Appendix A: Compactness Criterion
The following result which is originally due to [14] in the finite dimensional case and which can be e.g. found in [9], provides a compactness criterion of square integrable cylindrical Wiener processes on a Hilbert space.
Theorem A.1
Let (Bt)t∈[0,T] be a cylindrical Wiener process on a separable Hilbert space with respect to a complete probability space \(({\Omega } ,\mathcal {F},\mu )\), where \(\mathcal {F}\) is generated by (Bt)t∈[0,T]. Further, let \({\mathscr{L}}_{HS}(\mathcal{H} ,\mathbb{R})\) be the space of Hilbert-Schmidt operators from to \(\mathbb {R}\) and let \(D: \mathbb {D}^{1,2}\longrightarrow L^{2}({\Omega } ;L^{2}([0,T])\otimes {\mathscr{L}}_{HS}(\mathcal{H},\mathbb {R}))\) be the Malliavin derivative in the direction of (Bt)t∈[0,T], where \(\mathbb {D}^{1,2}\) is the space of Malliavin differentiable random variables in L2(Ω).
Suppose that C is a self-adjoint compact operator on \(L^{2}([0,T])\otimes {\mathscr{L}}_{HS}(\mathcal{H},\mathbb {R})\) with dense image. Then for any c > 0 the set
$$ \mathcal{G}=\left\{ G\in \mathbb{D}^{1,2}:\left\Vert G\right\Vert_{L^{2}({\Omega} )}+\left\Vert C^{-1}DG\right\Vert_{L^{2}({\Omega};L^{2}([0,T])\otimes \mathcal{L}_{HS}(\mathcal{H},\mathbb{R}))}\leq c\right\} $$
is relatively compact in L2(Ω).
In this paper we aim at using a special case of the previous theorem, which is more suitable for explicit estimations. To this end we need the following auxiliary result from [14].
Lemma A.2
Denote by vs,s ≥ 0, with v0 = 1 the Haar basis of L2([0, 1]). Define for any \(0<\alpha <\frac {1}{2}\) the operator Aα on L2([0, 1]) by
$$ A_{\alpha }v_{s}=2^{i\alpha }v_{s},\quad \text{ if }s=2^{i}+j,\quad i\geq 0,\quad 0\leq j\leq 2^{i}, $$
and
Then for \(\alpha <\beta <\frac {1}{2}\) we have that
$$ \begin{array}{@{}rcl@{}} \left\Vert A_{\alpha }f\right\Vert_{L^{2}([0,1])}^{2} \leq 2(\left\Vert f\right\Vert_{L^{2}([0,1])}^{2}+\frac{1}{1-2^{-2(\beta -\alpha )}}{{\int}_{0}^{1}}{{\int}_{0}^{1}}\frac{\left\vert f(t)-f(u)\right\vert^{2}}{\left\vert t-u\right\vert^{1+2\beta }}dtdu). \end{array} $$
Theorem A.3
Let Dk be the Malliavin derivative in the direction of the k-th component of (Bt)t∈[0,T]. In addition, let \(0<\alpha _{k}<\beta _{k}<\frac {1}{2}\) and γk > 0 for all k ≥ 1. Define the sequence \(\mu _{s,k}=2^{-i\alpha _{k}}\gamma _{k}\), if s = 2i + j, i ≥ 0, 0 ≤ j ≤ 2i, k ≥ 1. Assume that μs,k→0 for \(s,k\longrightarrow \infty \). Let c > 0 and \(\mathcal {G}\) the collection of all \(G\in \mathbb {D}^{1,2}\) such that
$$ \left\Vert G\right\Vert_{L^{2}({\Omega} )}\leq c, $$
$$ \sum\limits_{k\geq 1}\gamma_{k}^{-2}\left\Vert D^{k}G\right\Vert_{L^{2}({\Omega};L^{2}([0,1]))}^{2}\leq c, $$
and
$$ \sum\limits_{k\geq 1}\frac{1}{(1-2^{-2(\beta_{k}-\alpha_{k})}){\gamma_{k}^{2}}}{{\int}_{0}^{1}}{{\int}_{0}^{1}}\frac{\left\Vert {D_{t}^{k}}G-{D_{u}^{k}}G\right\Vert_{L^{2}({\Omega})}^{2}}{\left\vert t-u\right\vert^{1+2\beta_{k}}}dtdu\leq c. $$
Then \(\mathcal {G}\) is relatively compact in L2(Ω).
Proof
As before denote by vs,s ≥ 0, with v0 = 1 the Haar basis of L2([0, 1]) and by \(e_{k}^{\ast }= \langle e_{k}, \cdot \rangle _{H}, k\geq 1\), an orthonormal basis of \({\mathscr{L}}_{HS}(\mathcal{H},\mathbb {R})\), where ek,k ≥ 0, is an orthonormal basis of . Define a self-adjoint compact operator C on \(L^{2}([0,1])\otimes {\mathscr{L}}_{HS}(\mathcal{H} ,\mathbb {R})\) with dense image by
$$ C(v_{s}\otimes e_{k}^{\ast })=\mu_{s,k}v_{s}\otimes e_{k}^{\ast },\quad s\geq 0,\quad k\geq 1. $$
Then it follows for \(G\in \mathbb {D}^{1,2}\) from Lemma A.2 that
$$ \begin{array}{@{}rcl@{}} &&\left\Vert C^{-1}DG\right\Vert_{L^{2}({\Omega} ;L^{2}([0,1])\otimes \mathcal{L}_{HS}(\mathcal{H},\mathbb{R}))}^{2} \\ &&~~=\sum\limits_{k\geq 1}\sum\limits_{s\geq 0}\mu_{s,k}^{-2}E[\left\langle DG,v_{s}\otimes e_{k}^{\ast }\right\rangle_{L^{2}([0,1])\otimes \mathcal{L}_{HS}(\mathcal{H},\mathbb{R}))}^{2}] \\ &&~~=\sum\limits_{k\geq 1}\gamma_{k}^{-2}\left\Vert A_{\alpha_{k}}D^{k}G\right\Vert_{L^{2}({\Omega} ;L^{2}([0,1]))}^{2} \\ &&~~\leq 2\sum\limits_{k\geq 1}\gamma_{k}^{-2}\left\Vert D^{k}G\right\Vert_{L^{2}({\Omega} ;L^{2}([0,1]))}^{2} \\ &&~~~~+2\sum\limits_{k\geq 1}\frac{1}{(1-2^{-2(\beta_{k}-\alpha_{k})}){\gamma_{k}^{2}}} {{\int}_{0}^{1}}{{\int}_{0}^{1}}\frac{\left\Vert {D_{t}^{k}}G-{D_{u}^{k}}G\right\Vert_{L^{2}({\Omega} )}^{2}}{\left\vert t-u\right\vert^{1+2\beta_{k}}}dtdu \\ &&~~\leq M \end{array} $$
for a constant \(M<\infty \). So using Theorem A.1 we obtain the result. □
Appendix B: Integration by Parts Formula
In this section we derive an integration by parts formula similar to [6] which is used in the proof of Theorem 4.10 to verify the conditions of the compactness criterion Theorem A.3. Before stating the integration by parts formula, we start by giving some definitions and notations frequently used during the course of this section.
Let n be a given integer. We consider the function \(f:[0,T]^{n}\times (\mathbb {R}^{d})^{n} \rightarrow \mathbb {R}\) of the form
$$ \begin{array}{@{}rcl@{}} f(s,z) = \prod\limits_{j=1}^{n} f_{j} (s_{j}, z_{j}), \quad s = (s_{1}, \dots, s_{n}) \!\in\! [0,T]^{n}, \quad z = (z_{1}, {\dots} ,z_{n}) \in (\mathbb{R}^{d})^{n}, \end{array} $$
(42)
where \(f_{j} : [0,T] \times \mathbb {R}^{d} \rightarrow \mathbb {R}\), \(j=1,\dots , n\), are compactly supported smooth functions. Further, we deal with the function \({\varkappa }: [0,T]^{n} \rightarrow \mathbb {R}\) which is of the form
$$ \begin{array}{@{}rcl@{}} {\varkappa} (s) = \prod\limits_{j=1}^{n} {\varkappa}_{j} (s_{j}), \quad s \in [0,T]^{n}, \end{array} $$
(43)
with integrable factors \({\varkappa }_{j} : [0,T] \rightarrow \mathbb {R}\), \(j=1,\dots , n\).
