April 29, 2025
In the present article we study strong approximation of solutions of scalar stochastic differential equations (SDEs) with bounded and \(\alpha\)-Hölder continuous drift coefficient and constant diffusion coefficient at time point \(1\). Recently, it was shown in [1] that for such SDEs the equidistant Euler scheme achieves an \(L^p\)-error rate of at least \((1+\alpha)/2\), up to an arbitrary small \(\varepsilon\), for all \(p\geq 1\) and all \(\alpha\in(0, 1]\) in terms of the number of evaluations of the driving Brownian motion \(W\). In this article we prove a matching lower error bound for \(\alpha\in(0, 1)\). More precisely, we show that for every \(\alpha\in(0, 1)\), the \(L^p\)-error rate \((1+\alpha)/2\) of the Euler scheme in [1] can not be improved in general by no numerical method based on finitely many evaluations of \(W\) at fixed time points. Up to now, this result was known in the literature only for \(\alpha=1\).
Additionally, we extend a result from [2] on sharp lower errror bounds for strong approximation of SDEs with a bounded drift coefficient of fractional Sobolev regularity \(\alpha\in (0,1)\) and constant diffusion coefficient at time point \(1\). We prove that for every \(\alpha\in (0,1)\), the \(L^p\)-error rate \((1 + \alpha)/2\) that was shown in [3] for the equidistant Euler scheme can, up to a logarithmic term, not be improved in general by no numerical method based on finitely many evaluations of W at fixed time points. This result was known from [2] only for \(\alpha\in (1/2,1)\) and \(p=2\).
For the proof of these lower bounds we use variants of the Weierstrass function as a drift coefficient and we employ the coupling of noise technique recently introduced in [4].
Consider a scalar additive noise driven stochastic differential equation (SDE) \[\label{sde0} \begin{align} dX_t & = \mu(X_t) \, dt + dW_t, \quad t\in [0,1],\\ X_0 & = x_0 \end{align}\tag{1}\] with deterministic initial value \(x_0\in{\mathbb{R}}\), drift coefficient \(\mu\colon{\mathbb{R}}\to{\mathbb{R}}\) and a one-dimensional driving Brownian motion \(W=(W_t)_{t\in[0,1]}\). Note that the SDE 1 has a unique strong solution if \(\mu\) is bounded and measurable, see [5].
In this article we study the complexity of strong approximation of the solution \(X\) of the SDE 1 at the final time point \(1\) by numerical methods based on finitely many evaluations of the driving Brownian motion \(W\).
A classical numerical method of this type is the Euler scheme with \(n\) equidistant steps, given by \(X^E_{n,0} = x_0\) and \[X^E_{n,(i+1)/n} = X^E_{n,i/n}+ \mu\bigl( X^E_{n,i/n}\bigr)\cdot 1/n+ W_{(i+1)/n} -W_{i/n}\] for \(i=1,\dots,n\).
Recently, it was proven in [1] that if the drift coefficient \(\mu\) is bounded and \(\alpha\)-Hölder continuous with \(\alpha\in(0,1]\) then the Euler scheme achieves for all \(p\in [1,\infty)\) an \(L^p\)-error rate of at least \((1+\alpha)/2-\) in terms of the number \(n\) of evaluations of \(W\), i.e., for all \(\varepsilon\in (0,\infty)\) there exists \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), \[\label{eul} {\mathbb{E}}\bigl[| X_1- X^E_{n,1}|^p\bigr]^{1/p} \le \frac{c}{n^{(1+\alpha)/2-\varepsilon}}.\tag{2}\]
This upper bound naturally leads to the question whether \((1+\alpha)/2-\) is the best possible \(L^p\)-error rate that can be achieved for approximation of \(X_1\) by numerical methods based on finitely many evaluations of \(W\) at fixed time points in \([0,1]\), or whether there exists a method from this class that achieves a better \(L^p\)-error rate than \((1+\alpha)/2\).
Up to now, the answer to this question was known in the literature only for \(\alpha=1\). More precisely, it follows from more general results in [6] and [7] that if the drift coefficient \(\mu\) has bounded, continuous derivatives up to order \(3\) on some open interval containing \(x_0\) and satisfies \(\mu'(x_0)\neq 0\), then the best possible \(L^1\)-error rate that can be achieved by any numerical method based on finitely many evaluations of \(W\) at fixed time points in \([0,1]\), is at most \(1\), i.e. there exists \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), \[\inf_{\substack{ t_1,\dots ,t_n \in [0,1]\\ g \colon {\mathbb{R}}^n \to {\mathbb{R}}\text{ measurable} \\ }} {\mathbb{E}}\bigl[|X_1-g(W_{t_1}, \ldots, W_{t_n})|\bigr]\geq \frac{c}{ n}.\] The assumptions from [6] and [7] are in particular satisfied for the SDE 1 with \(\mu=\cos\) and \(x_0\in{\mathbb{R}}\setminus \{\pi k\mid k\in \mathbb{Z}\}\). Since this choice of \(\mu\) is bounded and Lipschitz continuous, we conclude that for all \(p\in[1, \infty)\) the \(L^p\)-error rate \((1+\alpha)/2-\) of the Euler scheme in 2 can essentially not be outperformed in general by no numerical method based on finitely many evaluations of \(W\) if \(\alpha=1\).
In the present article, we treat the case \(\alpha\in (0,1)\). We show that for all \(\alpha\in (0,1)\) and all \(p\in[1, \infty)\) the \(L^p\)-error rate \((1+\alpha)/2-\) of the Euler scheme in 2 can essentially not be outperformed in general. To be more precise, for \(\alpha\in (0,1)\) let \[C^\alpha({\mathbb{R}}) = \biggl\{ f\colon {\mathbb{R}}\to{\mathbb{R}}\,\Bigl| \quad \sup_{x\not = y}\frac{|f(x)-f(y)|}{|x-y|^{\alpha}} <\infty \biggr\}\] be the space of \(\alpha\)-Hölder continuous functions on \({\mathbb{R}}\). The first main result of this article is the following lower bound.
Theorem 1. For every \(\alpha \in (0,1)\) there exist \(c\in (0,\infty)\) and a bounded \(\mu\in C^\alpha({\mathbb{R}})\) such that for all \(n\in{\mathbb{N}}\), \[\label{Mainlb} \inf_{\substack{ t_1,\dots ,t_n \in [0,1]\\ g \colon {\mathbb{R}}^n \to {\mathbb{R}}\text{ measurable} \\ }} {\mathbb{E}}\bigl[|X_1-g(W_{t_1}, \ldots, W_{t_n})|\bigr]\geq \frac{c}{n^{(1+\alpha)/2}}.\tag{3}\]
Theorem 1 also provides an essentially matching lower bound for the upper bound recently proven in [8] for strong approximation of SDEs 1 with bounded and measurable drift coefficient \(\mu\) by the Euler scheme. Indeed, it was shown in [8] that for such SDEs the Euler scheme with equidistant steps achieves for all \(p\in [1,\infty)\) an \(L^p\)-error rate of at least \(1/2-\) in terms of the number \(n\) of evaluations of \(W\), i.e. for all \(\varepsilon\in (0,\infty)\) there exists \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), \[\label{eul2} {\mathbb{E}}\bigl[| X_1- X^E_{n,1}|^p\bigr]^{1/p} \le \frac{c}{n^{1/2-\varepsilon}}.\tag{4}\] Choosing \(\alpha=2\varepsilon\) in Theorem 1 we obtain the lower bound \(c/n^{1/2+\varepsilon}\), and thus the \(L^p\)-error rate \(1/2-\) of the Euler scheme in 4 can essentially not be outperformed in general by no numerical method based on finitely many evaluations of \(W\) for SDEs 1 with bounded and measurable drift coefficient \(\mu\). Up to now, it was only known that an \(L^p\)-error rate better than \(3/4\) can not be achieved in general for such SDEs, see [2], [4], [9].
In this article, we furthermore study the complexity of strong approximation of \(X_1\) in the case when the drift coefficient \(\mu\) of the SDE 1 has fractional Sobolev regularity. To be more precise, for \(\alpha\in (0,1)\) and \(p\in [1,\infty)\) let \[W^{\alpha,p}({\mathbb{R}}) = \biggl\{ f\colon {\mathbb{R}}\to{\mathbb{R}}\,\Bigl| \, f \text{ is measurable and }\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|f(x)-f(y)|^p}{|x-y|^{1+\alpha p}}\, dx\, dy <\infty \biggr\}\] be the space of real-valued functions \(f\) on \({\mathbb{R}}\) that have Sobolev regularity of order \(\alpha\) with integrability exponent \(p\).
In [3] it was shown that if \(\mu\) is bounded and \(\mu \in W^{\alpha,p}({\mathbb{R}})\) for some \(\alpha\in (0,1)\) and \(p\in [1,\infty)\) then the Euler scheme with equidistant steps achieves an \(L^p\)-error rate of at least \((1+\alpha)/2-\) in terms of the number \(n\) of evaluations of \(W\), i.e. for all \(\varepsilon\in (0,\infty)\) there exists \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), \[\label{eul3} {\mathbb{E}}\bigl[| X_1- X^E_{n,1}|^p\bigr]^{1/p} \le \frac{c}{n^{(1+\alpha)/2-\varepsilon}}.\tag{5}\]
Recently, in [2] an essentially matching lower bound for \(\alpha\in(1/2, 1)\) and \(p=2\) was proven. More precisely, it was shown in [2] that for every \(\alpha\in (1/2,1)\) there exist \(c\in (0,\infty)\) and a bounded, Lebesgue integrable \(\mu\in W^{\alpha,2}({\mathbb{R}})\) such that for all \(n\in{\mathbb{N}}\), \[\inf_{\substack{ t_1,\dots ,t_n \in [0,1]\\ g \colon {\mathbb{R}}^n \to {\mathbb{R}}\text{ measurable} \\ }} {\mathbb{E}}\bigl[|X_1-g(W_{t_1}, \ldots, W_{t_n})|^2\bigr]^{1/2}\geq \frac{c}{ \ln (n+1)\cdot n^{(1+\alpha)/2}}.\] Furthermore, it follows from [9] that for the SDE 1 with \(\mu=1_{[0,1 ]}\) the best possible \(L^p\)-error rate that can be achieved by any numerical method based on finitely many evaluations of \(W\) at fixed time points in \([0,1]\), is at most \(3/4\) for all \(p\in[1, \infty)\), i.e. there exists \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), \[\inf_{\substack{ t_1,\dots ,t_n \in [0,1]\\ g \colon {\mathbb{R}}^n \to {\mathbb{R}}\text{ measurable} \\ }} {\mathbb{E}}\bigl[|X_1-g(W_{t_1}, \ldots, W_{t_n})|^p\bigr]^{1/p}\geq \frac{c}{ n^{3/4}}.\] Since \(1_{[0,1]}\in W^{\alpha,2}({\mathbb{R}})\) for all \(\alpha\in(0, 1/2)\), see e.g. [10], this result yields that the \(L^p\)-error rate \((1+\alpha)/2-\) of the Euler scheme in 2 can essentially not be outperformed in general for \(\alpha=1/2-\) and \(p=2\).
However, the sharpness of 5 in the case \(\alpha<1/2\) or \(p\not=2\), remained an open question up to now. In the present article we close this gap and show that in fact for all \(\alpha\in (0,1)\) and all \(p\in[1, \infty)\) the \(L^p\)-error rate \((1+\alpha)/2-\) of the Euler scheme in 5 can essentially not be outperformed in general. More formally, the following lower bound is our second main result.
Theorem 2. For every \(\alpha \in (0,1)\), every \(p \in [1,\infty)\) and every \(\varepsilon\in (0, \infty)\) there exist \(c\in (0,\infty)\) and a bounded \(\mu\in \bigcap_{q\ge \min(p,2)}W^{\alpha,q}({\mathbb{R}})\) with \(\mu\in C^\alpha({\mathbb{R}})\) and \(\mu\in \bigcap_{q\ge 1} L^q({\mathbb{R}})\) such that for all \(n\in{\mathbb{N}}\), \[\label{Mainlb95Sobolev} \inf_{\substack{ t_1,\dots ,t_n \in [0,1]\\ g \colon {\mathbb{R}}^n \to {\mathbb{R}}\text{ measurable} \\ }} {\mathbb{E}}\bigl[|X_1-g(W_{t_1}, \ldots, W_{t_n})|^p\bigr]^{1/p}\geq \frac{c}{ (\ln (n+1))^{\gamma_p + \varepsilon}\cdot n^{(1+\alpha)/2}},\tag{6}\] where \(\gamma_p=\tfrac{2}{\min(p,2)^2}\).
