\varepsilon} 1_{(\Omega_0^{(m,d^m)})^\complement}. \end{gather}\] Lemma 9 and Remark 29 imply that \(\max_t | \hat{Y}^{m}_t - Y_t |\) and \(\max_t | J_t |\) are bounded from above by \(\cap_{p\ge 1}L^p\) random variable. By using 42 and Condition 10, both terms of the right-hand side converge to \(0\) in \(L^p\). The proof is completed. ◻

Proof of Corollary . Recall that \(R^m_t\) \((0\leq t\leq 1)\) is defined by 33 . Since \(R^m_t=R^m_{\tau^m_{k-1}}+(R^m_t-R^m_{\tau^m_{k-1}})\) for \(\tau^m_{k-1}\leq t\leq \tau^m_k\), we have \[\begin{align} \max_{0\leq t\leq 1} |R^m_t| \leq \max_{t\in D_m} |R^m_t| + \max_{1\leq k\leq 2^m} \max_{\tau^m_{k-1}\leq t\leq \tau^m_k} |R^m_t-R^m_{\tau^m_{k-1}}|. \end{align}\] Since the first term is estimated in Theorem 12, we give an estimate of the second term. Let \(\tau^m_{k-1}\leq t\leq \tau^m_k\). We decompose \(R^m_t-R^m_{\tau^m_{k-1}}\) into two terms; \[\begin{align} \Phi^m_1(t) &= \hat{Y}^{m}_t-\hat{Y}^{m}_{\tau^m_{k-1}}-(Y_t-Y_{\tau^m_{k-1}}), & \Phi^m_2(t) &= J_{\tau^m_{k-1}}I^m_{\tau^m_{k-1}}-J_tI^m_t. \end{align}\] We have \[\begin{align} \Phi^m_1(t)&=\big\{\sigma(\hat{Y}^{m}_{\tau^m_{k-1}})-\sigma(Y_{\tau^m_{k-1}})\big\}B_{\tau^m_{k-1},t} +\big\{((D\sigma)[\sigma])(\hat{Y}^{m}_{\tau^m_{k-1}})-((D\sigma)[\sigma])(Y_{\tau^m_{k-1}})\big\} {\mathbb{B}}_{\tau^m_{k-1},t}\\ &\qquad +\big\{b(\hat{Y}^{m}_{\tau^m_{k-1}})-b(Y_{\tau^m_{k-1}})\big\}(t-\tau^m_{k-1})+ c(\hat{Y}^{m}_{\tau^m_{k-1}})d^{m}_{\tau^m_{k-1},t}+\big\{\hat{\epsilon}^{m}_{\tau^m_{k-1},t}-\epsilon^m_{\tau^m_{k-1},t}\big\}, \end{align}\] which implies \[\begin{align} |\Phi^m_1(t)| &\leq C \big\{ |\hat{Y}^{m}_{\tau^m_{k-1}}-Y_{\tau^m_{k-1}}|\Delta_m^{H^-} +\hat{X}\Delta_m^{2H^-} +\hat{X}\Delta_m^{3H^-} \big\}\\ &\leq C \big\{ |J_{\tau^m_{k-1}}I^m_{\tau^m_{k-1}}+R^m_{\tau^m_{k-1}}|\Delta_m^{H^-} +\hat{X}\Delta_m^{2H^-} \big\}. \end{align}\] Here \(C\) is a constant depending on \(\sigma\), \(b\), \(c\) and \(C(B)\). From this we obtain \[\begin{align} (2^m)^{2H-\frac{1}{2}+\varepsilon} |\Phi^m_1(t)| &\leq C \big( 1+\|J\|_{H^-} \big) \|(2^m)^{2H-\frac{1}{2}}I^m|_{D_m}\|_{H^-} \Delta_m^{H^--\varepsilon}\\ &\qquad + C \big\{ (2^m)^{2H-\frac{1}{2}+\varepsilon} \max_k|R^m_{\tau^m_k}| \big\} \Delta_m^{H^-} + C \hat{X} \Delta_m^{\frac{1}{2}-2H+2H^--\varepsilon}\\ &=: CX_{m,1} \Delta_m^{H^--\varepsilon} +CX_{m,2} \Delta_m^{H^-} +C \hat{X} \Delta_m^{\frac{1}{2}-2H+2H^--\varepsilon}. \end{align}\] We have \(I^m_t=I^m_{\tau^m_{k-1}}\) \((\tau^m_{k-1}\le t\le \tau^m_k)\), which implies \[\begin{align} (2^m)^{2H-\frac{1}{2}+\varepsilon}|\Phi^m_2(t)| = (2^m)^{\varepsilon} |J_{\tau^m_{k-1}}-J_t| |(2^m)^{2H-\frac{1}{2}}I^m_{\tau^m_{k-1}}| \le X_{m,1} \Delta_m^{H^--\varepsilon}. \end{align}\] Noting that the right-hand sides in the two estimates are independent of \(k\), we have \[\begin{align} (2^m)^{2H-\frac{1}{2}+\varepsilon} \max_{1\leq k\leq 2^m} \max_{\tau^m_{k-1}\leq t\leq \tau^m_k} |R^m_t-R^m_{\tau^m_{k-1}}| &\leq (C+1)X_{m,1} \Delta_m^{H^--\varepsilon} +CX_{m,2} \Delta_m^{H^-}\\ &\phantom{\leq} \qquad +C \hat{X} \Delta_m^{\frac{1}{2}-2H+2H^--\varepsilon} \end{align}\] We see that \(\sup_m\{\|X_{m,1}\|_{L^p},\|X_{m,2}\|_{L^p}, \|\hat{X}\|_{L^p}\}<\infty\) for all \(p\ge 1\) which follows from Lemma 20, Remark 29, Condition 10 and Theorem 12. Hence noting \(3H^--1\leq H^-\) and \(3H^--1\leq \frac{1}{2}-2H+2H^-\), we complete the proof. ◻