Let αj be a multi-index and \(D^{\alpha _{j}}\) its corresponding differential operator. For \(\alpha := (\alpha _{1}, \dots , \alpha _{n}) \in \mathbb {N}_{0}^{d\times n}\) we define the norm \(|\alpha | = {\sum }_{j=1}^{n} {\sum }_{k=1}^{d} \alpha _{j}^{(k)}\) and write
$$ \begin{array}{@{}rcl@{}} D^{\alpha} f(s,z) = \prod\limits_{j=1}^{n} D^{\alpha_{j}} f_{j} (s_{j}, z_{j}). \end{array} $$
Let k be an arbitrary integer. Given \((s,z) = (s_{1}, \dots , s_{kn} ,z_{1}, \dots , z_{n}) \in [0,T]^{kn} \times (\mathbb {R}^{d})^{n}\) and a shuffle permutation \(\sigma \in \mathcal {S}(n,n)\) we define the shuffled functions
$$ f_{\sigma}(s,z) := \prod\limits_{j=1}^{kn} f_{[\sigma (j)]} (s_{j}, z_{[\sigma (j)]}) $$
and
$$ {\varkappa}_{\sigma }(s) := \prod\limits_{j=1}^{kn} {\varkappa}_{[\sigma(j)]}(s_{j}), $$
where [j] is equal to (j − in) if (in + 1) ≤ j ≤ (i + 1)n, \(i=0, \dots , (k-1)\). For a multi-index α, we define
$$ \begin{array}{@{}rcl@{}} {\Psi}_{\alpha}^{f} (\theta, t, z, H, d) \!:=\! \left( \prod\limits_{k=1}^{d} \sqrt{(2\left\vert \alpha^{(k)}\right\vert)!}\! \right) \sum\limits_{\sigma \in \mathcal{S}(n,n)} {\int}_{{\Delta}_{\theta, t}^{2n}} \left\vert f_{\sigma}(s,z)\right\vert \vert {\Delta} s \vert^{-H \left( 1+\alpha_{[\sigma({\Delta})]}\right)} ds, \end{array} $$
(44)
and
$$ \begin{array}{@{}rcl@{}} {\Psi}_{\alpha}^{{\varkappa}}(\theta, t, H, d) := \left( \prod\limits_{k=1}^{d} \sqrt{(2\left\vert \alpha^{(k)}\right\vert)!} \right) \sum\limits_{\sigma \in \mathcal{S}(n,n)} {\int}_{{\Delta}_{\theta, t}^{2n}} \left\vert {\varkappa}_{\sigma}(s)\right\vert \vert {\Delta} s \vert^{-H \left( 1+\alpha_{[\sigma({\Delta})]} \right)} ds, \end{array} $$
(45)
where for any \(a,b \in \mathbb {R}\)
$$ \begin{array}{@{}rcl@{}} &&\vert {\Delta} s \vert^{H_{k} \left( a+b\cdot \alpha_{[\sigma({\Delta})]}^{(k)} \right)} := \vert s_{1} \vert^{H_{k} \left( a + b\left( \alpha_{[\sigma(1)]}^{(k)} + \alpha_{[\sigma(2n)]}^{(k)}\right) \right)} \prod\limits_{j=2}^{2n} \left| s_{j}-s_{j-1}\right|^{H_{k} \left( a + b\left( \alpha_{\lbrack \sigma(j)]}^{(k)} + \alpha_{\lbrack \sigma(j-1)]}^{(k)} \right) \right)}, \\ &&| {\Delta} s |^{H \left( a+b\cdot \alpha_{[\sigma({\Delta})]} \right)} := \prod\limits_{k=1}^{d} \vert {\Delta} s \vert^{H_{k} \left( a+b\cdot \alpha_{[\sigma({\Delta})]}^{(k)} \right)}. \end{array} $$
Theorem B.1
Suppose the functions \({\Psi }_{\alpha }^{f}(\theta , t, z, H, d)\) and \({\Psi }_{\alpha }^{{\varkappa }}(\theta , t, H, d)\) defined in Eqs. 44 and 45, respectively, are finite. Then,
$$ \begin{array}{@{}rcl@{}} {\Lambda}_{\alpha}^{f} (\theta, t, z) := (2\pi)^{-dn} {\int}_{(\mathbb{R}^{d})^{n}} {\int}_{{\Delta}_{\theta,t}^{n}} \prod\limits_{j=1}^{n} f_{j} (s_{j}, z_{j})(-iu_{j})^{\alpha_{j}} e^{-i\left\langle u_{j}, \widehat{B}_{s_{j}}^{d,H} - z_{j} \right\rangle }dsdu, \end{array} $$
(46)
where \(\widehat {B}_{t}^{d,H} := \left (\frac {B_{t}^{H_{1}}}{\sqrt {\mathfrak {K}_{H_{1}}}}, \dots , \frac {B_{t}^{H_{d}}}{\sqrt {\mathfrak {K}_{H_{d}}}} \right )^{\top }\) and \(\mathfrak {K}_{H_{k}}\) is the constant in Lemma 2.4, is a square integrable random variable in L2(Ω) and
$$ \begin{array}{@{}rcl@{}} \mathbb{E}\left[{\left\vert {\Lambda}_{\alpha }^{f} (\theta, t, z)\right\vert^{2}}\right] \leq \frac{T^{\frac{\vert \alpha \vert}{6}}}{(2\pi)^{dn}} {\Psi}_{\alpha}^{f} (\theta, t, z, H, d). \end{array} $$
(47)
Furthermore,
$$ \begin{array}{@{}rcl@{}} \mathbb{E}\left[\left|{{\int}_{(\mathbb{R}^{d})^{n}}{\Lambda}_{\alpha }^{{\varkappa} f}(\theta, t, z)dz}\right|\right] \leq \frac{T^{\frac{\vert \alpha \vert}{12}}}{\sqrt{2\pi}^{dn}} ({\Psi}_{\alpha }^{{\varkappa} }(\theta, t, H, d))^{\frac{1}{2}} \prod\limits_{j=1}^{n} \left\Vert f_{j}\right\Vert_{L^{1}(\mathbb{R}^{d};L^{\infty}([0,T]))}, \end{array} $$
(48)
and the integration by parts formula
$$ \begin{array}{@{}rcl@{}} {\int}_{{\Delta}_{\theta ,t}^{n}} D^{\alpha} f \left( s, \widehat{B}_{s}^{d,H} \right) ds = {\int}_{(\mathbb{R}^{d})^{n}} {\Lambda}_{\alpha}^{f}(\theta, t, z) dz, \end{array} $$
(49)
holds.
Proof
For notational simplicity we consider merely the case 𝜃 = 0 and write \({\Lambda }_{\alpha }^{f} (t, z) := {\Lambda }_{\alpha }^{f} (0, t, z)\). For any integrable function \(g:(\mathbb {R}^{d})^{n}\longrightarrow \mathbb {C}\) we have that
$$ \begin{array}{@{}rcl@{}} &&\left\vert {\int}_{(\mathbb{R}^{d})^{n}}g(u_{1},...,u_{n})du_{1}...du_{n} \right\vert^{2} \\ &&\quad ={\int}_{(\mathbb{R}^{d})^{n}}g(u_{1},...,u_{n})du_{1}...du_{n}{\int}_{(\mathbb{R}^{d})^{n}}\overline{g(u_{n+1},...,u_{2n})}du_{n+1}...du_{2n}\\ &&\quad ={\int}_{(\mathbb{R}^{d})^{n}}g(u_{1},...,u_{n})du_{1}...du_{n}(-1)^{dn}{\int}_{(\mathbb{R}^{d})^{n}}\overline{g(-u_{n+1},...,-u_{2n})} du_{n+1}...du_{2n}, \end{array} $$
where the change of variables (un+ 1,...,u2n)↦(−un+ 1,...,−u2n) was applied in the last equality. Thus,
$$ \begin{array}{@{}rcl@{}} \left\vert {\Lambda}_{\alpha }^{f}(t,z)\right\vert^{2} &=& (2\pi )^{-2dn}(-1)^{dn}{\int}_{(\mathbb{R}^{d})^{2n}}{\int}_{{\Delta}_{0,t}^{n}}\prod\limits_{j=1}^{n}f_{j}(s_{j},z_{j})(-iu_{j})^{\alpha_{j}}e^{-i\left\langle u_{j}, \widehat{B}^{d,H}_{s_{j}}-z_{j}\right\rangle }ds \\ &&\times {\int}_{{\Delta}_{0,t}^{n}}\prod\limits_{j=n+1}^{2n}f_{[j]}(s_{j},z_{[j]})(-iu_{j})^{\alpha_{\lbrack j]}}e^{-i\left\langle u_{j}, \widehat{B}^{d,H}_{s_{j}}-z_{[j]}\right\rangle}ds du \\ &=& (2\pi )^{-2dn}(-1)^{dn} ~ i^{\vert \alpha \vert} \sum\limits_{\sigma \in \mathcal{S}(n,n)}{\int}_{(\mathbb{R}^{d})^{2n}}\left( \prod\limits_{j=1}^{n}e^{-i\left\langle z_{j},u_{j}+u_{j+n}\right\rangle }\right) \\ &&\times {\int}_{{\Delta}_{0,t}^{2n}}f_{\sigma }(s,z) \left( \prod\limits_{j=1}^{2n}u_{\sigma(j)}^{\alpha_{\lbrack \sigma (j)]}} \right) \exp \left\{- i \sum\limits_{j=1}^{2n}\left\langle u_{\sigma (j)}, \widehat{B}_{s_{j}}^{d,H} \right\rangle \right\} ds du, \end{array} $$
where we applied shuffling in the sense of Eq. 9. Taking the expectation on both sides together with the independence of the fractional Brownian motions \(B^{H_{k}}\), k = 1,...,d, yields that