The proofs of Theorem 1 and Theorem 2 are constructive. For every \(\alpha \in (0,1)\), a possible choice of the drift coefficient \(\mu\) in Theorem 1 is given by the Weierstrass function \[\label{muAlpha} \mu_\alpha(x) = \sum_{j = 1}^\infty 2^{-\alpha j} \sin(2^j x), \quad x \in {\mathbb{R}},\tag{7}\] which is known to be bounded and \(\alpha\)-Hölder continuous, but not \((\alpha+\varepsilon)\)-Hölder continuous for no \(\varepsilon\in (0,\infty)\). For every \(\alpha \in (0,1)\) and every \(p\in[1, \infty)\), a possible choice of the drift coefficient \(\mu\) in Theorem 2 is given by the Weierstrass-type function \[\label{muSob} \mu_{\alpha,\beta}(x) = 1_{[-2\pi, 4\pi]}(x) \cdot \sum_{j = 1}^\infty {j^{-\beta}} 2^{-\alpha j} \sin(2^j x), \quad x \in {\mathbb{R}},\tag{8}\] with \(\beta \in (\tfrac{1}{\min(p,2)}, \infty)\).
The rest of the article is organised as follows. In Section 2, we introduce some notation. In Section 3 we provide the construction and properties of the bi-Lipschitz transformation used to transform the SDE 1 into an SDE with zero drift coefficient and bounded Lipschitz continuous diffusion coefficient and we prove \(L^p\)-error estimates for a Milstein-type approximation of the solution of the transformed SDE. Section 4 contains preliminary estimates that are used for both the proof of Theorem 1 and the proof of Theorem 2. The proofs of Theorem 1 and Theorem 2 are then carried out in Section 5 and Section 6, respectively. In the Appendix we provide an auxiliary moment estimate and we prove the properties of the Weierstrass-type function \(\mu_{\alpha,\beta}\) stated in Lemma 10.
For a set \(A\subset{\mathbb{R}}\) and a function \(f\colon A\to{\mathbb{C}}\) we put \(\|f\|_\infty = \sup_{x\in A} |f(x)|\).
We use \(\mathbf{i}\) to denote the imaginary unit in \({\mathbb{C}}\).
For \(f\colon {\mathbb{R}}\to{\mathbb{R}}\) with \(f_{|[0,2\pi]} \in L^2([0,2\pi])\) and \(j\in{\mathbb{Z}}\) we use \[\widehat f_j = \frac{1}{\sqrt{2\pi}}\int_0^{2\pi} f(x) \exp(-\mathbf{i}j x)\, dx,\quad j\in {\mathbb{Z}},\] to denote the \(j\)-th Fourier coefficient of \(f\).
In this section we provide the construction and properties of the bi-Lipschitz transformation used to transform the SDE 1 into an SDE with zero drift coefficient. This transformation is a well-known tool to remove the drift coefficient of an SDE, see e.g. [11].
Let \(\mu\colon {\mathbb{R}}\to {\mathbb{R}}\) be locally integrable and define \[\label{trans} G_\mu\colon {\mathbb{R}}\to{\mathbb{R}}, \, \, x\mapsto \int_0^x e^{-2\int_0^y \mu(z)\, dz}\, dy.\tag{9}\]
The following lemma is a slight generalization of [2].
Lemma 1. Let \(\mu : {\mathbb{R}}\rightarrow {\mathbb{R}}\) be locally integrable such that \(\sup_{y \in {\mathbb{R}}} |\int_0^y \mu(z) \, dz| < \infty\). Then \(G_\mu\) has the following properties.
\(G_\mu\) is continuously differentiable and there exist \(c_1,c_2\in (0,\infty)\) such that \(c_1 \le G_\mu'\le c_2\).
\(G_\mu\) is a bijection, \(G_\mu^{-1}\) is continuously differentiable and \(c_2^{-1} \le (G_\mu^{-1})'\le c_1^{-1}\).
If, additionally, \(\mu\) is bounded then
Proof. By the assumptions on \(\mu\), the mapping \[T\colon {\mathbb{R}}\to{\mathbb{R}},\,\, y\mapsto \int_0^y \mu(z)\, dz\] is continuous and bounded. As a consequence, \(G_\mu\) is continuously differentiable with \[G_\mu'(x) = e^{-2T(x)},\,\, x\in {\mathbb{R}}.\] Moreover, for every \(x\in {\mathbb{R}}\), \[e^{-2\|T\|_{\infty}} \le e^{-2T(x)} \le e^{2\|T\|_\infty},\] which completes the proof of part (i). Part (ii) is an immediate consequence of part (i).
Next, assume, additionally, that \(\mu\) is bounded. Then \(T\) is Lipschitz continuous and, by the fundamental theorem of calculus, \(T\) is differentiable Lebesgue-almost everywhere with weak derivative \(\mu\). Since the mapping \(S\colon{\mathbb{R}}\to {\mathbb{R}}\), \(y\mapsto e^{-2y}\) is continuously differentiable and \(T\) is bounded, we obtain that \(G_\mu'=S\circ T\) is Lipschitz continuous with weak derivative \((S'\circ T) \cdot \mu = -2G_\mu'\cdot \mu\). This proves the statement on \(G_\mu'\) in part (iii).
By (ii) and the Lipschitz continuity of \(G_\mu'\), we obtain the Lipschitz continuity of \(G_\mu'\circ G_\mu^{-1}\). Furthermore, there exists a Borel set \(A\subset {\mathbb{R}}\) such that \(\lambda(A^c) =0\) and for every \(x\in A\), the function \(G'_\mu\) is differentiable in \(x\) with derivative \(G_\mu''(x) = -2\mu(x)\cdot G_\mu'(x)\). We conclude that for every \(x\in G_\mu(A)\), the function \(G_\mu'\circ G_\mu^{-1}\) is differentiable in \(x\) with derivative \[(G_\mu'\circ G_\mu^{-1})'(x)=G_\mu''(G_\mu^{-1}(x))\cdot (G_\mu^{-1})'(x) = \frac{-2\mu(G_\mu^{-1}(x))G_\mu'(G_\mu^{-1}(x)) }{G_\mu'(G_\mu^{-1}(x))} = -2 \mu(G_\mu^{-1}(x)).\] By (i) and (ii) we obtain that \(\lambda((G_\mu(A))^c)= \lambda(G_\mu(A^c)) = \int_{A^c} G_\mu'(x)\lambda (dx) =0\), which finishes the proof of the statement on \(G_\mu'\circ G_\mu^{-1}\) in part (iii).
By (i) we obtain that for all \(x,y\in {\mathbb{R}}\), \[\begin{align} |(G_\mu^{-1})'(x) - (G_\mu^{-1})'(y)| & = \biggl|\frac{G_\mu'(G_\mu^{-1}(y)) - G_\mu'(G_\mu^{-1}(x))}{G_\mu'(G_\mu^{-1}(x))G_\mu'(G_\mu^{-1}(y))}\biggr| \le c_1^{-2} |G_\mu'\circ G_\mu^{-1}(y) - G_\mu'\circ G_\mu^{-1}(x)|, \end{align}\] which jointly with the Lipschitz continuity of \(G_\mu'\circ G_\mu^{-1}\) yields the Lipschitz continuity of \((G_\mu^{-1})'\). Finally, for every \(x\in G_\mu(A)\), the function \((G_\mu^{-1})' = 1/(G_\mu'\circ G_\mu^{-1})\) is differentiable in \(x\) with derivative \[(G_\mu^{-1})''(x) = -\frac{ (G_\mu'\circ G_\mu^{-1})'(x)}{(G_\mu'\circ G_\mu^{-1}(x))^2} = \frac{2\mu}{(G_\mu')^2}(G_\mu^{-1}(x)),\] which completes the proof of part (iii) and hereby finishes the proof of the lemma. ◻
Applying \(G_\mu\) to the solution \(X\) of the SDE 1 yields the solution \(Y\) of an SDE with zero drift coefficient and Lipschitz continuous diffusion coefficient. The following result is a slight generalization of [2].
Lemma 2. Let \(\mu : {\mathbb{R}}\rightarrow {\mathbb{R}}\) be measurable and bounded with \(\sup_{y \in {\mathbb{R}}} |\int_0^y \mu(z) \, dz| < \infty\). Then the SDE 1 has a unique strong solution \(X\) and the stochastic process \[Y=\bigl(Y_t = G_\mu(X_t)\bigr)_{t\in[0,1]}\] is the unique strong solution of the SDE \[\label{sde1NEW} \begin{align} dY_t & = b_\mu(Y_t) \, dW_t, \quad t\in [0,1],\\ Y_0 & = G_\mu(x_0), \end{align}\tag{10}\] where \(b_\mu = G_\mu'\circ G_\mu^{-1}\) is Lipschitz continuous with weak derivative \(b_\mu' = -2\mu\circ G_\mu^{-1}\).
The proof of Lemma 2 is almost identical to the proof of [2] using Lemma 1 in place of [2]. For convenience of the reader we provide a proof of it here.
Proof. By assumption, \(\mu\) is measurable and bounded, which implies existence and uniqueness of a strong solution of 1 , see [5]. By Lemma 1(i),(iii) we may apply a generalized Itô formula, see e.g. [12], to conclude that the stochastic process \((G_\mu(X_t))_{t\in[0,1]}\) is a strong solution of the SDE 10 . The properties of \(b_\mu\) are stated in Lemma 1 (iii). As a consequence of the Lipschitz continuity of \(b_\mu\), the strong solution of the SDE 10 is unique. ◻
For \(n\in{\mathbb{N}}\) let \(Y^M_n=(Y^M_{n,t})_{t\in [0,1]}\) denote the time-continuous Milstein-type scheme with step-size \(1/n\) associated to the SDE 10 given by \(Y^M_{n, 0} =G_\mu(x_0)\) and \[\label{mil} Y^M_{n, t} = Y^M_{n, \ell/n} + b_{\mu}(Y^M_{n, \ell/n})(W_{t} - W_{\ell/n}) + \frac{1}{2} b_{\mu} b_{\mu}'(Y^M_{n, \ell/n})((W_{t} - W_{\ell/n})^2 - (t-\ell/n))\tag{11}\] for all \(t\in (\ell/n,(\ell+1)/n]\) and every \(\ell \in \{0,1,\dots,n-1\}\). Next, we provide an \(L^p\)-error estimate for \(Y^M_n\).
The following result is a slight generalization of Theorem 7.5(a) in [13].
