Restoration of Well-Posedness of Infinite-Dimensional Singular ODE’s via Noise

Abstract

In this paper we aim at generalizing the results of A. K. Zvonkin (That removes the drift, 22, 129, 41) and A. Y. Veretennikov (Theory Probab. Appl., 24, 354, 39) on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

Appendices

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 (B_t)_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 (B_t)_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 (B_t)_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 v_s,s ≥ 0, with v₀ = 1 the Haar basis of L²([0, 1]). Define for any $0<\alpha <\frac {1}{2}$ the operator A_α 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

$$ A_{\alpha }1=1. $$

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 D^k be the Malliavin derivative in the direction of the k-th component of (B_t)_t∈[0,T]. In addition, let $0<\alpha _{k}<\beta _{k}<\frac {1}{2}$ and γ_k > 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 μ_s,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 v_s,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 e_k,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 α_j 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 (u_n+ 1,...,u_2n)↦(−u_n+ 1,...,−u_2n) 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 permA_k 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 S_m 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 H_k 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,

$$ \begin{array}{@{}rcl@{}} \prod\limits_{j=1}^{2n}\sigma_{j}^{-2\alpha_{[\sigma (j)]}^{(k)}} \leq \mathfrak{K}_{H_{k}}^{-2 \vert \alpha^{(k)} \vert} T^{4 H_{k} \vert \alpha^{(k)} \vert} \vert {\Delta} s \vert^{-2H_{k} \alpha_{[\sigma({\Delta})]}^{(k)}}. \end{array} $$

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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 f_j, 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 C_R. 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

$$ \begin{array}{@{}rcl@{}} -\sum\limits_{k=1}^{d} H_{k} \left( 1+2\alpha_{j}^{(k)}\right)+\left( H_{m} - \frac{1}{2} - \gamma_{m}\right)\geq \delta >-1 \end{array} $$

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for all $j= 1, {\dots } n$ and d ≥ 1, where γ_m ∈ (0,H_m) is sufficiently small. Then there exist constants C_T (depending on T) and K_d,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

$$ \frac{y^{-\alpha} -x^{-\alpha}}{(x-y)^{\gamma}} \leq C y^{-\alpha-\gamma}, $$

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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 C_H,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

$$ \begin{array}{@{}rcl@{}} {\Pi}_{\gamma}(m):= \frac{{\prod}_{j=1}^{n}{\Gamma} (w_{j} +1)}{\Gamma\left( {\sum}_{j=1}^{n} w_{j} + \left( H-\frac{1}{2}-\gamma \right){\sum}_{j=1}^{n} \varepsilon_{j} + n \right)}. \end{array} $$

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Proof

Recall, that for given exponents a,b > − 1 and some fixed s_j+ 1 > s_j 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 C_H,T := C ⋅ c_H, where c_H 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 π_γ(2n) is defined as in Eq. 61. We define the constant K_d,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 π_γ(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 C_T (depending on T) and K_d,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 C_H,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 < s_j < T and C_H,T := C ⋅ c_H, where c_H 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,...,X_n 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 = (a_ij)_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,...,Z_n 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 $\det \text {Cov}Z_{1},...,$ Z_n≠ 0.

Lemma C.4

Let a ∈ ℓ^p, $1 \leq p < \infty$. 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 $\mathcal {G}_{1} \subset \mathcal {G}_{2}$

$$ \begin{array}{@{}rcl@{}} \text{Var}\left( X\vert \mathcal{G}{_{1}}\right) \geq \text{Var}\left( {X\vert \mathcal{G}_{2}}\right). \end{array} $$