$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}\left[{\left\vert {\Lambda}_{\alpha }^{f}(t,z)\right\vert^{2}}\right] \\ &&\quad =(2\pi )^{-2dn}(-1)^{dn}~ i^{\vert \alpha \vert} \sum\limits_{\sigma \in \mathcal{S}(n,n)}{\int}_{(\mathbb{R}^{d})^{2n}}\left( \prod\limits_{j=1}^{n} e^{-i\left\langle z_{j},u_{j}+u_{j+n}\right\rangle }\right) \\ &&\qquad \times {\int}_{{\Delta}_{0,t}^{2n}}f_{\sigma }(s,z) \left( \prod\limits_{j=1}^{2n} u_{\sigma(j)}^{\alpha_{\lbrack \sigma (j)]}} \right) \exp\left\{ -\frac{1}{2} \text{Var}{\sum\limits_{j=1}^{2n} \left\langle u_{\sigma (j)}, \widehat{B}_{s_{j}}^{d,H} \right\rangle}\right\} ds du \\ &&\quad = (2\pi )^{-2dn}(-1)^{dn} ~ i^{\vert \alpha \vert} \sum\limits_{\sigma \in \mathcal{S}(n,n)}{\int}_{(\mathbb{R}^{d})^{2n}} \left( \prod\limits_{j=1}^{n} e^{-i\left\langle z_{j} ,u_{j} + u_{j+n}\right\rangle}\right) \\ &&\qquad \times {\int}_{{\Delta}_{0,t}^{2n}}f_{\sigma }(s,z) \left( \prod\limits_{j=1}^{2n} u_{\sigma(j)}^{\alpha_{\lbrack \sigma (j)]}} \right) \exp\left\{ -\frac{1}{2} \sum\limits_{k=1}^{d} \text{Var}{\sum\limits_{j=1}^{2n}u_{\sigma(j)}^{(k)} \frac{B_{s_{j}}^{H_{k}}}{\sqrt{\mathfrak{K}_{H_{k}}}}} \right\}ds du \\ &&\quad =(2\pi )^{-2dn}(-1)^{dn} ~ i^{\vert \alpha \vert} \sum\limits_{\sigma \in \mathcal{S}(n,n)}{\int}_{(\mathbb{R}^{d})^{2n}}\left( \prod\limits_{j=1}^{n} e^{-i\left\langle z_{j},u_{j}+u_{j+n}\right\rangle }\right) \\ &&\qquad \times {\int}_{{\Delta}_{0,t}^{2n}} f_{\sigma }(s,z) \left( \prod\limits_{j=1}^{2n} u_{\sigma(j)}^{\alpha_{\lbrack \sigma (j)]}} \right) \prod\limits_{k=1}^{d}\exp \left\{ -\frac{1}{2 \mathfrak{K}_{H_{k}}} (u_{\sigma}^{(k)})^{\top} {\Sigma}_{k} u_{\sigma}^{(k)} \right\} ds du, \end{array} $$
(50)
where \(u_{\sigma }^{(k)} = \left (u_{\sigma (1)}^{(k)}, \dots , u_{\sigma (2n)}^{(k)} \right )^{\top }\) and
$$ {\Sigma}_{k} = {\Sigma}_{k}(s):= \left( \mathbb{E}\left[{B_{s_{i}}^{H_{k}}B_{s_{j}}^{H_{k}}}\right]\right)_{1\leq i,j\leq 2n}. $$
Moreover, we obtain for every \(\sigma \in \mathcal {S}(n,n)\) that
$$ \begin{array}{@{}rcl@{}} &&{\int}_{{\Delta}_{0,t}^{2n}}\left\vert f_{\sigma }(s,z)\right\vert {\int}_{(\mathbb{R}^{d})^{2n}} \prod\limits_{k=1}^{d} \left( \left( \prod\limits_{j=1}^{2n}\left\vert u_{\sigma(j)}^{(k)}\right\vert^{\alpha_{[\sigma (j)]}^{(k)}} \right) e^{-\frac{1}{2 \mathfrak{K}_{H_{k}}} (u_{\sigma}^{(k)})^{\top} {\Sigma}_{k} u_{\sigma}^{(k)}} \right) du ds \\ &&\quad ={\int}_{{\Delta}_{0,t}^{2n}}\left\vert f_{\sigma }(s,z)\right\vert \prod\limits_{k=1}^{d} \left( {\int}_{\mathbb{R}^{2n}} \left( \prod\limits_{j=1}^{2n}\left\vert u_{j}^{(k)}\right\vert^{\alpha_{[\sigma (j)]}^{(k)}}\right) e^{-\frac{1}{2} \left\langle \frac{{\Sigma}_{k}}{\mathfrak{K}_{H_{k}}} u^{(k)},u^{(k)}\right\rangle } du^{(k)} \right) ds, \end{array} $$
(51)
where \(u^{(k)} := \left (u_{1}^{(k)}, \dots , u_{2n}^{(k)} \right )^{\top }.\) For every 1 ≤ k ≤ d we have by using substitution that
$$ \begin{array}{@{}rcl@{}} &&{\int}_{\mathbb{R}^{2n}} \left( \prod\limits_{j=1}^{2n}\left\vert u_{j}^{(k)}\right\vert^{\alpha_{[\sigma (j)]}^{(k)}}\right) e^{-\frac{1}{2}\left\langle \frac{{\Sigma}_{k}}{\mathfrak{K}_{H_{k}}} u^{(k)}, u^{(k)}\right\rangle} du^{(k)} \\ &&\quad =\frac{\mathfrak{K}_{H_{k}}^{n}}{(\det {\Sigma}_{k})^{1/2}}{\int}_{\mathbb{R}^{2n}} \left( \prod\limits_{j=1}^{2n} \left \vert \left\langle \sqrt{\mathfrak{K}_{H_{k}}} {\Sigma}_{k}^{-1/2}u^{(k)},\widetilde{e}_{j}\right\rangle \right\vert^{\alpha_{[\sigma (j)]}^{(k)}}\right) e^{-\frac{1}{2}\left\langle u^{(k)},u^{(k)}\right\rangle} du^{(k)}. \end{array} $$
(52)
Considering a standard Gaussian random vector \(Z\sim \mathcal {N}(0,\text{Id}_{2n})\), we get that
$$ \begin{array}{@{}rcl@{}} &&{\int}_{\mathbb{R}^{2n}} \left( \prod\limits_{j=1}^{2n}\left\vert \left\langle {\Sigma}_{k}^{-1/2}u^{(k)},\widetilde{e}_{j} \right\rangle \right\vert^{\alpha_{[\sigma(j)]}^{(k)}}\right) e^{-\frac{1}{2}\left\langle u^{(k)},u^{(k)}\right\rangle} du^{(k)} \\ &&\quad = (2\pi )^{n}\mathbb{E}\left[{\prod\limits_{j=1}^{2n}\left\vert \left\langle {\Sigma}_{k}^{-1/2}Z,\widetilde{e}_{j}\right\rangle \right\vert^{\alpha_{[\sigma(j)]}^{(k)}}}\right]. \end{array} $$
(53)
Using a Brascamp-Lieb type inequality which is due to Lemma C.1, we further get that
$$ \begin{array}{@{}rcl@{}} \mathbb{E}\left[{\prod\limits_{j=1}^{2n}\left\vert \left\langle {\Sigma}_{k}^{-1/2}Z, \widetilde{e}_{j}\right\rangle \right\vert^{\alpha_{[\sigma (j)]}^{(k)}}}\right] \leq \sqrt{\text{perm}{A_{k}}}=\sqrt{\sum\limits_{\pi \in S_{2\left\vert \alpha^{(k)}\right\vert }}\prod\limits_{i=1}^{2\left\vert \alpha^{(k)}\right\vert}a_{i,\pi (i)}^{(k)}}, \end{array} $$
where \(\left \vert \alpha ^{(k)}\right \vert :={\sum }_{j=1}^{n}\alpha _{j}^{(k)}\) and permAk is the permanent of the covariance matrix \(A_{k} =(a_{i,j}^{(k)})_{1\leq i,j \leq 2\left \vert \alpha ^{(k)}\right \vert }\) of the Gaussian random vector
$$ \begin{array}{@{}rcl@{}} \underset{\alpha_{[\sigma (1)]}^{(k)}\text{ times}}{\left( \underbrace{\left\langle {\Sigma}_{k}^{-1/2}Z,\widetilde{e}_{1}\right\rangle ,...,\left\langle {\Sigma}_{k}^{-1/2}Z, \widetilde{e}_{1}\right\rangle}\right.}, {\dots} ,\underset{\alpha_{[\sigma (2n)]}^{(k)}\text{ times}}{\left.\underbrace{\left\langle {\Sigma}_{k}^{-1/2}Z, \widetilde{e}_{2n}\right \rangle ,...,\left\langle {\Sigma}_{k}^{-1/2}Z, \widetilde{e}_{2n}\right\rangle}\right)}, \end{array} $$
and Sm denotes the permutation group of size m. Using an upper bound for the permanent of positive semidefinite matrices which is due to [3], we find that
$$ \begin{array}{@{}rcl@{}} \text{perm}{A_{k}}=\sum\limits_{\pi \in S_{2\left\vert \alpha^{(k)}\right\vert}}\prod\limits_{i=1}^{2\left\vert \alpha^{(k)}\right\vert }a_{i,\pi (i)}^{(k)}\leq \left( 2\left\vert \alpha^{(k)}\right\vert \right)!\prod\limits_{i=1}^{2\left\vert \alpha^{(k)}\right\vert }a_{i,i}^{(k)}. \end{array} $$
(54)
Now let \({\sum }_{l=1}^{j-1}\alpha _{[\sigma (l)]}^{(k)}+1 \leq i \leq {\sum }_{l=1}^{j}\alpha _{[\sigma (l)]}^{(k)}\) for some fixed j ∈{1,..., 2n}. Then
$$ \begin{array}{@{}rcl@{}} a_{i,i}^{(k)}=\mathbb{E}\left[{\left\langle {\Sigma}_{k}^{-1/2}Z,\widetilde{e}_{j} \right\rangle \left\langle {\Sigma}_{k}^{-1/2}Z,\widetilde{e}_{j} \right\rangle}\right]. \end{array} $$
Substitution gives moreover that
$$ \mathbb{E}[{\left\langle {\Sigma}_{k}^{-1/2}Z, \widetilde{e}_{j}\right\rangle \left\langle {\Sigma}_{k}^{-1/2}Z, \widetilde{e}_{j} \right\rangle}] = (\det {\Sigma}_{k})^{1/2}\frac{1}{(2\pi )^{n}}{\int}_{\mathbb{R}^{2n}}{u_{j}^{2}}\exp \left\lbrace -\frac{1}{2}\left\langle {\Sigma}_{k} u, u\right\rangle \right\rbrace du. $$
(55)
Applying Lemma C.2 we get