Proposition 1. Let \(\alpha \in (0,1)\) and let \(\mu \in C^\alpha({\mathbb{R}})\) be bounded with \(\sup_{y \in {\mathbb{R}}} |\int_0^y \mu(z) \, dz| < \infty\). Then for all \(p \in [1, \infty)\) there exists \(c \in (0, \infty)\) such that for all \(n \in {\mathbb{N}}\), \[\label{milest} {\mathbb{E}}\bigl[\,\|Y - Y^M_n\|_\infty^p\bigr]^{1/p} \leq \frac{c}{ n^{(1 + \alpha) / 2}}.\qquad{(1)}\]
Proof. We proceed similar to the proof of Theorem 7.5(a) in [13]. Without loss of generality, we may assume that \(p\ge 2\). For \(t\in [0,1]\) and \(n\in {\mathbb{N}}\) put \(\underline {t}_n= \lfloor nt \rfloor/n\). Then for all \(t\in [0,1]\) and all \(n\in {\mathbb{N}}\) we have \[\label{mil1} \begin{align} Y_t - Y^M_{n,t} & = \int_0^t \bigl(b_\mu(Y_s) - b_\mu(Y^M_{n,\underline {s}_n}) - b_\mu b_\mu'(Y^M_{n,\underline {s}_n})(W_s-W_{\underline {s}_n} )\bigr) \, dW_s\\ & = \int_0^t (b_\mu(Y_s) -b_\mu(Y^M_{n,s})) \, dW_s \\ & \qquad \qquad + \int_0^t \bigl(b_\mu(Y^M_{n,s})- b_\mu(Y^M_{n,\underline {s}_n}) - b_\mu'(Y^M_{n,\underline {s}_n})(Y^M_{n, s}-Y^M_{n,\underline {s}_n}) \bigr) \, dW_s \\ & \qquad\qquad + \int_0^t \Bigl( b_\mu(b_\mu')^2(Y^M_{n,\underline {s}_n} ) \int_{\underline {s}_n}^s (W_u-W_{\underline {s}_n} ) \, dW_u \Bigr) \, dW_s. \end{align}\tag{12}\]
Using the Burkholder-Davis-Gundy inequality, the Lipschitz continuity of \(b_\mu\), see Lemma 2, and the Hölder inequality we obtain that there exist \(c_1,c_2\in (0,\infty)\) such that for all \(t\in [0,1]\) and all \(n\in {\mathbb{N}}\), \[\label{mil2} \begin{align} {\mathbb{E}}\biggl[ \,\sup_{s\in [0,t] } \biggl| \int_0^s (b_\mu(Y_s) -b_\mu(Y^M_{n,s})) \, dW_s \biggr|^p \biggr] & \le c_1 {\mathbb{E}}\biggl[ \, \biggl| \int_0^t |b_\mu(Y_s) -b_\mu(Y^M_{n,s})|^2 \, ds \biggr|^{p/2} \biggr] \\ & \le c_2 {\mathbb{E}}\biggl[ \, \int_0^t |Y_s -Y^M_{n,s}|^p \, ds \biggr] \\ & \le c_2 \int_0^t {\mathbb{E}}\biggl[\sup_{u\in [0,s] } |Y_u -Y^M_{n,u}|^p\biggr ] \, ds. \end{align}\tag{13}\]
By Lemma 2, \(b_\mu\) is absolutely continuous with weak derivative \(b_\mu'=-2\mu\circ G_\mu^{-1}\). Using the latter fact as well as \(\mu \in C^\alpha({\mathbb{R}})\) and the Lipschitz continuity of \(G_\mu^{-1}\), see Lemma 1(ii), we obtain that there exist \(c_1, c_2\in (0,\infty)\) such that for all \(x,y\in {\mathbb{R}}\) with \(x\le y\), \[\begin{align} | b_\mu (y)- b_\mu(x) - b_\mu'(x)(y-x)|& = \biggl | \int_x^y (b_\mu'(u)-b_\mu'(x))\, du \biggr| \leq c_1 \int_x^y |G_\mu^{-1}(u)-G_\mu^{-1}(x)|^\alpha\, du\\ &\le c_2 \int_x^y (u-x)^\alpha \, du \le c_2 (y-x)^{1+\alpha}. \end{align}\] Using the latter estimate, the Burkholder-Davis-Gundy inequality and the Hölder inequality we conclude that there exist \(c_1,c_2\in (0,\infty)\) such that for all \(n\in {\mathbb{N}}\), \[\label{mil4} \begin{align} & {\mathbb{E}}\biggl[ \, \sup_{s\in [0,1]} \biggl| \int_0^s \bigl(b_\mu(Y^M_{n,s})- b_\mu(Y^M_{n,\underline {s}_n}) - b_\mu'(Y^M_{n,\underline {s}_n})(Y^M_{n,s}-Y^M_{n,\underline {s}_n}) \bigr) \, dW_s \biggr|^p \biggr] \\ & \qquad \qquad \le c_1 {\mathbb{E}}\biggl[ \, \biggl| \int_0^1 \bigl|b_\mu(Y^M_{n,s})- b_\mu(Y^M_{n,\underline {s}_n}) - b_\mu'(Y^M_{n,\underline {s}_n})(Y^M_{n,s}-Y^M_{n,\underline {s}_n}) \bigr|^2 \, ds \biggr|^{p/2} \biggr] \\ & \qquad \qquad \le c_2 \int_0^1 {\mathbb{E}}\bigl[ |Y^M_{n,s} -Y^M_{n,\underline {s}_n}|^{p(1+\alpha)}\bigr] \, ds . \end{align}\tag{14}\] Since \(\mu\) is bounded and \(G_\mu'\) is bounded, see Lemma 1(i), we get that \(b_\mu\) and \(b_\mu'\) are bounded as well. Hence there exist \(c_1,c_2\in (0,\infty)\) such that for all \(s\in [0,1]\) and all \(n\in {\mathbb{N}}\), \[\label{mil5} \begin{align} & {\mathbb{E}}\bigl[ |Y^M_{n,s} -Y^M_{n,\underline {s}_n}|^{p(1+\alpha)} \bigr] \\ & \qquad = {\mathbb{E}}\bigl[ \bigl| b_{\mu}(Y^M_{n, \underline {s}_n})(W_{s} - W_{\underline {s}_n}) + \frac{1}{2} b_{\mu} b_{\mu}'(Y^M_{n, \underline {s}_n})((W_{s} - W_{\underline {s}_n})^2 - (s-\underline {s}_n)) \bigr|^{p(1+\alpha)} \bigr] \\ & \qquad \le c_1 {\mathbb{E}}\bigl[ \bigl( |W_{s} - W_{\underline {s}_n}|+(W_{s} - W_{\underline {s}_n})^2 + 1/n \bigr)^{p(1+\alpha)} \bigr] \\ & \qquad \le \frac{c_2}{ n^{p(1+\alpha)/2}}. \end{align}\tag{15}\] Combining 14 and 15 , we conclude that there exists \(c\in (0,\infty)\) such that for all \(t\in [0,1]\) and all \(n\in {\mathbb{N}}\), \[\label{mil6} {\mathbb{E}}\biggl[ \, \sup_{s\in [0,1]} \biggl| \int_0^s \bigl(b_\mu(Y^M_{n,s})- b_\mu(Y^M_{n,\underline {s}_n}) - b_\mu'(Y^M_{n,\underline {s}_n})(Y^M_{n,s}-Y^M_{n,\underline {s}_n}) \bigr) \, dW_s \biggr|^p \biggr] \le \frac{c}{ n^{p(1+\alpha)/2}}.\tag{16}\]
Finally, by the boundedness of \(b_\mu\) and \(b_\mu'\), the Burkholder-Davis-Gundy inequality and the Hölder inequality, we obtain that there exist \(c_1,c_2,c_3\in (0,\infty)\) such that for all \(t\in[0,1]\) and all \(n\in {\mathbb{N}}\), \[\label{mil7} \begin{align} & {\mathbb{E}}\biggl[ \,\sup_{t\in [0,1] } \biggl| \int_0^t \Bigl( b_\mu(b_\mu')^2(Y^M_{n,\underline {s}_n} ) \int_{\underline {s}_n}^s (W_u-W_{\underline {s}_n} ) \, dW_u \Bigr) \, dW_s \biggr|^p \biggr] \\ & \qquad \le c_1 {\mathbb{E}}\biggl[ \, \biggl| \int_0^1 b_\mu^2(b_\mu')^4(Y^M_{n,\underline {s}_n} ) \Bigl(\int_{\underline {s}_n}^s (W_u-W_{\underline {s}_n} ) \, dW_u \Bigr)^2 \, ds \biggr|^{p/2} \biggr] \\ & \qquad \le c_2 \int_0^1 {\mathbb{E}}\bigl[| (W_s-W_{\underline {s}_n} )^2 - (s-\underline {s}_n)|^p\bigr] \, ds \\ & \qquad \le \frac{c_3}{ n^{p}}. \end{align}\tag{17}\] Combining 12 with 13 , 16 and 17 we obtain that there exists \(c\in (0,\infty)\) such that for all \(t\in [0,1]\) and all \(n\in {\mathbb{N}}\), \[\label{mil8} {\mathbb{E}}\biggl[\sup_{s\in [0,t] } |Y_s -Y^M_{n,s}|^p\biggr ] \le c \int_0^t {\mathbb{E}}\biggl[\sup_{u\in [0,s] } |Y_u -Y^M_{n,u}|^p\biggr ] \, ds + \frac{c }{n^{p(1+\alpha)/2}}.\tag{18}\] Since \(b_\mu\) and \(b_\mu'\) are bounded, it is straightforward to see that \({\mathbb{E}}\bigl[\,\|Y\|_\infty^p\bigr] + {\mathbb{E}}\bigl[\,\| Y^M_n\|_\infty^p\bigr]< \infty\). Hence, ?? is a consequence of 18 and the Gronwall inequality. ◻
In the following let \((\Omega,\mathcal{A},{\mathbb{P}})\) be a probability space, let \(W\colon [0,1]\times \Omega\to {\mathbb{R}}\) be a standard Brownian motion on \([0,1]\), let \(x_0\in{\mathbb{R}}\), let \(\mu\colon {\mathbb{R}}\to{\mathbb{R}}\) be measurable and bounded and let \(X\) denote the strong solution of the corresponding SDE 1 .
For every \(p\in [1,\infty)\) and every discretization \(\pi =\{t_1,\dots,t_n\}\) of \([0,1]\) with \(0 < t_1 <\dots <t_n = 1\) we use \[e_p(\pi) = \inf_{ g \colon {\mathbb{R}}^n \to {\mathbb{R}}\text{ measurable}} {\mathbb{E}}\bigl[ |X_1-g(W_{t_1},\dots,W_{t_n} )|^p\bigr]^{1/p}\] to denote the smallest possible \(L^p\)-error that can be achieved for approximating \(X_1\) based on \(W_{t_1},\dots,W_{t_n}\). To obtain a lower bound for \(e_p(\pi)\) we construct, similar to [4] and [2], a Brownian motion \(\widetilde{W}^\pi\colon [0,1]\times \Omega\to{\mathbb{R}}\) that is coupled with \(W\) at the points \(t_1,\dots,t_n\) and we analyse the \(L^p\)-distance of the solution \(\widetilde{X}^\pi\) of the SDE 1 with driving Brownian motion \(\widetilde{W}^\pi\) and \(X\) at the final time, see Lemma 3.
Throughout the rest of the paper we put \[t_0 =0.\] Let \(\overline{W}^\pi\colon [0,1]\times \Omega\to{\mathbb{R}}\) denote the piecewise linear interpolation of \(W\) on \([0,1]\) at the points \(t_0, \ldots, t_{n}\), i.e. \[\overline{W}^\pi_t=\tfrac{t-t_{i-1}}{t_i-t_{i-1}}\,W_{t_i}+\tfrac{t_i-t}{t_i-t_{i-1}}\, W_{t_{i-1}}, \quad t\in [t_{i-1}, t_i],\] for \(i\in\{1, \ldots, n\}\), and put \[B^\pi=W-\overline{W}^\pi.\] It is well known that \((B^\pi_t)_{t\in [t_{i-1}, t_i]}\) is a Brownian bridge on \([t_{i-1}, t_i]\) for every \(i\in\{1, \ldots, n\}\) and that the stochastic processes \((B^\pi_t)_{t\in [t_{0}, t_1]}, \ldots,(B^\pi_t)_{t\in [t_{n-1}, t_{n}]}, \overline{W}^\pi\) are independent. Without loss of generality, we may assume that \((\Omega,\mathcal{A},{\mathbb{P}})\) is rich enough to carry for every \(i\in\{1, \ldots, n\}\) a Brownian bridge \((\widetilde{B}^\pi_t)_{t\in [t_{i-1}, t_i]}\) on \([t_{i-1}, t_i]\) such that \((\widetilde{B}^\pi_t)_{t\in [t_{0}, t_1]}, \ldots,(\widetilde{B}^\pi_t)_{t\in [t_{n-1}, t_{n}]}, W\) are independent. Put \(\widetilde{B}^\pi=(\widetilde{B}^\pi_t)_{t\in[0,1]}\) and define a Brownian motion \(\widetilde{W}^\pi\colon [0,1]\times \Omega\to{\mathbb{R}}\) by \[\widetilde{W}^\pi= \overline{W}^\pi+\widetilde{B}^\pi.\] We use \[\label{extra1} \widetilde{X}^\pi = (\widetilde{X}_t^\pi )_{t\in[0,1]}\tag{19}\] to denote the strong solution of the SDE 1 with driving Brownian motion \(\widetilde{W}^\pi\) in place of \(W\) and \[\mathcal{F}^{W,\widetilde{W}^\pi} = \bigl(\mathcal{F}^{W,\widetilde{W}^\pi}_t = \sigma(\{(W_s,\widetilde{W}^\pi_s)\colon s\in [0,t]\})\bigr)_{t\in [0,1]}\] to denote the filtration generated by the process \((W,\widetilde{W}^\pi)\).
Clearly, \[\label{qx1} \forall i\in\{1,\dots,n\}\colon \, W_{t_i} = \widetilde{W}^\pi_{t_i}\tag{20}\] and it is easy to check that \[\label{qx2} \forall i\in\{1,\dots,n\}\colon \, \mathcal{F}^{W,\widetilde{W}^\pi}_{t_i} \text{ and } \sigma\bigl(\{(W_t-W_{t_i},\widetilde{W}^\pi_t-\widetilde{W}^\pi_{t_i})\colon t\in[t_i,1]\}\bigr)\text{ are independent}.\tag{21}\]
For all \(n\in{\mathbb{N}}\) we put \[\Pi^n = \bigl\{\{t_1,\dots,t_n\}\colon 0<t_1< \dots <t_n=1\bigr\}\] and we define \[\Pi = \bigcup_{n\in{\mathbb{N}}} \Pi^n.\]
Lemma 3. Let \(\mu\colon{\mathbb{R}}\to {\mathbb{R}}\) be measurable and bounded. Then, for every \(p\in[1, \infty)\) and every \(\pi\in\Pi\), \[e_p(\pi) \geq \frac{1}{2}\, {\mathbb{E}}[|X_1-\widetilde{X}^\pi_1|^p]^{1/p}.\]
Proof. The proof is similar to the proof of [4]. ◻
Next, we use the transformation \(G_\mu\), see 9 , to switch from a solution of the SDE 1 to a solution of the SDE 10 , see Lemma 2.
