t \in[0 ,T]$, we will assume there exists $C < \infty$ such that $$\begin{aligned}
 \label{R1DBound}
\left\|  R(t, \cdot) - R (s, \cdot) \right\| _{q-var; [0, T]} \leq C \left|  t - s\right| ^{\frac{1}{\rho}}.\end{aligned}$$ This bound will be used to control the $L^{2} (\Omega)$ norm of the iterated integrals. An immediate consequence of the bound is illustrated in the following lemma (Lemma 2.14 in [@cl2018]).

\[RdotRhoVar\] Let $X$ be a continuous, centered Gaussian process in $\mathbb{R}$ and assume its covariance function satisfies $$\begin{aligned}
\left\|  R (t , \cdot) - R (s , \cdot)\right\| _{q-var; [0, T]} \leq C \left|
t -s\right| ^{\frac{1}{\rho}}, \quad\forall s < t \in[0, T],\end{aligned}$$ for some $q, \rho\geq1$. Then

(i) $R(t) := R(t, t)$ is of bounded $\rho$-variation.

(ii) For $p > 2 \rho$, $X$ has a $\frac{1}{p}$-Hölder continuous modification.

Volterra processes and fractional Brownian motion
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A *Volterra kernel* $K$ is a square-integrable function $K:\left[ 0,T\right]^2 \rightarrow \mathbb{R}$ such that $K(t,s) = 0 \;\forall s\geq t$. Associated with any Volterra kernel is a lower triangular, Hilbert-Schmidt operator $\mathbb{K}: L^{2}\left(  \left[  0,T \right] \right) \rightarrow L^{2}\left(  \left[  0,T\right]  \right)  $ given by $$\begin{aligned}
\mathbb{K} \left(  f\right)  \left(  \cdot\right)  =\int_{0}^{T}K\left(  \cdot,s\right) f\left(  s\right)  \text{ for all }f\in L^{2}\left(  