$$ \begin{array}{@{}rcl@{}} {\int}_{\mathbb{R}^{2n}}{u_{j}^{2}}\exp \left\lbrace -\frac{1}{2}\left\langle {\Sigma}_{k} u, u\right\rangle \right\rbrace du &=& \frac{(2\pi )^{(2n-1)/2}}{(\det {\Sigma}_{k})^{1/2}}{\int}_{\mathbb{R}} v^{2}\exp \left\lbrace -\frac{1}{2}v^{2} \right\rbrace dv\frac{1}{{\sigma_{j}^{2}}} \\ &=& \frac{(2\pi )^{n}}{(\det {\Sigma}_{k})^{1/2}}\frac{1}{{\sigma_{j}^{2}}}, \end{array} $$
(56)
where \({\sigma _{j}^{2}}:=\text {Var}\left ({B_{s_{j}}^{H_{k}}\left \vert B_{s_{1}}^{H_{k}}...,B_{s_{2n}}^{H_{k}}\text { without }B_{s_{j}}^{H_{k}} \right .}\right ).\)
Subsequently, we aim at the application of the strong local non-determinism property of the fractional Brownian motions, cf. Lemma 2.4, i.e. for all 0 < r < t ≤ T exists a constant \(\mathfrak {K}_{H_{k}}\) depending on Hk and T such that
$$ \text{Var}\left( {B_{t}^{H_{k}}\left\vert B_{s}^{H_{k}},\left\vert t-s\right\vert \geq r \right.}\right) \geq \mathfrak{K}_{H_{k}} r^{2H_{k}}. $$
Hence, we get due to Lemmas C.5 and C.6 that
$$ (\det {\Sigma}_{k}(s))^{1/2}\geq \mathfrak{K}_{H_{k}}^{\frac{(2n-1)}{2}}\left\vert s_{1}\right\vert^{H_{k}}\left\vert s_{2}-s_{1}\right\vert^{H_{k}}...\left\vert s_{2n}-s_{2n-1}\right\vert^{H_{k}}, $$
(57)
and
$$ \begin{array}{@{}rcl@{}} {\sigma_{1}^{2}} &\geq& \mathfrak{K}_{H_{k}} \left\vert s_{2} - s_{1} \right\vert^{2H_{k}}, \\ {\sigma_{j}^{2}} &\geq& \mathfrak{K}_{H_{k}} \min \left\lbrace \left\vert s_{j}-s_{j-1}\right\vert^{2H_{k}},\left\vert s_{j+1}-s_{j}\right\vert^{2H_{k}}\right\rbrace, ~ 2\leq j \leq 2n-1, \\ \sigma_{2n}^{2} &\geq& \mathfrak{K}_{H_{k}} \left\vert s_{2n}-s_{2n-1}\right\vert^{2H_{k}}. \end{array} $$
Thus,
$$ \begin{array}{@{}rcl@{}} \prod\limits_{j=1}^{2n}\sigma_{j}^{-2\alpha_{[\sigma (j)]}^{(k)}} \leq \mathfrak{K}_{H_{k}}^{-2 \vert \alpha^{(k)} \vert} T^{4 H_{k} \vert \alpha^{(k)} \vert} \vert {\Delta} s \vert^{-2H_{k} \alpha_{[\sigma({\Delta})]}^{(k)}}. \end{array} $$
(58)
Concluding from Eqs. 54, 55, 56, and 58 we have that
$$ \begin{array}{@{}rcl@{}} \text{perm}{(A_{k})} &\leq& \left( 2\left\vert \alpha^{(k)}\right\vert\right)! \prod\limits_{i=1}^{2\left\vert \alpha^{(k)}\right\vert }a_{i,i}^{(k)} \\ &\leq& \left( 2\left\vert \alpha^{(k)}\right\vert \right)! \prod\limits_{j=1}^{2n} \left( (\det {\Sigma}_{k})^{1/2} \frac{1}{(2\pi )^{n}}\frac{(2\pi )^{n}}{(\det {\Sigma}_{k})^{1/2}}\frac{1}{{\sigma_{j}^{2}}} \right)^{\alpha_{[\sigma (j)]}^{(k)}} \\ &\leq& \left( 2\left\vert \alpha^{(k)}\right\vert \right)! \mathfrak{K}_{H_{k}}^{-2 \vert \alpha^{(k)} \vert} T^{4 H_{k} \vert \alpha^{(k)} \vert} \vert {\Delta} s \vert^{-2H_{k} \alpha_{[\sigma({\Delta})]}^{(k)}}. \end{array} $$
Consequently,
$$ \begin{array}{@{}rcl@{}} \mathbb{E}\left[\prod\limits_{j=1}^{2n}\left\vert \left\langle {\Sigma}_{k}^{-1/2}Z, \widetilde{e}_{j} \right\rangle\right\vert^{\alpha_{\lbrack \sigma (j)}^{(k)}}\right] \leq \sqrt{\left( 2\left\vert \alpha^{(k)}\right\vert \right)!} \mathfrak{K}_{H_{k}}^{-\vert \alpha^{(k)} \vert} T^{2 H_{k} \vert \alpha^{(k)}\vert} \vert {\Delta} s \vert^{-H_{k} \alpha_{[\sigma({\Delta})]}^{(k)}}. \end{array} $$
Therefore we get from Eqs. 50, 51, 52, 53, and 57 that
$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}\left[{\left\vert {\Lambda}_{\alpha }^{f}(t,z)\right\vert^{2}}\right] \\ &&\quad \leq (2\pi)^{-2dn} \sum\limits_{\sigma \in \mathcal{S}(n,n)}{\int}_{{\Delta}_{0,t}^{2n}}\left\vert f_{\sigma }(s,z)\right\vert \prod\limits_{k=1}^{d} \left( {\int}_{\mathbb{R}^{2n}}\left\vert u^{(k)}\right\vert^{\alpha^{(k)}} e^{-\frac{1}{2\mathfrak{K}_{H_{k}}}\left\langle {\Sigma}_{k} u^{(k)},u^{(k)}\right\rangle} du^{(k)} \right) ds \\ &&\quad \leq (2\pi)^{-dn} \sum\limits_{\sigma \in \mathcal{S}(n,n)} {\int}_{{\Delta}_{0,t}^{2n}} \left\vert f_{\sigma }(s,z)\right\vert \prod\limits_{k=1}^{d} \left( \frac{\mathfrak{K}_{H_{k}}^{n+\vert \alpha^{(k)}\vert}}{\left( \det {\Sigma}_{k}(s) \right)^{\frac{1}{2}}} \mathbb{E}\!\left[{ \prod\limits_{j=1}^{2n} \left\vert \left\langle {\Sigma}_{k}^{-\frac{1}{2}} Z, \widetilde{e}_{j} \right\rangle \right\vert^{\alpha_{\sigma(j)}^{(k)}}} \right]\right) \!ds \\ &&\quad \leq (2\pi)^{-dn} \sum\limits_{\sigma \in \mathcal{S}(n,n)}{\int}_{{\Delta}_{0,t}^{2n}}\left\vert f_{\sigma }(s,z)\right\vert \left( \prod\limits_{k=1}^{d} \vert {\Delta} s \vert^{-H_{k}} \mathfrak{K}_{H_{k}}^{\vert \alpha^{(k)}\vert + \frac{1}{2}} \right)\\ &&\qquad \times \prod\limits_{k=1}^{d} \left( \sqrt{(2\left\vert \alpha^{(k)}\right\vert )!} \mathfrak{K}_{H_{k}}^{-\vert \alpha^{(k)} \vert} T^{2 H_{k} \vert \alpha^{(k)} \vert} \vert {\Delta} s \vert^{-H_{k} \alpha_{[\sigma({\Delta})]}^{(k)}} \right) ds \\ &&\quad \leq (2\pi)^{-dn} T^{\frac{\vert \alpha \vert}{6}} \left( \prod\limits_{k=1}^{d} \sqrt{\mathfrak{K}_{H_{k}}} \sqrt{\left( 2\left\vert \alpha^{(k)}\right\vert \right)!} \right) \sum\limits_{\sigma \in \mathcal{S}(n,n)} {\int}_{{\Delta}_{0,t}^{2n}} \left\vert f_{\sigma }(s,z)\right\vert \vert {\Delta} s \vert^{- H \left( 1 + \alpha_{[\sigma({\Delta})]} \right)} ds. \end{array} $$
Since \(\sup _{k \geq 1} \mathfrak {K}_{H_{k}} \in (0,1)\), inequality Eq. 47 holds.
Next we prove the estimate Eq. 48. With inequality Eq. 47, we get that
$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}\left[\left|{{\int}_{(\mathbb{R}^{d})^{n}}{\Lambda}_{\alpha}^{{\varkappa} f}(\theta ,t,z)dz}\right|\right] \leq {\int}_{(\mathbb{R}^{d})^{n}} \mathbb{E}\left[\left|{{\Lambda}_{\alpha}^{{\varkappa} f}(\theta ,t,z)}\right|^{2}\right]^{\frac{1}{2}} dz \\ &&\quad \leq \frac{T^{\frac{\vert \alpha \vert}{12}}}{\sqrt{2\pi}^{dn}} {\int}_{(\mathbb{R}^{d})^{n}}({\Psi}_{\alpha }^{{\varkappa} f}(\theta,t,z,H,d))^{\frac{1}{2}}dz. \end{array} $$
Taking the supremum over [0,T] with respect to each function fj, i.e.
$$ \left\vert f_{[\sigma (j)]}(s_{j},z_{[\sigma (j)]})\right\vert \leq \underset{s_{j}\in [0,T]}{\sup}\left\vert f_{[\sigma (j)]}(s_{j},z_{[\sigma(j)]})\right\vert , ~ j=1,...,2n, $$
yields that