Let \[\label{extra2} Y=(G_\mu(X_t))_{t\in[0,1]}\tag{22}\] and for every \(\pi\in \Pi\) define \[\label{extra3} \widetilde{Y}^\pi=(G_\mu(\widetilde{X}^\pi_t))_{t\in[0,1]}.\tag{23}\]
Lemma 4. Let \(\mu\colon{\mathbb{R}}\to {\mathbb{R}}\) be measurable and bounded with \(\sup_{y \in {\mathbb{R}}} |\int_0^y \mu(z) \, dz| < \infty\). Then, for every \(p\in[1,\infty)\) there exists \(c\in (0,\infty)\) such that for all \(\pi \in \Pi\), \[\label{gL2} {\mathbb{E}}\bigl[|X_1-\widetilde{X}^\pi_1|^p\bigr]^{1/p} \geq c\, {\mathbb{E}}[|Y_1-\widetilde{Y}^\pi_1|^p]^{1/p}.\tag{24}\]
Proof. By Lemma 1(i), the transformation \(G_\mu\) is Lipschitz continuous, which obviously implies the claimed estimate. ◻
For every \(n\in{\mathbb{N}}\) we define \[\widetilde{\Pi}^n = \bigl\{ \{t_1,\dots, t_{5n}\}\colon 0<t_1 <\dots < t_{5n}=1, \{j/(4n)\colon j\in\{1,\dots, 4n\}\} \subset \{t_1,\dots,t_{5n} \}\bigr\}.\] Clearly, every \(\pi\in \widetilde{\Pi}^n\) satisfies \[\label{ndisc1} \max_{i\in \{1,\dots,5n\}} (t_i -t_{i-1}) \le 1/(4n).\tag{25}\] Moreover, it is easy to check that \[\label{ndisc2} \forall \pi \in \Pi^n \, \exists \widetilde{\pi}\in \widetilde{\Pi}^n\colon \, \pi \subset \widetilde{\pi}\tag{26}\] and \[\label{ndisc3} \forall \pi \in \widetilde{\Pi}^n\colon \, \#\bigl\{ i\in \{2,\dots, 5n\}\colon t_{i-1}\ge 1/2\text{ and } t_i-t_{i-1} = 1/(4n) \} \ge n.\tag{27}\]
Lemma 5. Let \(\alpha\in (0,1)\) and let \(\mu \in C^\alpha({\mathbb{R}})\) be bounded with \(\sup_{t \in {\mathbb{R}}} |\int_0^t \mu(z) \, dz| < \infty\). Then there exist \(c_1,c_2\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), all \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\) and all \(i\in \{1,\dots,5n\}\), \[\label{iter1} \begin{align} {\mathbb{E}}\bigl[|Y_{t_i} - \widetilde{Y}^\pi_{t_i}|^2\bigr] & \ge \left(1-\frac{c_1}{n}\right)\, {\mathbb{E}}\bigl[|Y_{t_{i-1}} - \widetilde{Y}^\pi_{t_{i-1}}|^2\bigr] \\ & \qquad\qquad + c_2\, {\mathbb{E}}\bigl[|(X_{t_i}- X_{t_{i-1}}) - (\widetilde{X}^\pi_{t_i}- \widetilde{X}^\pi_{t_{i-1}})|^2\bigr] - \frac{c_1}{n^{2+2\alpha}}. \end{align}\tag{28}\]
Proof. The proof of Lemma 5 is similar to the proof of Lemma 7 in [2]. Let \(n\in{\mathbb{N}}\) and \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\) and put \[\Delta_{i} = {\mathbb{E}}\bigl[|Y_{t_i} - \widetilde{Y}^\pi_{t_i}|^2\bigr]^{1/2}\] for \(i\in\{0,\dots,5n\}\). Proceeding exactly as in the proof of [2], using Lemma 1 in place of [2] and Lemma 2 in place of [2], we obtain that there exist \(c_1,c_2 \in (0,\infty)\) that neither depend on \(n\) nor on \(\pi\) such that for every \(i\in \{1,\dots,5n\}\), \[\label{f31} \Delta_i^2 \ge \left(1-\frac{c_1}{n}\right)\, \Delta_{i-1}^2 + c_2\, {\mathbb{E}}\bigl[|(X_{t_i}- X_{t_{i-1}}) - (\widetilde{X}^\pi_{t_i}- \widetilde{X}^\pi_{t_{i-1}})|^2\bigr] -2{\mathbb{E}}[\beta_i^2],\tag{29}\] where \(\beta_i\) is a random variable that satisfies \[\label{q3} \beta_i^2 \le \|\mu\|_\infty^2\, \|G_\mu'\|_\infty^2 \,\bigl( |X_{t_i}-\widetilde{X}^\pi_{t_i} |^2 + |X_{t_{i-1}}-\widetilde{X}^\pi_{t_{i-1}}|^2\bigr)^2,\tag{30}\] see [2].
Let \(Y^M_{4n}\) and \(\widetilde{Y}^{\pi,M}_{4n}\) denote the continuous-time Milstein-type schemes with \(4n\) equidistant steps for \(Y\) and \(\widetilde{Y}^\pi\), respectively, see 11 . Observe that by 20 we have \(Y^M_{4n,t} = \widetilde{Y}^{\pi,M}_{4n,t}\) for every \(t\in \{j/(4n)\colon j\in\{0,\dots, 4n\}\}\). For \(i\in \{0,\dots,5n\}\) put \(t_i^*=\lfloor t_i 4n\rfloor/(4n)\). Then, for every \(i\in \{0,\dots,5n\}\), \[\begin{align} X_{t_i}-\widetilde{X}^\pi_{t_i} & = (X_{t_i^*} - \widetilde{X}^\pi_{t_i^*}) + (X_{t_i} - X_{t_i^*} ) - (\widetilde{X}^\pi_{t_i} - \widetilde{X}^\pi_{t_i^*} ) \\ & =(G_\mu^{-1}(Y_{t_i^*}) -G_\mu^{-1}(Y^M_{4n,t_i^*})) + (G_\mu^{-1}(\widetilde{Y}^{\pi,M}_{4n,t_i^*})-G_\mu^{-1}(\widetilde{Y}^\pi_{t_i^*})) + \int_{t_i^*}^{t_i} ( \mu(X_s) - \mu(\widetilde{X}^\pi_s))\, ds, \end{align}\] which implies \[\label{rst44} |X_{t_i}-\widetilde{X}^\pi_{t_i}| \le \|(G_\mu^{-1})'\|_\infty\, \bigl( \|Y -Y^M_{4n}\|_\infty + \|\widetilde{Y}^\pi-\widetilde{Y}^{\pi,M}_{4n}\|_\infty\bigr) + \frac{\|\mu\|_\infty}{2n}.\tag{31}\]
By Proposition 1 we may thus conclude that for every \(p\in [1,\infty)\) there exists \(c \in (0,\infty)\) that neither depends on \(n\) nor on \(\pi\) such that for every \(i\in \{0,\dots,5n\}\), \[\label{rst444} {\mathbb{E}}\bigl[ |X_{t_i}-\widetilde{X}^\pi_{t_i}|^p\bigr] \le \frac{c}{n^{p(1+\alpha)/2}}.\tag{32}\] Combining this estimate for \(p=4\) with the bound 30 we obtain that there exists \(c \in (0,\infty)\) that neither depends on \(n\) nor on \(\pi\) such that for every \(i\in \{0,\dots,5n\}\), \[\label{q4} {\mathbb{E}}\bigl[\beta_i^2\bigr] \le \frac{c}{n^{2+2\alpha}}.\tag{33}\] Combining 29 with 33 completes the proof of the lemma. ◻
Lemma 6. Let \(\alpha \in (0,1)\) and let \(\mu \in C^\alpha({\mathbb{R}})\) be bounded with \(\sup_{t \in {\mathbb{R}}} |\int_0^t \mu(z) \, dz| < \infty\). Then there exists \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), all \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\) and all \(i\in \{1,\dots,5n\}\), \[\label{iter2a} \begin{align} & {\mathbb{E}}\bigl[|(X_{t_i}- X_{t_{i-1}}) - (\widetilde{X}^{\pi}_{t_i}- \widetilde{X}^{\pi}_{t_{i-1}})|^2\bigr] \\ & \qquad \ge \frac{1}{2} {\mathbb{E}}\Bigl[ \Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu (X_{t_{i-1} } +W_t - W_{t_{i-1} } ) - \mu (X_{t_{i-1}} +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt\Bigr|^2\Bigr] -\frac{c}{ n^{2+\alpha(1+\alpha)}}. \end{align}\tag{34}\]
Proof. Let \(n\in{\mathbb{N}}\) and \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\). Let \(i \in \{1, \dots, n\}\). Throughout this proof we use \(c\in (0,\infty)\) to denote a positive constant that neither depends on \(n\) nor on \(\pi\) nor on \(i\). The value of \(c\) may vary from occurence to occurence.
We have \[\begin{align} &(X_{t_i}- X_{t_{i-1}}) - (\widetilde{X}^{\pi}_{t_i}- \widetilde{X}^{\pi}_{t_{i-1}}) \\ & \qquad = \int_{t_{i-1}}^{t_i} \bigl (\mu (X_{t_{i-1} } +W_t - W_{t_{i-1} } ) - \mu (X_{t_{i-1}} +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \\ & \qquad \qquad + \int_{t_{i-1}}^{t_i} \bigl (\mu (X_t) - \mu (X_{t_{i-1}} + W_t - W_{t_{i-1}}) \bigr)\, dt\\ & \qquad \qquad \qquad + \int_{t_{i-1}}^{t_i} \bigl (\mu (X_{t_{i-1} } + \widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1} } ) - \mu (\widetilde{X}^{\pi}_t) \bigr)\, dt \end{align}\] and therefore \[\label{eq7951NEW} \begin{align} & {\mathbb{E}}\bigl[|(X_{t_i}- X_{t_{i-1}}) - (\widetilde{X}^{\pi}_{t_i}- \widetilde{X}^{\pi}_{t_{i-1}})|^2\bigr] \\ & \qquad \ge \frac{1}{2} {\mathbb{E}}\Bigl[ \Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu (X_{t_{i-1} } +W_t - W_{t_{i-1} } ) - \mu (X_{t_{i-1}} +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt\Bigr|^2\Bigr] \\ & \qquad \qquad - 2\Bigl({\mathbb{E}}\Bigl[ \Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu (X_t) - \mu (X_{t_{i-1}} + W_t - W_{t_{i-1}}) \bigr)\, dt\Bigr|^2\Bigr] \\ & \qquad \qquad \qquad \qquad + {\mathbb{E}}\Bigl[ \Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu (X_{t_{i-1} } + \widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1} } ) - \mu (\widetilde{X}^{\pi}_t) \bigr)\, dt\Bigr|^2\Bigr] \Bigr). \end{align}\tag{35}\] Employing the \(\alpha\)-Hölder continuity and boundedness of \(\mu\) as well as 25 we get \[\label{eq7952NEW} \begin{align} &{\mathbb{E}}\Bigl[ \Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu (X_t) - \mu (X_{t_{i-1}} + W_t - W_{t_{i-1}}) \bigr)\, dt\Bigr|^2\Bigr] \\ & \qquad \le c\, {\mathbb{E}}\Bigl[\Bigl( \int_{t_{i-1}}^{t_i} \Bigl|\int_{t_{i-1}}^t \mu(X_u) \, du\Bigr|^{\alpha}\, dt \Bigr)^2\Bigr] \le c\, \Bigl(\int_{t_{i-1}}^{t_i} (t-t_{i-1})^\alpha\, dt\Bigr)^2 \le \frac{c}{ n^{2 + 2\alpha}}. \end{align}\tag{36}\] Similarly, by the \(\alpha\)-Hölder continuity and boundedness of \(\mu\) as well as 25 and 32 we obtain \[\label{eq7953NEW} \begin{align} &{\mathbb{E}}\Bigl[ \Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu (X_{t_{i-1} } + \widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1} } ) - \mu (\widetilde{X}^{\pi}_t) \bigr)\, dt\Bigr|^2\Bigr] \\ & \qquad \le c\, {\mathbb{E}}\Bigl[ \Bigl( \int_{t_{i-1}}^{t_i} \Bigl | X_{t_{i-1}} - \widetilde{X}^{\pi}_{t_{i-1}} - \int_{t_{i-1}}^t \mu(\widetilde{X}^{\pi}_u) \, du\Bigr|^\alpha \, dt\Bigr)^2\Bigr] \\ & \qquad \le c\, {\mathbb{E}}\Bigl[ \Bigl( \int_{t_{i-1}}^{t_i} \bigl( |X_{t_{i-1}} - \widetilde{X}^{\pi}_{t_{i-1}}|^{\alpha} + (t-t_{i-1})^{\alpha}\bigr) \, dt \Bigr)^2\Bigr]\\ & \qquad \le \frac{c}{n^2}\Bigl({\mathbb{E}}\bigl[|X_{t_{i-1}} - \widetilde{X}^{\pi}_{t_{i-1}}|^{2\alpha}\bigr] + \frac{1}{n^{2\alpha}}\Bigr) \\ & \qquad \le \frac{c}{n^2}\Bigl( \frac{1}{n^{(1+\alpha) \alpha}}+ \frac{1}{n^{2\alpha}}\Bigr). \end{align}\tag{37}\] Combining 35 with 36 and 37 and using the fact that \(\alpha^2 \le \alpha\) completes the proof. ◻
We proceed with providing a lower bound for the first term on the right-hand side in 34 .