$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}\left[\left|{{\int}_{(\mathbb{R}^{d})^{n}}{\Lambda}_{\alpha}^{{\varkappa} f}(\theta ,t,z)dz}\right|\right] \\ &&~\leq \frac{T^{\frac{\vert \alpha \vert}{12}}}{\sqrt{2\pi}^{dn}} \underset{\sigma \in \mathcal{S}(n,n)}{\max}{\int}_{(\mathbb{R}^{d})^{n}}\left( \prod\limits_{j=1}^{2n}\left\Vert f_{[\sigma (j)]}(\cdot,z_{[\sigma (j)]})\right\Vert_{L^{\infty }([0,T])}\right)^{\frac{1}{2}}dz \\ &&~\times \left( \prod\limits_{k=1}^{d}\sqrt{(2\left\vert \alpha^{(k)}\right\vert )!}\sum\limits_{\sigma \in \mathcal{S}(n,n)}{\int}_{{\Delta}_{\theta,t}^{2n}}\left\vert {\varkappa}_{\sigma }(s)\right\vert \vert {\Delta} s \vert^{- H \left( 1 + \alpha_{[\sigma ({\Delta})]} \right)} ds \right)^{\frac{1}{2}} \\ &&~= \frac{T^{\frac{\vert \alpha \vert}{12}}}{\sqrt{2\pi}^{dn}} \underset{\sigma \in \mathcal{S}(n,n)}{\max}{\int}_{(\mathbb{R}^{d})^{n}}\left( \prod\limits_{j=1}^{2n}\left\Vert f_{[\sigma (j)]}(\cdot,z_{[\sigma (j)]})\right\Vert_{L^{\infty }([0,T])}\right)^{\frac{1}{2}} dz ~({\Psi}_{\alpha }^{{\varkappa} }(\theta, t, H, d))^{\frac{1}{2}} \\ &&~= \frac{T^{\frac{\vert \alpha \vert}{12}}}{\sqrt{2\pi}^{dn}} {\int}_{(\mathbb{R}^{d})^{n}}\prod\limits_{j=1}^{n}\left\Vert f_{j}(\cdot ,z_{j})\right\Vert_{L^{\infty }([0,T])}dz ~ ({\Psi}_{\alpha }^{{\varkappa} }(\theta, t, H, d))^{\frac{1}{2}} \\ &&~= \frac{T^{\frac{\vert \alpha \vert}{12}}}{\sqrt{2\pi}^{dn}} \left( \prod\limits_{j=1}^{n}\left\Vert f_{j} \right\Vert_{L^{1}(\mathbb{R}^{d};L^{\infty }([0,T]))} \right) ({\Psi}_{\alpha }^{{\varkappa} }(\theta, t, H, d))^{\frac{1}{2}}. \end{array} $$
Finally, we show the integration by parts formula Eq. 49. Note that a priori one cannot interchange the order of integration in Eq. 46, since e.g. for m = 1, f ≡ 1 one gets an integral of the Donsker-Delta function which is not a random variable in the usual sense. Therefore, we define for R > 0,
$$ \begin{array}{@{}rcl@{}} {\Lambda}^{f}_{\alpha,R} (\theta,t,z) := (2\pi)^{-dn} {\int}_{B(0,R)} {\int}_{{\Delta}^{n}_{\theta,t}} \prod\limits_{j=1}^{n} f_{j}(s_{j},z_{j}) (-iu_{j})^{\alpha_{j}} e^{-i \langle u_{j}, \widehat{B}_{s_{j}}^{d,H}-z_{j}\rangle } ds dv, \end{array} $$
where \(B(0,R):=\{v\in (\mathbb {R}^{d})^{n} : |v|<R\}\). This yields
$$ |{\Lambda}^{f}_{\alpha,R} (\theta,t,z)| \leq C_{R} {\int}_{{\Delta}^{n}_{\theta,t}} \prod\limits_{j=1}^{n} |f_{j}(s_{j}, z_{j})| ds $$
for a sufficient constant CR. Under the assumption that the above right-hand side is integrable over \((\mathbb {R}^{d})^{n}\), similar computations as above show that \({\Lambda }^{f}_{\alpha ,R}(\theta ,t,z)\to {\Lambda }^{f}_{\alpha } (\theta ,t,z)\) in L2(Ω) as \(R\to \infty \) for all 𝜃,t and z. By Lebesgue’s dominated convergence theorem and the fact that the Fourier transform is an automorphism on the Schwarz space, we obtain
$$ \begin{array}{@{}rcl@{}} &&{\int}_{(\mathbb{R}^{d})^{n}} {\Lambda}^{f}_{\alpha}(\theta,t,z) dz = \underset{R\to \infty}{\lim} {\int}_{(\mathbb{R}^{d})^{n}} {\Lambda}^{f}_{\alpha, R} (\theta,t,z)dx\\ &&~ = \underset{R\to \infty}{\lim} (2\pi)^{-dn}{\int}_{(\mathbb{R}^{d})^{n}} {\int}_{B(0,R)} {\int}_{{\Delta}_{\theta,t}^{n}} \prod\limits_{j=1}^{n} f_{j}(s_{j}, z_{j}) (-iu_{j})^{\alpha_{j}} e^{-i \left\langle u_{j}, \widehat{B}_{s_{j}}^{d,H}-z_{j} \right\rangle} dzduds\\ &&~ = \underset{R\to \infty}{\lim} {\int}_{{\Delta}_{\theta,t}^{n}} {\int}_{B(0,R)} (2\pi)^{-dn} {\int}_{(\mathbb{R}^{d})^{n}} \prod\limits_{j=1}^{n} f_{j}(s_{j}, z_{j})e^{i \langle u_{j},z_{j} \rangle} dz (-iu_{j})^{\alpha_{j}} e^{-i \left\langle u_{j}, \widehat{B}_{s_{j}}^{d,H} \right\rangle} duds\\ &&~ = \underset{R\to \infty}{\lim} {\int}_{{\Delta}_{\theta,t}^{n}} {\int}_{B(0,R)} \prod\limits_{j=1}^{n} \widehat{f}_{j}(s,-u_{j}) (-iu_{j})^{\alpha_{j}} e^{-i \left\langle u_{j}, \widehat{B}_{s_{j}}^{d,H} \right\rangle} duds\\ &&~ = {\int}_{{\Delta}_{\theta,t}^{n}} D^{\alpha} f\left( s,\widehat{B}_{s}^{d,H}\right)ds \end{array} $$
which is exactly the integration by parts formula Eq. 49. □
Applying Theorem B.1 we obtain the following crucial estimate (compare [1, 2, 6], and [7]):
Proposition B.2
Let the functions f and ϰ be defined as in Eqs. 42 and 43, respectively. Further, let \( 0\leq \theta ^{\prime } <\theta <t \leq T\) and for some m ≥ 1
$$ {\varkappa}_{j}(s)=(K_{H_{m}}(s,\theta )-K_{H_{m}}(s,\theta^{\prime}))^{\varepsilon_{j}},~\theta <s<t, $$
for every j = 1,...,n with \((\varepsilon _{1},...,\varepsilon _{n})\in \{0,1\}^{n}\). Let \(\alpha \in ({\mathbb {N}_{0}^{d}})^{n}\) be a multi-index. Assume there exists δ such that
$$ \begin{array}{@{}rcl@{}} -\sum\limits_{k=1}^{d} H_{k} \left( 1+2\alpha_{j}^{(k)}\right)+\left( H_{m} - \frac{1}{2} - \gamma_{m}\right)\geq \delta >-1 \end{array} $$
(59)
for all \(j= 1, {\dots } n\) and d ≥ 1, where γm ∈ (0,Hm) is sufficiently small. Then there exist constants CT (depending on T) and Kd,H (depending on d and H), such that for any 0 ≤ 𝜃 < t ≤ T we have
$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}\left[\left|{{\int}_{{\Delta}_{\theta ,t}^{n}}\left( \prod\limits_{j=1}^{n} D^{\alpha_{j}}f_{j}(s_{j}, \widehat{B}_{s_{j}}){\varkappa}_{j}(s_{j})\right) ds}\right|\right] \\ &&~ \leq \frac{K_{d,H}^{n} \cdot T^{\frac{\vert \alpha \vert}{12}}}{\sqrt{2\pi}^{dn}} \left( C_{T} \left( \frac{\theta -\theta^{\prime}}{\theta \theta^{\prime}}\right)^{\gamma_{m}} \theta^{(H_{m}-\frac{1}{2} - \gamma_{m})}\right)^{{\sum}_{j=1}^{n} \varepsilon_{j}} \prod\limits_{j=1}^{n} \left\Vert f_{j}(\cdot ,z_{j})\right\Vert_{L^{1}(\mathbb{R}^{d};L^{\infty }([0,T]))} \\ &&~ \times \frac{\left( {\prod}_{k=1}^{d} \left( 2\left\vert \alpha^{(k)}\right\vert\right)!\right)^{\frac{1}{4}}(t-\theta )^{-{\sum}_{k=1}^{d} H_{k} \left( n+2\left\vert \alpha^{(k)}\right\vert \right)+\left( H_{m}-\frac{1}{2}-\gamma_{m}\right){\sum}_{j=1}^{n} \varepsilon_{j}+n}}{\Gamma(2n - {\sum}_{k=1}^{d}H_{k}(2n+4\left\vert \alpha^{(k)}\right\vert )+2(H_{m} - \frac{1}{2}-\gamma_{m}){\sum}_{j=1}^{n} \varepsilon_{j})^{\frac{1}{2}}}. \end{array} $$
In order to prove this result we need the following two auxiliary results.
Lemma B.3
Let \(H \in \left (0,\frac {1}{2}\right )\) and t ∈ [0,T] be fixed. Then, there exists \(\beta \in \left (0,\frac {1}{2}\right )\) and a constant C > 0 independent of H such that
$$ \begin{array}{@{}rcl@{}} {{\int}_{0}^{t}} {{\int}_{0}^{t}} \frac{|K_{H}(t, \theta^{\prime}) - K_{H}(t,\theta)|^{2}}{|\theta^{\prime}-\theta|^{1+2\beta}}d\theta d\theta^{\prime} \leq C < \infty. \end{array} $$
Proof
Let \(0 \leq \theta ^{\prime }<\theta \leq t\) be fixed. Write
$$ K_{H} (t,\theta) - K_{H}(t,\theta^{\prime}) = c_{H}\left[f_{t}(\theta) - f_{t}(\theta^{\prime}) + \left( \frac{1}{2}-H\right) \left( g_{t}(\theta) - g_{t}(\theta^{\prime})\right)\right], $$
where \(f_{t} (\theta ):= \left (\frac {t}{\theta } \right )^{H-\frac {1}{2}} (t-\theta )^{H-\frac {1}{2}}\) and \(g_{t}(\theta ) := {\int \limits }_{\theta }^{t} \frac {f_{u} (\theta )}{u}du\).