Lemma 7. Let \(\mu\colon{\mathbb{R}}\to {\mathbb{R}}\) be measurable and bounded and let \(f\colon{\mathbb{R}}\rightarrow {\mathbb{R}}\) be measurable, \(2\pi\)-periodic and bounded. Then there exists \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), all \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\) and all \(i\in \{1,\dots,5n\}\) with \(t_{i-1} \ge 1/2\), \[\begin{align} & {\mathbb{E}}\Bigl[\Bigl| \int_{t_{i-1}}^{t_i} \bigl (f (X_{t_{i-1} } +W_t - W_{t_{i-1} } ) - f (X_{t_{i-1}} +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \Bigr|^2\Big] \\ & \qquad\qquad \qquad\qquad \ge c (t_i - t_{i-1})^{2} \sum_{j \in {\mathbb{Z}}} |\widehat f_j|^2 A((t_i - t_{i-1})j^2), \end{align}\] where \[\label{Axx} A(x) = \int_0^1\int_t^1 \exp(-x(u-t)/2) (1- \exp(- xt(1-u))) \, du \,dt\tag{38}\] for \(x\in{\mathbb{R}}\).
Proof. Let \(n\in{\mathbb{N}}\), let \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\) and let \(i \in \{1, \dots, 5n\}\) with \(t_{i-1} \ge 1/2\).
Since \(\mu\) is measurable and bounded, we may apply [14] to obtain that, for every \(t\in (0,1]\), the distribution of \(X_{t}\) has a Lebesgue density \(p_{t}\) and that there exists \(c\in (0,\infty)\) such that \[\inf_{t\in [1/2,1]} \inf_{x\in [0,2\pi]} p_{t}(x) \ge c.\] Hence, by 21 \[\label{rsv1NEW} \begin{align} & {\mathbb{E}}\Bigl[\Bigl| \int_{t_{i-1}}^{t_i} \bigl (f (X_{t_{i-1} } +W_t - W_{t_{i-1} } ) - f (X_{t_{i-1}} +\widetilde{W}^\pi_t - \widetilde{W}^\pi_{t_{i-1}}) \bigr)\, dt \Bigr|^2\Big] \\ & \qquad\qquad = {\mathbb{E}}\Bigl[\int_{{\mathbb{R}}} \Bigl| \int_{t_{i-1}}^{t_i} \bigl (f (x +W_t - W_{t_{i-1} } ) - f (x +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \Bigr|^2 p_{t_{i-1}}(x)\, dx \Big]\\ & \qquad \qquad \ge c\, {\mathbb{E}}\Bigl[\int_{0}^{2\pi} \Bigl| \int_{t_{i-1}}^{t_i} \bigl (f (x +W_t - W_{t_{i-1} } ) - f (x +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \Bigr|^2dx\Big]. \end{align}\tag{39}\]
Define \(g\colon \Omega\times {\mathbb{R}}\to {\mathbb{R}}\) by \[g(\omega,x) = \int_{t_{i-1}}^{t_i} \bigl (f (x +W_t(\omega) - W_{t_{i-1} }(\omega) ) - f (x +\widetilde{W}^{\pi}_t(\omega) - \widetilde{W}^{\pi}_{t_{i-1}}(\omega)) \bigr)\, dt\] for \((\omega,x)\in \Omega\times {\mathbb{R}}\). Since \(f\) is measurable and bounded we obtain that for every \(\omega\in\Omega\), the function \(g(\omega,\cdot)\colon {\mathbb{R}}\to{\mathbb{R}}\) is measurable and bounded as well. Moreover, using the \(2\pi\)-periodicity of \(f\), it is easy to see that for every \(\omega\in\Omega\) and every \(j\in{\mathbb{Z}}\), the \(j\)-th Fourier coefficient \(\widehat g_j(\omega)\) of \(g(\omega,\cdot)\) satisfies \[\label{cvbNEW} \widehat g_j(\omega) = \widehat f_j \int_{t_{i-1}}^{t_i} \bigl (\exp(-\mathbf{i}j (W_t(\omega) - W_{t_{i-1}}(\omega) )) - \exp(-\mathbf{i}j (\widetilde{W}^{\pi}_t (\omega) - \widetilde{W}^{\pi}_{t_{i-1}}(\omega) )) \bigr) \, dt.\tag{40}\] By Parseval’s theorem we obtain for every \(\omega\in \Omega\), \[\label{vc23NEW} \begin{align} &\int_{0}^{2\pi}\Bigl[\Bigl| \int_{t_{i-1}}^{t_i} \bigl (f(x +W_t (\omega)- W_{t_{i-1} }(\omega) ) - f (x +\widetilde{W}^{\pi}_t (\omega)- \widetilde{W}^{\pi}_{t_{i-1}}(\omega)) \bigr)\, dt \Bigr|^2\Big] dx \\ & \qquad \qquad = \sum_{j \in {\mathbb{Z}}} |\widehat g_j(\omega)|^2. \end{align}\tag{41}\]
Let \(j \in {\mathbb{Z}}\). Since \(\overline{W}^\pi, B^\pi, \widetilde{B}^\pi\) are independent we have \({\mathbb{P}}^{(W,\widetilde{W}^\pi)} = {\mathbb{P}}^{(\widetilde{W}^\pi,W)}= {\mathbb{P}}^{(-\widetilde{W}^\pi,-W)}\). Therefore, by 40
\[\begin{align} {\mathbb{E}}\bigl[|\widehat g_j|^2\bigr] & =4 |\widehat f_j|^2 \int_{t_{i-1}}^{t_i} \int_{t}^{t_i} {\mathbb{E}}[ \exp(-\mathbf{i}j (W_t - W_{t_{i-1}})) \overline{\exp(-\mathbf{i}j (W_u - W_{t_{i-1}}))}] \\ & \qquad \qquad \qquad \qquad \qquad - {\mathbb{E}}[ \exp(-\mathbf{i}j (W_t - W_{t_{i-1}})) \overline{\exp(-\mathbf{i}j(\widetilde{W}^\pi_u - \widetilde{W}^\pi_{t_{i-1}}))}] \, du \, dt \\ & = 4|\widehat f_j|^2 \int_{t_{i-1}}^{t_i} \int_{t}^{t_i} \bigl({\mathbb{E}}[\exp(\mathbf{i}j (W_t - W_u))] - {\mathbb{E}}[\exp(\mathbf{i}j (W_t - \widetilde{W}_u^\pi))]\bigr) \, du \, dt. \end{align}\] Let \(t_{i-1} \le t \le u \le t_i\). Then \(W_t - W_u \sim N(0, u-t)\). Moreover, \[W_t - \widetilde{W}^\pi_u = \frac{t - u}{t_i - t_{i-1}}(W_{t_i} - W_{t_{i-1}}) + B^\pi_t - \widetilde{B}^\pi_u,\] which yields \(W_t - \widetilde{W}^\pi_u\sim N(0, \sigma^2)\) with \[\sigma^2 = \frac{(t-u)^2}{t_i - t_{i-1}} + \frac{(t- t_{i-1})(t_i - t)}{t_i - t_{i-1}} + \frac{(u- t_{i-1})(t_i - u)}{t_i - t_{i-1}} = (u-t) + 2\frac{(t - t_{i-1})(t_i-u)}{t_i - t_{i-1}}.\] Hence, \[\label{bb33NEW} \begin{align} {\mathbb{E}}\bigl[ |\widehat g_j|^2\bigr] & = 4|\widehat f_j|^2 \int_{t_{i-1}}^{t_i} \int_{t}^{t_i} \exp \bigl(-j^2 (u-t)/2\bigr) \cdot \Bigl(1- \exp\Bigl(- j^2 \frac{(t - t_{i-1})(t_i-u)}{t_i - t_{i-1}}\Bigr)\Big) \,du \, dt \\ & = 4|\widehat f_j|^2 (t_i - t_{i-1})^2 \\ &\qquad \cdot \int_0^1\int_t^1 \exp(-j^2 (t_i - t_{i-1})(u-t)/2) \bigl(1- \exp(- j^2 (t_i - t_{i-1}) t(1-u))\bigr) \,du \, dt\\ & = 4|\widehat f_j|^2 (t_i - t_{i-1})^2 A((t_i - t_{i-1})j^2). \end{align}.\tag{42}\] Combining 39 with 41 and 42 completes the proof of the lemma. ◻
We first state properties of the Weierstrass function \(\mu_\alpha\) that are crucial for the proof of Theorem 1.
Lemma 8. Let \(\alpha\in (0,1)\). The function \(\mu_\alpha\) is bounded, \(2\pi\)-periodic with \(\int_0^{2\pi} \mu_\alpha(x)\, dx =0\) and satisfies \(\mu_\alpha \in C^\alpha({\mathbb{R}})\).
Proof. It is straightforward to check that \(\mu_\alpha\) is bounded, \(2\pi\)-periodic and \(\int_0^{2\pi} \mu_\alpha(x)\, dx =0\). For the proof of \(\mu_\alpha \in C^\alpha({\mathbb{R}})\) see e.g. [15]. ◻
Lemma 9. Let \(\alpha \in (0,1)\) and let \(\mu = \mu_\alpha\). Then there exists \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), all \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\) and all \(i\in \{1,\dots,5n\}\) with \(t_{i-1} \ge 1/2\), \[{\mathbb{E}}\Bigl[\Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu_\alpha (X_{t_{i-1} } +W_t - W_{t_{i-1} } ) - \mu_\alpha (X_{t_{i-1}} +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \Bigr|^2\Big] \ge c\, (t_i - t_{i-1})^{2+\alpha}.\]
Proof. Let \(n\in{\mathbb{N}}\), let \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\) and let \(i \in \{1, \dots 5n\}\) with \(t_{i-1} \ge 1/2\). Throughout this proof \(c\in (0,\infty)\) denotes a positive constant, which neither depends on \(n\) nor on \(\pi\) nor on \(i\) and may change its value from line to line.
By Lemma 8, the function \(\mu_\alpha\) is measurable, bounded and \(2\pi\)-periodic. Moreover, by the fact that \(\sin(z) = (\exp(\mathbf{i}z) - \exp(-\mathbf{i}z) )/(2\mathbf{i})\) for all \(z\in {\mathbb{R}}\), we obtain that for all \(x \in {\mathbb{R}}\), \[\label{similar} \begin{align} \mu_\alpha(x) = \sum_{j = 1}^\infty 2^{-\alpha j} \frac{\exp(\mathbf{i}2^j x) - \exp(-\mathbf{i}2^j x)}{2 \mathbf{i}} = \sum_{j\in {\mathbb{Z}}\setminus \{0\}} \frac{\operatorname{sgn}(j) 2^{- \alpha |j|} }{2 \mathbf{i}}\exp\bigl(\mathbf{i}\operatorname{sgn}(j) 2^{|j|}x\bigr). \end{align}\tag{43}\] We may thus apply Lemma 7 with \(\mu = \mu_\alpha\) and \(f = \mu_\alpha\) to obtain \[\begin{align} & {\mathbb{E}}\Bigl[\Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu_\alpha (X_{t_{i-1} } +W_t - W_{t_{i-1} } ) - \mu_\alpha (X_{t_{i-1}} +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \Bigr|^2\Big] \\ & \qquad \qquad \qquad \ge c (t_i - t_{i-1})^{2} \sum_{j \in {\mathbb{Z}}\setminus \{0\}} \frac{2^{-2\alpha |j| }}{4} A((t_i - t_{i-1}) 2^{2|j|}) \\ & \qquad \qquad \qquad \ge c (t_i - t_{i-1})^{2} 2^{-2\alpha j^* } A((t_i - t_{i-1}) 2^{2j^*}), \end{align}\] where \(j^\ast = \lceil -\log_2(\sqrt{t_i - t_{i-1}}) \rceil\). Clearly, we have \(1/(t_i-t_{i-1}) \le 2^{2j^*} \le 4/(t_i-t_{i-1})\) and therefore, \[\label{similar2} \begin{align} & 2^{- 2\alpha j^\ast} A((t_i - t_{i-1}) 2^{2j^\ast}) \\ &\qquad \ge 2^{-2\alpha} (t_i - t_{i-1})^{\alpha} \int_0^1\int_t^1 \exp(-2 (u-t)) (1- \exp(- t(1-u))) \,du \, dt \\ & \qquad \ge c (t_i - t_{i-1})^{\alpha}, \end{align}\tag{44}\] which finishes the proof of the lemma. ◻
We are ready to proceed with the proof of Theorem 1.
Proof of Theorem 1. Let \(\alpha\in (0,1)\). By Lemma 8 we have that \(\mu_\alpha\) is bounded, \(2\pi\)-periodic and satisfies \(\mu_\alpha \in C^\alpha({\mathbb{R}})\). Moreover, since \(\int_0^{2\pi}\mu_\alpha(x)\, dx=0\), we obtain \[\sup_{y\in{\mathbb{R}}}\Bigl|\int_0^y \mu_\alpha (z) \, dz \Bigr| \le 2\pi \|\mu_\alpha\|_\infty < \infty.\] We may thus apply Lemma 5, Lemma 6 and Lemma 9 to obtain that there exist \(c_1,c_2, c_3\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), all \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\) and all \(i\in \{1,\dots,5n\}\) with \(t_{i-1}\ge 1/2\),
\[\label{iter2NEW} {\mathbb{E}}\bigl[|Y_{t_i} - \widetilde{Y}^\pi_{t_i}|^2\bigr] \ge \left(1-\frac{c_1}{n}\right)\, {\mathbb{E}}\bigl[|Y_{t_{i-1}} - \widetilde{Y}^\pi_{t_{i-1}}|^2\bigr] + c_2\,(t_i-t_{i-1})^{2+\alpha} - \frac{c_3}{n^{2+\alpha(1+\alpha)}}.\tag{45}\]
Let \(n\in{\mathbb{N}}\) with \(n>c_1\) and let \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\). Choose the unique \(r(\pi) \in \{1, \dots, 5n\}\) with \(t_{r(\pi)} = \frac{1}{2}\). Iteratively applying 45 for \(i=5n, \ldots, r(\pi)+1\) we obtain \[\begin{align} &{\mathbb{E}}\bigl[|Y_{1} - \widetilde{Y}^{\pi}_{1}|^2\bigr] \ge \left(1- \frac{c_1}{n}\right)^{5n-r(\pi)} {\mathbb{E}}\bigl[|Y_{t_{r(\pi)}} - \widetilde{Y}^{\pi}_{t_{r(\pi)}}|^2\bigr] \\ & \qquad\qquad \qquad\qquad+ c_2 \sum_{i=r(\pi)+1}^n\left(1- \frac{c_1}{n}\right)^{n-i} (t_i - t_{i-1})^{2 + \alpha} - (5n-r(\pi))\cdot \frac{c_3}{n^{2+\alpha(1+\alpha)}} \end{align}\] and hence, using 27 , \[\label{thm2952NEW} \begin{align} {\mathbb{E}}\bigl[|Y_{1} - \widetilde{Y}^{\pi}_{1}|^2\bigr]&\ge c_2 \left(1 - \frac{c_1}{n}\right)^n \sum_{i=r(\pi)+1}^n(t_i - t_{i-1})^{2 + \alpha} -5n\cdot \frac{c_3}{n^{2+\alpha(1+\alpha)}} \\ & \ge c_2 \left(1 - \frac{c_1}{n}\right)^n \frac{1}{4^{2+\alpha} n^{1+\alpha}} -5n\cdot \frac{c_3}{n^{2+\alpha(1+\alpha)}}. \end{align}\tag{46}\]
Since \(\lim_{n \rightarrow \infty} (1 - \frac{c_1}{n})^n = e^{-c_1}\), we obtain by 46 that there exist \(c\in (0,\infty)\) and \(n^\ast\in{\mathbb{N}}\) such that for every \(n\in{\mathbb{N}}\) with \(n\ge n^\ast\) and all \(\pi\in\widetilde{\Pi}^n\), \[\label{aux11NEW} \begin{align} {\mathbb{E}}\bigl[|Y_{1} - \widetilde{Y}^{\pi}_{1}|^2\bigr] \ge \frac{c}{n^{1+\alpha}}. \end{align}\tag{47}\] Moreover, by Lemma 1 and 32 we obtain that for every \(p\in [1,\infty\)) there exist \(c_1,c_2\in (0,\infty)\) such that for every \(n\in{\mathbb{N}}\) and every \(\pi\in \widetilde{\Pi}^n\), \[\label{aux12NEW} \begin{align} \bigl({\mathbb{E}}\bigl[|Y_{1} - \widetilde{Y}^{\pi}_{1}|^p\bigr] \bigr)^{1/p} \le c_1 \bigl({\mathbb{E}}\bigl[|X_{1} - \widetilde{X}^{\pi}_{1}|^p\bigr] \bigr)^{1/p} \le \frac{c_2}{n^{(1+\alpha)/2}}. \end{align}\tag{48}\] By 47 and 48 we may apply Lemma 12 in the appendix with \(Z=Y_{1} - \widetilde{Y}^{\pi}_{1}\), \(p=1\) and \(r_1=r_2 =n^{-(1+\alpha)/2}\) to obtain that there exist \(c\in (0,\infty)\) and \(n^\ast\in{\mathbb{N}}\) such that for every \(n\in{\mathbb{N}}\) with \(n\ge n^\ast\) and every \(\pi\in \widetilde{\Pi}^n\), \[\label{thm2953NEW} \begin{align} {\mathbb{E}}\bigl[|Y_{1} - \widetilde{Y}^{\pi}_{1}|\bigr] \ge \frac{c}{n^{(1 + \alpha)/2}}. \end{align}\tag{49}\]
Using 26 as well as Lemma 3 and Lemma 4 we obtain that there exists \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), \[\label{end1} \begin{align} \inf_{\pi\in\Pi^n} e_1(\pi) & \ge \inf_{\pi\in\Pi^{\max(n,n^\ast)}}e_1(\pi) \\ & \ge \inf_{\pi\in\widetilde{\Pi}^{\max(n,n^\ast)}}e_1(\pi) \\ & \ge \frac{1}{2} \inf_{\pi\in\widetilde{\Pi}^{\max(n,n^\ast)}} {\mathbb{E}}\bigl[|X_1-\widetilde{X}^{\pi}_1|\bigr] \\ & \ge c \inf_{\pi\in\widetilde{\Pi}^{\max(n,n^\ast)}} {\mathbb{E}}\bigl[|Y_1-\widetilde{Y}^{\pi}_1|\bigr]. \end{align}\tag{50}\] Combining 50 with 49 completes the proof of Theorem 1. ◻
We first provide properties of the function \(\mu_{\alpha, \beta}\) that are crucial for the proof of Theorem 2.
Lemma 10. Let \(\alpha \in (0,1)\). Then for all \(\beta \in (0, \infty)\) we have
\(\mu_{\alpha, \beta}\) is bounded,
\(\mu_{\alpha, \beta}\in C^\alpha({\mathbb{R}})\),
\(\mu_{\alpha, \beta}\in \cap_{q \geq 1} L^q({\mathbb{R}})\).
Moreover, for all \(p \in [1,2]\) and all \(\beta \in (1/p, \infty)\) we have
The proof of Lemma 10 is shifted to the appendix.
Lemma 11. Let \(\alpha \in (0,1), \beta \in (0, \infty)\) and let \(\mu = \mu_{\alpha, \beta}\). Then there exist \(c_1, c_2, c_3\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), all \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\) and all \(i\in \{1,\dots, 5n\}\) with \(t_{i-1} \ge 1/2\), \[\label{iter3} \begin{align} &{\mathbb{E}}\Bigl[\Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu_{\alpha, \beta}(X_{t_{i-1} } +W_t - W_{t_{i-1} } ) - \mu_{\alpha, \beta}(X_{t_{i-1}} +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \Bigr|^2\Big] \\ & \qquad \qquad \ge \frac{c_1}{(-\log_2(t_i - t_{i-1}) + 1)^{2\beta}}(t_i - t_{i-1})^{2+\alpha} - c_2 e^{-c_3 n}. \end{align}\tag{51}\]
Proof. Let \(n\in{\mathbb{N}}\), let \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\) and let \(i\in \{1,\dots, 5n\}\) with \(t_{i-1} \ge 1/2\). Throughout this proof \(c_1,c_2,\dots\in (0,\infty)\) denote positive constants, which neither depend on \(n\) nor on \(\pi\) nor on \(i\).
Since \(\mu = \mu_{\alpha, \beta}\) is bounded, see Lemma 10, we can use [14] to derive, similar to 39 , that \[\label{rsv195Sob} \begin{align} & {\mathbb{E}}\Bigl[\Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu_{\alpha, \beta}(X_{t_{i-1} } +W_t - W_{t_{i-1} } ) - \mu_{\alpha, \beta}(X_{t_{i-1}} + \widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \Bigr|^2\Big] \\ & \qquad \qquad \ge c_1 \int_{0}^{2\pi} {\mathbb{E}}\Bigl[\Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu_{\alpha, \beta}(x +W_t - W_{t_{i-1} } ) - \mu_{\alpha, \beta}(x +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \Bigr|^2\Big] dx. \end{align}\tag{52}\]
Define \(f_{\alpha, \beta}\colon{\mathbb{R}}\to{\mathbb{R}}\) by \[\label{muSobx} f_{\alpha, \beta}(x) = \sum_{j = 1}^\infty {j^{-\beta}}2^{-\alpha j} \sin(2^j x), \qquad x \in {\mathbb{R}},\tag{53}\] and note that \(\mu_{\alpha, \beta}= 1_{[-2\pi, 4\pi]} f_{\alpha, \beta}\). Moreover, for all \(x \in [0, 2\pi]\) and all \(y,z\in [-2\pi,2\pi]\) we have \[\label{cvbn} \mu_{\alpha, \beta}(x+z) - \mu_{\alpha, \beta}(x+y) = f_{\alpha, \beta}(x+z) - f_{\alpha, \beta}(x+y).\tag{54}\]
Put \[B= \Bigl\{\,\sup_{t \in [t_{i-1}, t_i]} |W_t - W_{t_{i-1}}| \le 2\pi\Bigr\} \cap \Bigl\{\,\sup_{t \in [t_{i-1}, t_i]} |\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}| \le 2\pi\Bigr\}.\] Then, by 54 and the boundedness of \(\mu_{\alpha, \beta}\), \[\label{nccc3} \begin{align} & \int_{0}^{2\pi} {\mathbb{E}}\Bigl[\Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu_{\alpha, \beta}(x +W_t - W_{t_{i-1} } ) - \mu_{\alpha, \beta}(x +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \Bigr|^2\Big] dx \\ & \qquad \ge \int_{0}^{2\pi} {\mathbb{E}}\Bigl[1_{B}\Bigl| \int_{t_{i-1}}^{t_i} \bigl (\mu_{\alpha, \beta}(x +W_t - W_{t_{i-1} } ) - \mu_{\alpha, \beta}(x +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \Bigr|^2\Big] dx \\ & \qquad \ge \int_{0}^{2\pi} {\mathbb{E}}\Bigl[\Bigl| \int_{t_{i-1}}^{t_i} \bigl (f_{\alpha, \beta}(x +W_t - W_{t_{i-1} } ) - f_{\alpha, \beta}(x +\widetilde{W}_t - \widetilde{W}_{t_{i-1}}) \bigr)\, dt \Bigr|^2\Big] dx - c_2 {\mathbb{P}}(B^c). \end{align}\tag{55}\]
Using standard results for the Brownian motion and 25 we get \[\label{mmm3} \begin{align} {\mathbb{P}}(B^c) & \le 2{\mathbb{P}}\Bigl(\sup_{t \in [t_{i-1}, t_i]} |W_t - W_{t_{i-1}}| >2 \pi\Bigl) = 2{\mathbb{P}}\Bigl( \sup_{t \in [0,1]} |W_t| > \frac{2\pi}{\sqrt{t_i - t_{i-1}}}\Bigl) \\ & \le 4 {\mathbb{P}}(W_1 > 4\pi \sqrt{n}) \le c_3 e^{-c_4 n}. \end{align}\tag{56}\]