We continue with the estimation of \(K_{H} (t,\theta ) - K_{H}(t,\theta ^{\prime })\). First, observe that there exists a constant 0 < C < 1 such that
$$ \frac{y^{-\alpha} -x^{-\alpha}}{(x-y)^{\gamma}} \leq C y^{-\alpha-\gamma}, $$
(60)
for every \(0<y<x<\infty \) and \(\alpha :=\left (\frac {1}{2}-H\right ) \in \left (0,\frac {1}{2}\right )\) as well as \(0 < \gamma < \frac {1}{2}-\alpha \). Indeed, rewriting Eq. 60 yields using the substitution \(z:= \frac {x}{y}\), \(z\in (1,\infty )\),
$$ \begin{array}{@{}rcl@{}} \frac{y^{-\alpha} -x^{-\alpha}}{(x-y)^{\gamma}} y^{\alpha + \gamma} = \frac{1-z^{-\alpha}}{(z-1)^{\gamma}} =: g(z). \end{array} $$
Furthermore, since α + γ < 1 we get that
$$ \begin{array}{@{}rcl@{}} \underset{z\to 1}{\lim} g(z) = \underset{z\to 1}{\lim} \frac{1-z^{-\alpha}}{(z-1)^{\gamma}} = \underset{z\to 1}{\lim} \frac{1+\alpha z^{-\alpha-1}}{\gamma(z-1)^{\gamma-1}} = 0, \end{array} $$
and
$$ \begin{array}{@{}rcl@{}} \underset{z\to \infty}{\lim} g(z) = 0. \end{array} $$
Moreover, for \(2 \leq z \leq \infty \) we get the upper bound
$$ \begin{array}{@{}rcl@{}} 0 \leq g(z) \leq \frac{1-z^{-\alpha}}{(z-1)^{\gamma}} < \frac{1}{1} = 1, \end{array} $$
and for 1 < z < 2 we have that
$$ \begin{array}{@{}rcl@{}} g(z) = \frac{z^{\alpha}-1}{(z-1)^{\gamma} z^{\alpha}} < \frac{z-1}{(z-1)^{\gamma} (z-1)^{\alpha}} = (z-1)^{1-\gamma-\alpha} \leq 1. \end{array} $$
This shows inequality Eq. 60 which then implies for 0 < γ < H that
$$ \begin{array}{@{}rcl@{}} f_{t}(\theta) - f_{t}(\theta^{\prime}) &=& \left( \frac{t}{\theta} (t-\theta)\right)^{H-\frac{1}{2}}-\left( \frac{t}{\theta^{\prime}} (t-\theta^{\prime})\right)^{H-\frac{1}{2}} \\ &&\lesssim \left( \frac{t}{\theta}(t-\theta)\right)^{H-\frac{1}{2} -\gamma}t^{2\gamma }\frac{(\theta-\theta^{\prime})^{\gamma }}{(\theta \theta^{\prime})^{\gamma }} \lesssim \left( t-\theta \right)^{H-\frac{1}{2}-\gamma} \frac{(\theta -\theta^{\prime})^{\gamma}}{(\theta \theta^{\prime})^{\gamma}}. \end{array} $$
Further,
$$ \begin{array}{@{}rcl@{}} g_{t}(\theta )-g_{t}(\theta^{\prime}) &=& {\int}_{\theta }^{t}\frac{f_{u}(\theta)-f_{u}(\theta^{\prime})}{u}du-{\int}_{\theta^{\prime}}^{\theta }\frac{f_{u}(\theta^{\prime})}{u}du \\ &\leq& {\int}_{\theta }^{t}\frac{f_{u}(\theta )-f_{u}(\theta^{\prime})}{u}du \\ &\lesssim& \frac{(\theta -\theta^{\prime})^{\gamma }}{(\theta \theta^{\prime})^{\gamma }}{\int}_{\theta }^{t}\frac{(u-\theta )^{H-\frac{1}{2}-\gamma }}{u}du \\ &\leq& \frac{(\theta -\theta^{\prime})^{\gamma }}{(\theta \theta^{\prime})^{\gamma }} \theta^{H-\frac{1}{2}-\gamma }{\int}_{1}^{\infty }\frac{(v-1)^{H-\frac{1}{2} -\gamma }}{v} dv \\ &\lesssim& \frac{(\theta-\theta^{\prime})^{\gamma }}{(\theta \theta^{\prime})^{\gamma }}\theta^{H-\frac{1}{2}-\gamma } \\ &\lesssim& \frac{(\theta -\theta^{\prime})^{\gamma }}{(\theta \theta^{\prime})^{\gamma }}\theta^{H-\frac{1}{2}-\gamma }(t-\theta )^{H-\frac{1}{2} - \gamma}. \end{array} $$
Consequently, we get for γ ∈ (0,H), \(0<\theta ^{\prime }<\theta <t\leq T\), that
$$ K_{H}(t,\theta)-K_{H}(t,\theta^{\prime})\leq C \cdot c_{H} \frac{(\theta-\theta^{\prime})^{\gamma }}{(\theta \theta^{\prime})^{\gamma }}\theta^{H-\frac{1}{2} - \gamma }(t-\theta )^{H-\frac{1}{2}-\gamma }, $$
where C > 0 is a constant merely depending on T. Thus
$$ \begin{array}{@{}rcl@{}} &&{{\int}_{0}^{t}}{\int}_{0}^{\theta }\frac{(K_{H}(t,\theta)-K_{H}(t,\theta^{\prime}))^{2}}{|\theta -\theta^{\prime}|^{1+2\beta }}d\theta^{\prime}d\theta \\ &&~~~~~\lesssim {{\int}_{0}^{t}}{\int}_{0}^{\theta}\frac{|\theta -\theta^{\prime}|^{-1-2\beta +2\gamma }}{(\theta \theta^{\prime})^{2\gamma }}\theta^{2H-1-2\gamma }(t-\theta)^{2H-1-2\gamma }d\theta^{\prime}d\theta \\ &&~~~~~ = {{\int}_{0}^{t}}\theta^{2H-1-4\gamma }(t-\theta )^{2H-1-2\gamma }{\int}_{0}^{\theta}|\theta -\theta^{\prime}|^{-1-2\beta +2\gamma }(\theta^{\prime})^{-2\gamma}d\theta^{\prime}d\theta \\ &&~~~~= {{\int}_{0}^{t}}\theta^{2H-1-4\gamma -2\beta}(t-\theta )^{2H-1-2\gamma }\frac{\Gamma(-2\beta +2\gamma ){\Gamma} (-2\gamma +1)}{\Gamma (-2\beta +1)} d\theta \\ &&~~~\lesssim {{\int}_{0}^{t}}\theta^{2H-1-4\gamma -2\beta }(t-\theta )^{2H-1-2\gamma }d\theta \\ &&~= \frac{\Gamma (2H-2\gamma ){\Gamma} (2H-4\gamma -2\beta )}{\Gamma(4H-6\gamma -2\beta )}t^{4H-6\gamma -2\beta -1}<\infty, \end{array} $$
for sufficiently small γ and β. On the other hand, we have that
$$ \begin{array}{@{}rcl@{}} &&{{\int}_{0}^{t}}{\int}_{\theta}^{t}\frac{(K_{H}(t,\theta)-K_{H}(t,\theta^{\prime}))^{2}}{|\theta -\theta^{\prime}|^{1+2\beta }}d\theta^{\prime}d\theta \\ &&~~~~~~~\lesssim {{\int}_{0}^{t}}\theta^{2H-1-4\gamma }(t-\theta)^{2H-1-2\gamma }{\int}_{\theta}^{t}\frac{|\theta -\theta^{\prime}|^{-1-2\beta +2\gamma }}{(\theta^{\prime})^{2\gamma }}d\theta^{\prime}d\theta \\ &&~~~~~~~\leq {{\int}_{0}^{t}}\theta^{2H-1-6\gamma }(t-\theta)^{2H-1-2\gamma } {\int}_{\theta}^{t} |\theta -\theta^{\prime}|^{-1-2\beta +2\gamma } d\theta^{\prime}d\theta \\ &&~~~~~~~\lesssim {{\int}_{0}^{t}}\theta^{2H-1-6\gamma }(t-\theta )^{2H-1 -2\beta }d\theta \lesssim t^{4H-6\gamma -2\beta -1}. \end{array} $$
Therefore,
$$ \begin{array}{@{}rcl@{}} {{\int}_{0}^{t}}{{\int}_{0}^{t}}\frac{(K_{H}(t,\theta )-K_{H}(t,\theta^{\prime}))^{2}}{|\theta -\theta^{\prime}|^{1+2\beta }}d\theta^{\prime}d\theta <\infty . \end{array} $$
□
Lemma B.4
Let \(H \in \left (0, \frac {1}{2} \right )\), 0 ≤ 𝜃 < t ≤ T and \((\varepsilon _{1},\dots , \varepsilon _{n})\in \{0,1\}^{n}\) be fixed. Assume \(w_{j}+\left (H-\frac {1}{2}-\gamma \right ) \varepsilon _{j}>-1\) for all \(j=1,\dots ,n\). Then there exists a finite constant CH,T > 0 depending only on H and T such that for γ ∈ (0,H)
$$ \begin{array}{@{}rcl@{}} &&{\int}_{{\Delta}_{\theta,t}^{n}} \prod\limits_{j=1}^{n} (K_{H}(s_{j},\theta) - K_{H}(s_{j},\theta^{\prime}))^{\varepsilon_{j}} |s_{j}-s_{j-1}|^{w_{j}} ds \\ &&~\leq \left( C_{H,T} \left( \frac{\theta-\theta^{\prime}}{\theta \theta^{\prime}}\right)^{\gamma} \theta^{\left( H-\frac{1}{2} - \gamma\right)} \right)^{{\sum}_{j=1}^{n} \varepsilon_{j}} {\Pi}_{\gamma}(n) (t-\theta)^{{\sum}_{j=1}^{n} \left( w_{j} + \left( H-\frac{1}{2}-\gamma\right) \varepsilon_{j} \right) +n}, \end{array} $$
where
$$ \begin{array}{@{}rcl@{}} {\Pi}_{\gamma}(m):= \frac{{\prod}_{j=1}^{n}{\Gamma} (w_{j} +1)}{\Gamma\left( {\sum}_{j=1}^{n} w_{j} + \left( H-\frac{1}{2}-\gamma \right){\sum}_{j=1}^{n} \varepsilon_{j} + n \right)}. \end{array} $$
(61)
Proof
Recall, that for given exponents a,b > − 1 and some fixed sj+ 1 > sj we have
$$ {\int}_{\theta}^{s_{j+1}} (s_{j+1}-s_{j})^{a} (s_{j}-\theta)^{b} ds_{j} =\frac{\Gamma\left( a+1\right){\Gamma} \left( b+1\right)}{\Gamma \left( a+b+2\right)} (s_{j+1}-\theta)^{a+b+1}. $$
Due to Lemma B.3 we have that for every γ ∈ (0,H), \(0<\theta ^{\prime }<\theta <s_{j}\leq T\),
$$ \begin{array}{@{}rcl@{}} K_{H}(s_{j},\theta )-K_{H}(s_{j},\theta^{\prime})\leq C_{H,T} \frac{(\theta-\theta^{\prime})^{\gamma }}{(\theta \theta^{\prime})^{\gamma }}\theta^{H- \frac{1}{2}-\gamma }(s_{j}-\theta )^{H-\frac{1}{2}-\gamma }, \end{array} $$
for CH,T := C ⋅ cH, where cH is the constant in Eq. 14 and C > 0 is some constant merely depending on T. Consequently, we get that
$$ \begin{array}{@{}rcl@{}} &&{\int}_{\theta}^{s_{2}} |K_{H}(s_{1},\theta)-K_{H}(s_{1},\theta^{\prime})|^{\varepsilon_{1}} |s_{2}-s_{1}|^{w_{2}}|s_{1}-\theta|^{w_{1}}ds_{1} \\ &&~~~~~~\leq C_{H,T}^{\varepsilon_{1}} \frac{(\theta-\theta^{\prime})^{\gamma\varepsilon_{1} }}{(\theta \theta^{\prime})^{\gamma\varepsilon_{1}}}\theta^{\left( H-\frac{1}{2}-\gamma\right)\varepsilon_{1}}{\int}_{\theta}^{s_{2}}|s_{2}-s_{1}|^{w_{2}}|s_{1}-\theta|^{w_{1}+\left( H-\frac{1}{2}-\gamma\right)\varepsilon_{1}}ds_{1} \\ &&~~~~~~= C_{H,T}^{\varepsilon_{1}} \frac{(\theta-\theta^{\prime})^{\gamma\varepsilon_{1} }}{(\theta \theta^{\prime})^{\gamma\varepsilon_{1}}}\theta^{\left( H-\frac{1}{2}-\gamma\right)\varepsilon_{1} } \frac{\Gamma\left( \hat{w}_{1}\right){\Gamma}\left( \hat{w}_{2}\right)}{\Gamma\left( \hat{w}_{1}+\hat{w}_{2}\right)}(s_{2}-\theta)^{w_{1}+w_{2}+\left( H-\frac{1}{2}-\gamma\right)\varepsilon_{1}+1}, \end{array} $$
where
$$ \hat{w}_{1} := w_{1}+\left( H-\frac{1}{2}-\gamma\right)\varepsilon_{1}+1, \quad \hat{w}_{2}:=w_{2}+1. $$
Noting that
$$ \begin{array}{@{}rcl@{}} \prod\limits_{j=1}^{n-1} \frac{\Gamma \left( {\sum}_{l=1}^{j} w_{l} + \left( H-\frac{1}{2}-\gamma \right){\sum}_{l=1}^{j} \varepsilon_{l} + j\right){\Gamma} \left( w_{j+1}+1\right)}{\Gamma \left( {\sum}_{l=1}^{j+1} w_{l} + \left( H-\frac{1}{2}-\gamma \right){\sum}_{l=1}^{j} \varepsilon_{l} + j + 1 \right)} \leq {\Pi}_{\gamma}(n). \end{array} $$
and iterative integration yields the desired formula. □
Finally, we are able to give the proof of Proposition B.2.