Similar to 43 we have for every \(x\in{\mathbb{R}}\), \[\label{represent} f_{\alpha, \beta}(x) = \sum_{j\in {\mathbb{Z}}\setminus \{0\}} \frac{\operatorname{sgn}(j) |j|^{-\beta} 2^{- \alpha |j|} }{ 2 \mathbf{i}}\exp\bigl(\mathbf{i}\operatorname{sgn}(j) 2^{|j|}x\bigr).\tag{57}\] Since \(\mu_{\alpha, \beta}\) and \(f_{\alpha, \beta}\) are measurable and bounded and \(f_{\alpha, \beta}\) is \(2\pi\)-periodic we may thus apply Lemma 7 to obtain \[\label{ccvv23} \begin{align} & {\mathbb{E}}\Bigl[\Bigl| \int_{t_{i-1}}^{t_i} \bigl (f_{\alpha, \beta}(X_{t_{i-1} } +W_t - W_{t_{i-1} } ) - f_{\alpha, \beta}(X_{t_{i-1}} +\widetilde{W}^{\pi}_t - \widetilde{W}^{\pi}_{t_{i-1}}) \bigr)\, dt \Bigr|^2\Big] \\ & \qquad\qquad \qquad\qquad \ge c_5 (t_i - t_{i-1})^{2} \sum_{j \in {\mathbb{Z}}} | \widehat{ ( f_{\alpha, \beta}) }_j|^2 A((t_i - t_{i-1})j^2)\\ & \qquad\qquad \qquad\qquad = c_5 (t_i - t_{i-1})^{2} \sum_{j \in {\mathbb{Z}}\setminus\{0\}}\frac{|j|^{-2\beta}2^{-2\alpha |j|}}{4}A((t_i - t_{i-1})2^{2|j|})\\ & \qquad\qquad \qquad\qquad \geq c_6 (t_i - t_{i-1})^{2} (j^*)^{-2\beta}2^{-2\alpha j^*}A((t_i - t_{i-1})2^{2j^*}) \end{align}\tag{58}\] with \(A\colon{\mathbb{R}}\to{\mathbb{R}}\) given by 38 and \(j^\ast = \lceil -\log_2(\sqrt{t_i - t_{i-1}}) \rceil\). We conclude as in 44 that \[\label{dft2} (j^\ast)^{-2\beta}2^{- 2\alpha j^\ast} A((t_i - t_{i-1}) 2^{2j^\ast}) \ge \frac{c_7}{(-\log_2(t_i - t_{i-1}) + 1)^{2\beta}} (t_i - t_{i-1})^{\alpha}.\tag{59}\]
Combining 52 with 55 , 56 , 58 and 59 completes the proof of the lemma. ◻
Proof of Theorem 2. Let \(\alpha\in (0,1)\) and let \(\beta\in (0,\infty)\). By Lemma 10 we have that \(\mu_{\alpha, \beta}\) is bounded and satisfies \(\mu_{\alpha, \beta}\in C^\alpha({\mathbb{R}})\cap L^1({\mathbb{R}})\). We may thus use Lemma 5, Lemma 6 and Lemma 11 as well as 25 and 27 to derive, similar to 46 and 47 that there exist \(n^\ast\in {\mathbb{N}}\) and \(c_1,\ldots ,c_6\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\) with \(n\ge n^\ast\) and all \(\pi=\{t_1,\dots,t_{5n}\}\in\widetilde{\Pi}^n\) with \(0<t_1<\dots <t_{5n}=1\),
\[\label{iter2NEWx} \begin{align} {\mathbb{E}}\bigl[|Y_{1} - \widetilde{Y}^{\pi}_{1}|^2\bigr] & \ge c_2 \left(1 - \frac{c_1}{n}\right)^n \sum_{i=r(\pi)+1}^n \frac{(t_i - t_{i-1})^{2 +\alpha}}{(-\log_2(t_i - t_{i-1}) + 1)^{2\beta}} - 5n \cdot \frac{c_3}{n^{2 + \alpha(1+\alpha)}} \\ &\ge c_4 \left(1 - \frac{c_1}{n}\right)^n \cdot \frac{1}{(\log_2(4n) + 1)^{2\beta} n^{1 + \alpha}} - \frac{c_5}{n^{1 + \alpha(1+\alpha)}}\\ & \ge \frac{c_6 }{(\ln(n + 1))^{2\beta} n^{1 + \alpha}}, \end{align}\tag{60}\] where \(r(\pi)\) is the unique index in \(\{1,\dots,5n\}\) such that \(t_{r(\pi)}= 1/2\).
Furthermore, by Lemma 1 and 32 we derive, similar to 48 , that for every \(p\in [1,\infty)\) there exists \(c\in (0,\infty)\) such that for every \(n\in{\mathbb{N}}\) and every \(\pi\in \widetilde{\Pi}^n\), \[\label{aux12NEWx} \begin{align} \bigl({\mathbb{E}}\bigl[|Y_{1} - \widetilde{Y}^{\pi}_{1}|^p\bigr] \bigr)^{1/p} \le \frac{c}{n^{(1+\alpha)/2}}. \end{align}\tag{61}\]
By 60 and 61 we may apply Lemma 12 to obtain that there exists \(n^\ast\in {\mathbb{N}}\) and for every \(p\in[1,2]\) and \(q\in (1,\infty)\) there exists \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\) with \(n\ge n^\ast\) and all \(\pi\in\widetilde{\Pi}^n\), \[\label{NEWxx} \bigl( {\mathbb{E}}\bigl[|Y_{1} - \widetilde{Y}^{\pi}_{1}|^p\bigr]\bigr)^{1/p} \ge \frac{c}{(\ln(n + 1))^{\beta\cdot 2q/p} \, n^{(1 + \alpha)/2}}.\tag{62}\]
For \(p\in [1,\infty)\) and \(\varepsilon \in (0,\infty)\) let \(p^\ast = \min(2,p)\), choose \(\widetilde{\varepsilon}\in (0,\infty)\) such that \[2\widetilde{\varepsilon}/p^\ast+ 2\widetilde{\varepsilon}/(p^\ast)^2 + 2\widetilde{\varepsilon}^2/p^\ast \le \varepsilon\] and let \(\beta = 1/p^\ast +\widetilde{\varepsilon}\) and \(q= 1+\widetilde{\varepsilon}\). Then \(\beta\in (1/p^\ast,\infty)\) and by Lemma 10 we have \(\mu_{\alpha, \beta}\in \cap_{\tilde{p} \geq p^\ast} W^{\alpha,\tilde{p}}({\mathbb{R}})\). Since \(\beta \cdot 2q/p^\ast = 2/(p^\ast)^2 + 2\widetilde{\varepsilon}/p^\ast+ 2\widetilde{\varepsilon}/(p^\ast)^2 + 2\widetilde{\varepsilon}^2/p^\ast \le 2/(p^\ast)^2 + \varepsilon\) we conclude by 62 that there exist \(n^\ast\in {\mathbb{N}}\) and \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\) with \(n\ge n^\ast\) and all \(\pi\in\widetilde{\Pi}^n\), \[\label{NEWxxx} \bigl( {\mathbb{E}}\bigl[|Y_{1} - \widetilde{Y}^{\pi}_{1}|^p\bigr]\bigr)^{1/p} \ge \frac{c}{(\ln(n+1))^{2/(p^\ast)^2+\varepsilon}\, n^{(1 + \alpha)/2}}.\tag{63}\] Next, use 26 as well as Lemma 3 and Lemma 4 to obtain that for every \(p\in [1,\infty)\) there exists \(c\in (0,\infty)\) such that for all \(n\in{\mathbb{N}}\), \[\label{end1x} \begin{align} \inf_{\pi\in\Pi^n} e_p(\pi) & \ge \inf_{\pi\in\Pi^{\max(n,n^\ast)}}e_p(\pi) \\ &\ge \inf_{\pi\in\widetilde{\Pi}^{\max(n,n^\ast)}}e_p(\pi)\\ & \ge \frac{1}{2} \inf_{\pi\in\widetilde{\Pi}^{\max(n,n^\ast)}} \bigl( {\mathbb{E}}\bigl[|X_1-\widetilde{X}^{\pi}_1|^p\bigr] \bigr)^{1/p}\\ & \ge c \inf_{\pi\in\widetilde{\Pi}^{\max(n,n^\ast)}} \bigl({\mathbb{E}}\bigl[|Y_1-\widetilde{Y}^{\pi}_1|^p\bigr] \bigr)^{1/p}. \end{align}\tag{64}\] Combining 63 with 64 completes the proof of Theorem 2. ◻
Lemma 12. Let \(c_1,r_1,r_2\in (0,\infty)\) with \(r_1\le r_2\) and let \(Z\) be a real-valued random variable such that \(({\mathbb{E}}[Z^2])^{1/2} \ge c_1 r_1\) and for every \(p\in [1,\infty)\) there exists \(c_2(p)\in (0,\infty)\) such that \(({\mathbb{E}}[|Z|^p])^{1/p} \le c_2(p) r_2\). Then, for every \(p\in [1,2]\) and every \(q\in (1,\infty)\), \[({\mathbb{E}}[|Z|^p])^{1/p} \ge \Bigl(\frac{c_1}{c_2(\gamma)}\Bigr)^{2q/p} c_2(\gamma) \Bigl(\frac{r_1}{r_2}\Bigr)^{2q/p}r_2\] with \(\gamma = (2 - p / q) \cdot q / (q - 1)\in [1,\infty)\).
Proof. Let \(p\in [1,2]\) and let \(q \in (1, \infty)\). By the Hölder inequality, \[\begin{align} c_1^2 r_1^2 & \le {\mathbb{E}}\bigl[Z^2\bigr] = {\mathbb{E}}\bigl[|Z|^{p/q}\cdot |Z|^{2-p/q}\bigr] \le \bigl({\mathbb{E}}\bigl[|Z|^p\bigr]\bigr)^{1/ q} \cdot \bigl({\mathbb{E}}\bigl[|Z|^\gamma \bigr]\bigr)^{(q - 1)/ q}\\ & \le \bigl({\mathbb{E}}\bigl[|Z|^p\bigr]\bigr)^{1/ q} \cdot (c_2(\gamma))^{2-p/q} r_2^{2-p/q} \end{align}\] and therefore \[\bigl({\mathbb{E}}\bigl[|Z|^p\bigr]\bigr)^{1/ q} \ge \Bigl(\frac{c_1}{c_2(\gamma)}\Bigr)^2c_2(\gamma)^{p/q} \Bigl(\frac{r_1}{r_2}\Bigr)^{2} r_2^{p/q},\] which completes the proof of the lemma. ◻
Proof of Lemma 10. Let \(\beta \in(0, \infty)\). Recall the definition 53 of the function \(f_{\alpha, \beta}\) and note that \[\label{l1} \mu_{\alpha, \beta}=1_{[-2\pi, 4\pi]}\cdot f_{\alpha, \beta}.\tag{65}\]
Clearly, for all \(x\in{\mathbb{R}}\), \[\label{l4} |\mu_{\alpha, \beta}(x)|\leq |f_{\alpha, \beta}(x)|\leq \sum_{j = 1}^\infty 2^{-\alpha j} <\infty,\tag{66}\] which implies (i).
We next prove (ii). First, we show that \(f_{\alpha, \beta}\in C^\alpha({\mathbb{R}})\). To this end, we proceed similarly to the proof of [15]. For \(h \in (-1,1)\setminus\{0\}\) put \(j_h=\lceil \log_2(1/|h|) \rceil\). Then there exists \(c\in(0,\infty)\) such that for all \(x \in \mathbb{R}\) and all \(h \in (-1,1)\setminus\{0\}\) we have \[\label{lemSob1951} \begin{align} |f_{\alpha, \beta}(x + h) - f_{\alpha, \beta}(x)| &= \Bigl|\sum_{j = 1}^\infty j^{-\beta} 2^{-\alpha j} \bigl(\sin(2^j (x + h)) - \sin(2^j x)\bigr)\Bigr| \\ &\le \sum_{j = 1}^{j_h} 2^{-\alpha j} \bigl|\sin(2^j (x + h)) - \sin(2^j x)\bigr| + 2 \sum_{j = j_h + 1}^\infty 2^{-\alpha j} \\ &\le \sum_{j = 1}^{j_h} 2^{(1 - \alpha)j} |h| + 2 \sum_{j = j_h + 1}^\infty 2^{-\alpha j} \\ &= 2^{1 - \alpha}|h| \cdot \frac{2^{(1 - \alpha)j_h} - 1}{2^{1-\alpha} - 1} + 2\cdot \frac{2^{-\alpha(j_h + 1)}}{1 - 2^{-\alpha}} \\ &\le 2|h|\cdot \frac{2^{(1 - \alpha)(\log_2(1/|h|) + 1)} }{2^{1-\alpha} - 1} + \frac{2}{1 - 2^{-\alpha}} \cdot 2^{-\alpha(\log_2(1/|h|) + 1)}\\ & = |h|^\alpha\Bigl( \frac{2^{2 - \alpha}}{2^{1-\alpha} - 1} + \frac{2^{1-\alpha}}{1-2^{-\alpha}} \Bigr)\\ & \le c |h|^\alpha. \end{align}\tag{67}\] Since \(f_{\alpha, \beta}\) is bounded, see 66 , we may thus conclude that \(f_{\alpha, \beta}\in C^\alpha({\mathbb{R}})\). Observe that \(f_{\alpha, \beta}(-2\pi)=f_{\alpha, \beta}(4\pi)=0\) and thus \(\mu_{\alpha, \beta}\) is continuous. Hence, \(\mu_{\alpha, \beta}\in C^\alpha({\mathbb{R}})\) by the construction of \(\mu_{\alpha, \beta}\).