Proof Proof of Proposition B.2
The integration by parts formula Eq. 49 yields that
$$ {\int}_{{\Delta}_{\theta ,t}^{n}}\left( \prod\limits_{j=1}^{n} D^{\alpha_{j}}f_{j}(s_{j},\widehat{B}_{s_{j}}){\varkappa}_{j}(s_{j})\right) ds={\int}_{\mathbb{R}^{dn}}{\Lambda}_{\alpha}^{{\varkappa} f}(\theta ,t,z)dz. $$
Taking the expectation and applying Theorem B.1 we get that
$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}\left[\left|{{\int}_{{\Delta}_{\theta ,t}^{n}}\left( \prod\limits_{j=1}^{n} D^{\alpha_{j}}f_{j}(s_{j}, \widehat{B}_{s_{j}}){\varkappa}_{j}(s_{j})\right) ds}\right|\right] \\ &&\quad \leq \frac{T^{\frac{\vert \alpha \vert}{12}}}{\sqrt{2\pi}^{dn}} ({\Psi}_{\alpha }^{{\varkappa} }(\theta, t, H, d))^{\frac{1}{2}} \prod\limits_{j=1}^{n} \left\Vert f_{j}\right\Vert_{L^{1}(\mathbb{R}^{d};L^{\infty}([0,T]))}, \end{array} $$
where
$$ \begin{array}{@{}rcl@{}} &&{\Psi}_{\alpha }^{{\varkappa} }(\theta, t, H, d) := \left( \prod\limits_{k=1}^{d}\sqrt{(2\left\vert \alpha^{(k)}\right\vert )!} \right) \\ &&\quad \times\sum\limits_{\sigma \in \mathcal{S}(n,n)}{\int}_{{\Delta}_{0,t}^{2n}} \vert {\Delta} s \vert^{- H \left( 1 + \alpha_{[\sigma({\Delta})]} \right)} \prod\limits_{j=1}^{2n} (K_{H_{m}}(s_{j},\theta)-K_{H_{m}}(s_{j},\theta^{\prime} ))^{\varepsilon_{\lbrack \sigma (j)]}}ds. \end{array} $$
Under the assumption \(-{\sum }_{k=1}^{d}H_{k} (1+ \alpha _{\lbrack \sigma (j)]}^{(k)} + \alpha _{\lbrack \sigma (j-1)]}^{(k)})+(H_{m} - \frac {1}{2}-\gamma _{m})\varepsilon _{\lbrack \sigma (j)]}>-1\) for all j = 1,..., 2n, we can apply Lemma B.4 and thus get
$$ \begin{array}{@{}rcl@{}} &&{\Psi}_{\alpha }^{{\varkappa} }(\theta, t, H, d) \\ &&\quad \leq \sum\limits_{\sigma \in \mathcal{S}(n,n)} \left( C_{T} \left( \frac{\theta -\theta^{\prime}}{\theta \theta^{\prime}}\right)^{\gamma_{m}} \theta^{(H_{m}-\frac{1}{2}-\gamma_{m})} \right)^{{\sum}_{j=1}^{2n} \varepsilon_{[\sigma(j)]}} {\Pi}_{\gamma }(2n) \\ &&\qquad \times \left( \prod\limits_{k=1}^{d}\sqrt{(2\left\vert \alpha^{(k)}\right\vert )!} \right) (t-\theta )^{-{\sum}_{k=1}^{d}H_{k} \left( 2n+4\left\vert \alpha^{(k)}\right\vert \right)+(H_{m}-\frac{1}{2}-\gamma_{m}){\sum}_{j=1}^{2n}\varepsilon_{\lbrack \sigma (j)]}+2n}, \end{array} $$
where πγ(2n) is defined as in Eq. 61. We define the constant Kd,H by
$$ \begin{array}{@{}rcl@{}} K_{d,H} := 2 \underset{j= 1, {\dots} ,2n}{\sup} {\Gamma} \left( 1-\sum\limits_{k=1}^{d} H_{k} \left( 1+ \alpha_{\lbrack \sigma (j)]}^{(k)} + \alpha_{\lbrack \sigma (j-1)]}^{(k)}\right)\right) \end{array} $$
(62)
and thus an upper bound of πγ(2n) is given by
$$ \begin{array}{@{}rcl@{}} {\Pi}_{\gamma }(2n) \leq \frac{K_{d,H}^{2n}}{2^{2n} {\Gamma}\left( -{\sum}_{k=1}^{d} H_{k}\left( 2n+4\left\vert \alpha^{(k)}\right\vert \right)+\left( H_{m} - \frac{1}{2}-\gamma_{m} \right){\sum}_{j=1}^{2n}\varepsilon_{\lbrack \sigma(j)]}+2n \right)}. \end{array} $$
Note that \({\sum }_{j=1}^{2n}\varepsilon _{\lbrack \sigma (j)]} = 2{\sum }_{j=1}^{n}\varepsilon _{j}\) and
$$ \begin{array}{@{}rcl@{}} \# \mathcal{S}(n,n) = \dbinom{2n}{n} = \frac{2^{2n}}{\sqrt{\pi}} \frac{\Gamma\left( n+\frac{1}{2} \right)}{\Gamma(n+1)} \leq 2^{2n}. \end{array} $$
Hence, it follows that
$$ \begin{array}{@{}rcl@{}} &&({\Psi}_{k}^{{\varkappa} }(\theta, t, H, d))^{\frac{1}{2}} \\ &&\quad \leq K_{d,H}^{n} \left( C_{T} \left( \frac{\theta -\theta^{\prime}}{\theta \theta^{\prime}}\right)^{\gamma_{m} }\theta^{(H_{m}-\frac{1}{2}-\gamma_{m})}\right)^{{\sum}_{j=1}^{n} \varepsilon_{j}} \\ &&\qquad \times \frac{\left( {\prod}_{k=1}^{d}\left( 2\left\vert \alpha^{(k)}\right\vert \right)!\right)^{\frac{1}{4}}(t-\theta )^{-{\sum}_{k=1}^{d}H_{k} \left( n+2\left\vert \alpha^{(k)}\right\vert \right)+\left( H_{m}-\frac{1}{2}-\gamma_{m} \right){\sum}_{j=1}^{n} \varepsilon_{j}+n}}{\Gamma\left( 2n - {\sum}_{k=1}^{d} H_{k} \left( 2n+4\left\vert \alpha^{(k)}\right\vert \right)+2\left( H_{m} - \frac{1}{2}-\gamma_{m} \right){\sum}_{j=1}^{n} \varepsilon_{j} \right)^{\frac{1}{2}}}, \end{array} $$
□
Proposition B.5
Let the functions f and ϰ be defined as in Eqs. 42 and 43, respectively. Let 0 ≤ 𝜃 < t ≤ T and
$$ {\varkappa}_{j}(s)=(K_{H_{m}}(s,\theta ))^{\varepsilon_{j}},\theta <s<t, $$
for every \(j=1, \dots , n\) with \((\varepsilon _{1}, \dots , \varepsilon _{n})\in \{0,1\}^{n}\). Let \(\alpha \in ({\mathbb {N}_{0}^{d}})^{n}\) be a multi-index and suppose that there exists δ such that
$$ -\sum\limits_{k=1}^{d} H_{k} \left( 1+2\alpha_{j}^{(k)}\right)+\left( H_{m}-\frac{1}{2}\right) \geq \delta >-1 $$
for all \(j= 1, \dots , n\) and d ≥ 1. Then there exist constants CT (depending on T) and Kd,H (depending on d and H) such that for any 0 ≤ 𝜃 < t ≤ T we have
$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}\left[\left|{{\int}_{{\Delta}_{\theta ,t}^{n}}\left( \prod\limits_{j=1}^{n} D^{\alpha_{j}}f_{j}(s_{j}, \widehat{B}_{s_{j}}){\varkappa}_{j}(s_{j})\right) ds}\right|\right] \\ &&~ \leq \frac{K_{d,H}^{n} \cdot T^{\frac{\vert \alpha \vert}{12}}}{\sqrt{2\pi}^{dn}} \left( C_{T} \theta^{(H_{m}-\frac{1}{2})}\right)^{{\sum}_{j=1}^{n} \varepsilon_{j}} \prod\limits_{j=1}^{n} \left\Vert f_{j}(\cdot ,z_{j})\right\Vert_{L^{1}(\mathbb{R}^{d};L^{\infty }([0,T]))} \\ &&\quad \times \frac{\left( {\prod}_{k=1}^{d} \left( 2\left\vert \alpha^{(k)}\right\vert\right)!\right)^{\frac{1}{4}}(t-\theta )^{-{\sum}_{k=1}^{d} H_{k} \left( n+2\left\vert \alpha^{(k)}\right\vert \right)+ \left( H_{m}-\frac{1}{2}\right){\sum}_{j=1}^{n} \varepsilon_{j}+n}}{\Gamma(2n-{\sum}_{k=1}^{d}H_{k}(2n+4\left\vert \alpha^{(k)}\right\vert )+2(H_{m} - \frac{1}{2}){\sum}_{j=1}^{n} \varepsilon_{j})^{\frac{1}{2}}}. \end{array} $$
The proof of Proposition B.5 is similar to the one of Proposition B.2 by using the subsequent lemma instead of Lemma B.4 and thus it is omitted in this manuscript.
Lemma B.6
Let \(H \in \left (0,\frac {1}{2}\right )\), 0 ≤ 𝜃 < t ≤ T and \((\varepsilon _{1},\dots , \varepsilon _{n})\in \{0,1\}^{n}\) be fixed. Assume \(w_{j}+\left (H-\frac {1}{2}\right ) \varepsilon _{j}>-1\) for all \(j=1,\dots ,n\). Then there exists a finite constant CH,T > 0 depending only on H and T such that
$$ \begin{array}{@{}rcl@{}} &&{\int}_{{\Delta}_{\theta,t}^{n}} \prod\limits_{j=1}^{n} (K_{H}(s_{j},\theta))^{\varepsilon_{j}} |s_{j}-s_{j-1}|^{w_{j}} ds \\ &&~\leq \left( C_{H,T} \theta^{\left( H-\frac{1}{2}\right)} \right)^{{\sum}_{j=1}^{n} \varepsilon_{j}} {\Pi}_{0}(n) (t-\theta)^{{\sum}_{j=1}^{n} \left( w_{j} + \left( H-\frac{1}{2}\right) \varepsilon_{j} \right) + n}, \end{array} $$
where π0 is defined in Eq. 61.
Proof
Using similar arguments as in the proof of Lemma B.3 we get the following estimate
$$ |K_{H}(s_{j},\theta)| \leq C_{H,T} |s_{j}-\theta|^{H-\frac{1}{2}}\theta^{H-\frac{1}{2}} $$
for every 0 < 𝜃 < sj < T and CH,T := C ⋅ cH, where cH is the constant in Eq. 14 and C > 0 is some constant merely depending on T. Thus,
$$ \begin{array}{@{}rcl@{}} &&{\int}_{\theta}^{s_{2}} (K_{H}(s_{1},\theta))^{\varepsilon_{1}}|s_{2}-s_{1}|^{w_{2}}|s_{1}-\theta|^{w_{1}}ds_{1} \\ &&\quad \leq C_{H,T}^{\varepsilon_{1}} \theta^{\left( H-\frac{1}{2}\right)\varepsilon_{1}} {\int}_{\theta}^{s_{2}} |s_{2}-s_{1}|^{w_{2}} |s_{1}-\theta|^{w_{1}+\left( H-\frac{1}{2}\right)\varepsilon_{1}}ds_{1} \\ &&\quad = C_{H,T}^{\varepsilon_{1}} \theta^{\left( H-\frac{1}{2}\right)\varepsilon_{1}} \frac{\Gamma\left( w_{1} +\left( H-\frac{1}{2}\right)\varepsilon_{1}+1\right){\Gamma}\left( w_{2}+1\right)}{\Gamma\left( w_{1} + w_{2} + \left( H-\frac{1}{2}\right)\varepsilon_{1}+2\right)}(s_{2}-\theta)^{w_{1}+w_{2}+\left( H-\frac{1}{2}\right)\varepsilon_{1}+1}. \end{array} $$
Proceeding similar to the proof of Lemma B.4 yields the desired estimate. □
Remark B.7
Note that
$$ \prod\limits_{k=1}^{d} \left( 2\left\vert \alpha^{(k)}\right\vert \right)! \leq \sqrt{2\pi}^{d} e^{\frac{\vert \alpha \vert}{2}} \frac{\Gamma\left( \frac{5}{2} \vert \alpha \vert +1\right)}{\sqrt{5 \pi \vert \alpha \vert}}. $$
Indeed, since for n ≥ 1 sufficiently large we have by Stirling’s formula that
$$ \begin{array}{@{}rcl@{}} \sqrt{2\pi n} \left( \frac{n}{e} \right)^{n} \leq n! \leq e^{\frac{1}{12n}} \sqrt{2\pi n} \left( \frac{n}{e} \right)^{n}, \end{array} $$
we get by assuming without loss of generality that |α(k)|≥ 1 for all 1 ≤ k ≤ d, that
$$ \begin{array}{@{}rcl@{}} \prod\limits_{k=1}^{d} \left( 2 \vert \alpha^{(k)} \vert \right)! &\leq& \prod\limits_{k=1}^{d} e^{\frac{1}{24 \vert \alpha^{(k)} \vert}} \sqrt{4 \pi \vert \alpha^{(k)} \vert} \left( \frac{2 \vert \alpha^{(k)} \vert}{e} \right)^{2 \vert \alpha^{(k)} \vert} \\ &\leq& e^{\frac{d}{24}} \sqrt{\frac{8}{5}\pi}^{d} \prod\limits_{k=1}^{d} \left( \frac{5}{2} \vert \alpha^{(k)} \vert \right)^{\frac{\vert \alpha^{(k)} \vert}{2}} \left( \frac{\frac{5}{2} \vert \alpha^{(k)} \vert}{e} \right)^{2 \vert \alpha^{(k)} \vert} \\ &\leq& \sqrt{2\pi}^{d} \prod\limits_{k=1}^{d} \left( \frac{\frac{5}{2} \vert \alpha \vert}{e} \right)^{\frac{5}{2} \vert \alpha^{(k)} \vert} e^{\frac{\vert \alpha^{(k)} \vert}{2}} \\ &\leq& \sqrt{2\pi}^{d} e^{\frac{\vert \alpha \vert}{2}} \left( \frac{\frac{5}{2} \vert \alpha \vert}{e} \right)^{\frac{5}{2} \vert \alpha \vert} \leq \sqrt{2\pi}^{d} e^{\frac{\vert \alpha \vert}{2}} \frac{\Gamma\left( \frac{5}{2} \vert \alpha \vert +1\right)}{\sqrt{5 \pi \vert \alpha \vert}}. \end{array} $$
Appendix C: Technical Results
The following technical result can be found in [26].
Lemma C.1
Assume that X1,...,Xn are real centered jointly Gaussian random variables, and \({\Sigma } =(\mathbb {E}[X_{j}X_{k}])_{1\leq j,k\leq n}\) is the covariance matrix, then
$$ \begin{array}{@{}rcl@{}} \mathbb{E}[|{ X_{1}|\vert ... \vert X_{n}}|] \leq \sqrt{\text{perm}{\Sigma}}, \end{array} $$
where perm (A) is the permanent of a matrix A = (aij)1≤i,j≤n defined by
$$ \begin{array}{@{}rcl@{}} \text{perm}{A}=\sum\limits_{\pi \in \mathcal{S}_{n}} \prod\limits_{j=1}^{n}a_{j,\pi (j)} \end{array} $$
for the symmetric group \(\mathcal {S}_{n}\).
The next lemma corresponds to [12, Lemma 2]:
Lemma C.2
Let Z1,...,Zn be mean zero Gaussian random variables which are linearly independent. Then for any measurable function \(g:\mathbb {R}\longrightarrow \mathbb {R}_{+}\) we have that
$$ \begin{array}{@{}rcl@{}} &{\int}_{\mathbb{R}^{n}}g(v_{1}) e^{-\frac{1}{2} \text{Var}{{\sum}_{j=1}^{n}v_{j}Z_{j}}} dv_{1}...dv_{n} =\frac{(2\pi )^{\frac{n-1}{2}}}{(\det \text{Cov}{Z_{1},...,Z_{n}})^{\frac{1}{2}}}{\int}_{\mathbb{R}} g\left( \frac{v}{\sigma_{1}}\right) e^{-\frac{v^{2}}{2}} dv, \end{array} $$
where \({\sigma _{1}^{2}}:=\text {Var}{Z_{1}\left \vert Z_{2},...,Z_{n}\right .}\).
Remark C.3
Note that here linearly independence is meant in the sense that \(\det \text {Cov}Z_{1},...,\) Zn≠ 0.
Lemma C.4
Let a ∈ ℓp, \(1 \leq p < \infty \). Then, for every n ≥ 1 and d ≥ 1
$$ \begin{array}{@{}rcl@{}} \sum\limits_{k_{1},\dots, k_{n} = 1}^{d} \prod\limits_{j=1}^{n} a_{k_{j}} = \left( \sum\limits_{k=1}^{d} a_{k} \right)^{n}, \end{array} $$
(63)
and
$$ \begin{array}{@{}rcl@{}} \underset{d \to \infty}{\lim} \sum\limits_{k_{1},\dots, k_{n}}^{d} \prod\limits_{j=1}^{n} \vert a_{k_{j}} \vert^{p} = \left( \Vert a \Vert_{\ell^{p}} \right)^{n}. \end{array} $$
(64)
Proof
We proof Eq. 63 by induction. For n = 1 the result holds. Therefore we assume that Eq. 63 holds for n and we show that it also holds for n + 1. Thus, we get by the induction hypothesis that
$$ \begin{array}{@{}rcl@{}} \sum\limits_{k_{1},\dots, k_{n+1} = 1}^{d} \prod\limits_{j=1}^{n+1} a_{k_{j}} &=& \sum\limits_{k_{n+1}=1}^{d} a_{k_{n+1}} \left( \sum\limits_{k_{1},\dots, k_{n} = 1}^{d} \prod\limits_{j=1}^{n} a_{k_{j}} \right) \\ &=& \sum\limits_{k_{n+1}=1}^{d} a_{k_{n+1}} \left( \sum\limits_{k = 1}^{d} a_{k} \right)^{n} = \left( \sum\limits_{k = 1}^{d} a_{k} \right)^{n+1}. \end{array} $$
Equation 64 is an immediate consequence of Eq. 63 and the continuity of the function f(x) = xn for fixed n ≥ 1. □
The subsequent lemmas are due to [4].
Lemma C.5
Let \((X_{1},\dots ,X_{n})\) be a mean-zero Gaussian random vector. Then,
$$ \begin{array}{@{}rcl@{}} \det\left( \text{Cov}\left( {X_{1},\dots,X_{n}}\right)\right) = \text{Var}\left( {X_{1}}\right) \text{Var}\left( {X_{2} \vert X_{1}} {\cdots} \text{Var}({X_{n} \vert X_{n-1},{\dots} ,X_{1}}\right). \end{array} $$
Lemma C.6
For any square integrable random variable X and σ-algebras \(\mathcal {G}_{1} \subset \mathcal {G}_{2}\)
$$ \begin{array}{@{}rcl@{}} \text{Var}\left( X\vert \mathcal{G}{_{1}}\right) \geq \text{Var}\left( {X\vert \mathcal{G}_{2}}\right). \end{array} $$