Since \(\mu_{\alpha, \beta}\) is continuous and has compact support we conclude that \(\mu_{\alpha, \beta}\in L^q({\mathbb{R}})\) for all \(q\geq 1\), which shows (iii).
Finally, we prove (iv). Let \(p\in[1,2]\) and \(\beta\in(1/p, \infty)\). Below we show that \[\label{l2} \mu_{\alpha, \beta}\in W^{\alpha,p}({\mathbb{R}}).\tag{68}\] Using (ii) and 68 we obtain that for all \(q\geq p\) there exists \(c\in(0, \infty)\) such that \[\begin{align} \int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|\mu_{\alpha, \beta}(x)-\mu_{\alpha, \beta}(y)|^q}{|x-y|^{1+\alpha q}}\, dx\, dy &=\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|\mu_{\alpha, \beta}(x)-\mu_{\alpha, \beta}(y)|^p}{|x-y|^{1+\alpha p}}\cdot \Bigl(\frac{|\mu_{\alpha, \beta}(x)-\mu_{\alpha, \beta}(y)|}{|x-y|^{\alpha}}\Bigr)^{q-p}\, dx\, dy\\ &\leq c \int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|\mu_{\alpha, \beta}(x)-\mu_{\alpha, \beta}(y)|^p}{|x-y|^{1+\alpha p}}\, dx\, dy < \infty, \end{align}\] and hence \(\mu_{\alpha, \beta}\in W^{\alpha,q}({\mathbb{R}})\) for all \(q\geq p\).
For the proof of 68 we consider the function \[\nu=1_{[0, 2\pi]}\cdot f_{\alpha, \beta}.\] We show below that \[\label{l3} I =\int_{-2\pi}^{4\pi}\int_{-2\pi}^{4\pi} \frac{|\nu(x) - \nu(y)|^p}{|x-y|^{1+\alpha p}} \, dy \, dx < \infty.\tag{69}\] Applying [16] with \(\Omega=(-2\pi, 4\pi)\), \(u=\nu_{|\Omega}\) and \(K=[0, 2\pi]\) we conclude that \(\nu \in W^{\alpha,p}({\mathbb{R}})\). Hence, also \(\nu(\cdot +2\pi), \nu(\cdot -2\pi)\in W^{\alpha,p}({\mathbb{R}})\). Finally, using the fact that \(f_{\alpha, \beta}(0)=f_{\alpha, \beta}(2\pi)=0\) and the \(2\pi\)-periodicity of \(f_{\alpha, \beta}\) we obtain \[\mu_{\alpha, \beta}=1_{[-2\pi, 0]}\cdot f_{\alpha, \beta}+1_{[0, 2\pi]}\cdot f_{\alpha, \beta}+1_{[2\pi, 4\pi]}\cdot f_{\alpha, \beta}=\nu(\cdot+2\pi)+\nu+ \nu(\cdot-2\pi),\] which yields 68 .
It remains to prove 69 . Clearly, \[I=I_1+2I_2,\] where \[I_1=\int_{0}^{2\pi}\int_{0}^{2\pi} \frac{|f_{\alpha, \beta}(x) - f_{\alpha, \beta}(y)|^p}{|x-y|^{1+\alpha p}} \, dy \, dx, \quad I_2=\int_{0}^{2\pi}\int_{[-2\pi, 0] \cup [2\pi, 4\pi] } \frac{|f_{\alpha, \beta}(x)|^p}{|x-y|^{1+\alpha p}} \, dy \, dx.\]
Using the fact that \(f_{\alpha, \beta}(0)=f_{\alpha, \beta}(2\pi)=0\) and \(f_{\alpha, \beta}\in C^\alpha({\mathbb{R}})\) we obtain that there exists \(c\in(0, \infty)\) such that for all \(x \in [0, 2\pi]\), \[|f_{\alpha, \beta}(x)|=|f_{\alpha, \beta}(x) - f_{\alpha, \beta}(0)| = |f_{\alpha, \beta}(x) - f_{\alpha, \beta}(2\pi)| \le c \min(x^{\alpha}, (2\pi - x)^{\alpha}).\] Hence, there exists \(c\in(0, \infty)\) such that \[\begin{align} I_2&=\frac{1}{\alpha p}\int_0^{2\pi} |f_{\alpha, \beta}(x)|^p\cdot \Bigl(\frac{1}{x^{\alpha p}}-\frac{1}{(x+2\pi)^{ \alpha p}} + \frac{1}{(2\pi - x)^{\alpha p}}-\frac{1}{(4\pi-x)^{\alpha p}}\Bigr) \, dx \\ &\le c \int_0^{2\pi}\min(x^{\alpha p}, (2\pi - x)^{\alpha p})\cdot \Bigl(\frac{1}{x^{\alpha p}} + \frac{1}{(2\pi - x)^{\alpha p}}\Bigr) \, dx \le 4\pi c < \infty. \end{align}\]
For the proof of \(I_1 < \infty\) observe that \[\begin{align} I_1 &= \int_0^{2\pi} \int_{-x}^{2\pi - x} \frac{|f_{\alpha, \beta}(x + y) - f_{\alpha, \beta}(x)|^p}{|y|^{1+\alpha p}} \, dy \, dx \\ &\le \int_{-2\pi}^{2\pi} \frac{1}{|y|^{1+ \alpha p}} \int_0^{2\pi} |f_{\alpha, \beta}(x + y) - f_{\alpha, \beta}(x)|^p \, dx \, dy. \end{align}\] Below we show that there exist \(\delta \in (0, 1)\) and \(c \in (0, \infty)\) such that for all \(y \in (-\delta, \delta)\setminus\{0\}\), \[\label{lemSob1956} \begin{align} \int_0^{2\pi} |f_{\alpha, \beta}(x + y) - f_{\alpha, \beta}(x)|^2 \, dx \le \frac{c|y|^{2\alpha }}{(-\log_2(|y|))^{2\beta}}. \end{align}\tag{70}\] Using the latter estimate, the boundedness of \(f_{\alpha, \beta}\) and the Hölder inequality we obtain that there exist \(\delta \in (0, 1)\) and \(c_1, \ldots, c_4\in(0, \infty)\) such that \[\begin{align} I_1 &\le c_1\int_{\delta}^{2\pi} \frac{1}{|y|^{1+ \alpha p}} dy + \int_{-\delta}^{\delta} \frac{1}{|y|^{1+ \alpha p}} \int_0^{2\pi} |f_{\alpha, \beta}(x + y) - f_{\alpha, \beta}(x)|^p \, dx \, dy \\ &\le c_2 + c_3\int_{-\delta}^{\delta} \frac{1}{|y|^{1+ \alpha p}} \cdot \Bigl(\int_0^{2\pi} |f_{\alpha, \beta}(x + y) - f_{\alpha, \beta}(x)|^2 \, dx\Bigr)^{p/2} \, dy \\ &\le c_2 + c_4 \int_{-\delta}^{\delta} \frac{1}{|y| \cdot(-\log_2(|y|))^{\beta p}} \, dy. \end{align}\] Since \(\beta p > 1\) we conclude that \(I_1<\infty\).
Next, we derive 70 . Using the representation 57 of \(f_{\alpha, \beta}\) we obtain by the Parseval’s identity that for all \(y\in(-2\pi, 2\pi)\), \[\label{l9} \begin{align} &\int_0^{2\pi} |f_{\alpha, \beta}(x + y) - f_{\alpha, \beta}(x)|^2 \, dx \\ &\qquad \qquad = \int_0^{2\pi} \Bigl|\sum_{j \in{\mathbb{Z}}\setminus\{0\}} \operatorname{sgn}(j)|j|^{-\beta}2^{- \alpha |j|}\cdot \frac{\exp(\mathbf{i}\operatorname{sgn}(j) 2^{|j|}y) - 1}{2 \mathbf{i}} \cdot \exp(\mathbf{i}\operatorname{sgn}(j) 2^{|j|}x)\Bigr|^2 \, dx \\ & \qquad \qquad = \frac{\pi}{2} \sum_{j \in{\mathbb{Z}}\setminus\{0\}} a_j(y), \end{align}\tag{71}\] where \[a_j(y)=|j|^{-2\beta}2^{- 2\alpha |j|}\cdot |\exp(\mathbf{i}\operatorname{sgn}(j) 2^{|j|}y) - 1|^2\] for \(j \in{\mathbb{Z}}\setminus\{0\}\) and \(y\in(-2\pi, 2\pi)\). Clearly, there exists \(c\in(0, \infty)\) such that for all \(y \in (-1,1)\setminus\{0\}\), \[\label{lemSob1953} \begin{align} \sum_{ |j| \ge -\log_2(|y|)} a_j(y) & \le 8 \sum_{j = \lceil -\log_2(|y|)\rceil}^\infty j^{-2\beta}2^{- 2\alpha j} \\ &\le \frac{8}{(-\log_2(|y|))^{2\beta}} \cdot |y|^{2\alpha } \cdot \sum_{j=0}^\infty 2^{-2\alpha j}=\frac{c|y|^{2\alpha}}{(-\log_2(|y|))^{2\beta}}. \end{align}\tag{72}\] Moreover, using the inequality \[|e^{ix}-1|\leq |x|, \quad x\in{\mathbb{R}},\] and the fact that there exists \(\kappa\in{\mathbb{N}}\) such that the function \[(0, \infty) \ni x \mapsto x^{-2\beta} 2^{2(1-\alpha)x}\in{\mathbb{R}}\] is monotonically increasing on \([\kappa, \infty)\), we obtain that there exists \(c\in(0, \infty)\) such that for all \(y \in (-1,1)\setminus\{0\}\), \[\label{lemSob1954} \begin{align} \sum_{0 < |j| < -\log_2(|y|)} a_j(y) &\le 2 \sum_{j=1}^{\lceil -\log_2(|y|) \rceil} j^{-2\beta} 2^{-2\alpha j} 2^{2j}|y|^2 \\ & \le 2|y|^2 \sum_{j=1}^{\kappa} j^{-2\beta} 2^{2(1-\alpha)j} + 2|y|^2 \int_1^{\lceil -\log_2(|y|) \rceil+1} x^{-2\beta} 2^{2(1-\alpha)x} \, dx\\ & \le c|y|^2 + 2|y|^2 \int_1^{ -\log_2(|y|) +2} x^{-2\beta} 2^{2(1-\alpha)x} \, dx. \end{align}\tag{73}\] Clearly, \[\label{l7} \lim_{y \rightarrow 0} |y|^2 \Big/ \frac{|y|^{2\alpha}}{(-\log_2(|y|))^{2\beta}}=\lim_{y \rightarrow 0} |y|^{2(1-\alpha)} (-\log_2(|y|))^{2\beta}=0.\tag{74}\] Furthermore, by the rule of L’Hôpital, \[\label{l8} \begin{align} &\lim_{y \rightarrow 0} |y|^2 \int_1^{ -\log_2(|y|) + 2} x^{-2\beta} 2^{2(1-\alpha)x} \, dx \Big/ \frac{|y|^{2\alpha}}{(-\log_2(|y|))^{2\beta}} \\ & \quad= \lim_{y \downarrow 0} \int_1^{ -\log_2(y) + 2} x^{-2\beta} 2^{2(1-\alpha)x} \, dx \Big/ \frac{y^{2\alpha - 2}}{(-\log_2(y))^{2\beta}} \\ &\quad= \lim_{y \downarrow 0} \Biggl( \frac{-2^{4(1-\alpha)}}{\ln(2)} \frac{1}{(-\log_2(y) + 2)^{2\beta}}y^{2\alpha -3}\Biggr) \Big/ \Biggl(\frac{(2\alpha - 2)(-\log_2(y)) + 2\beta /\ln(2)}{(-\log_2(y))^{2\beta+1}}y^{2\alpha-3}\Biggr) \\ &\quad= \frac{2^{4(1-\alpha)}}{\ln(2)} \lim_{y \downarrow 0} \frac{(-\log_2(y))^{2\beta+1}}{(-\log_2(y) + 2)^{2\beta}\cdot ((2-2\alpha)(-\log_2(y)) - 2\beta /\ln(2))} \\ &\quad = \frac{2^{4(1-\alpha)}}{\ln(2)(2 - 2\alpha)}. \end{align}\tag{75}\] Combining 71 to 75 yields 70 . This completes the proof of the lemma. ◻