\left[  0,T\right] \right) .\end{aligned}$$ Given a standard Brownian motion $B$ and a Volterra kernel $K,$we define a Volterra process $X = \left( X_{t} \right)_{t\in\left[ 0,T\right] }$ as the Itô integral $$\begin{aligned}
 \label{voltrep}
X_{t}=\int_{0}^{T}K(t,s)\,\mathrm{d}B_{s};\end{aligned}$$ this is a centered Gaussian process with covariance function $$\begin{aligned}
R(s,t) = \int_{0}^{t\wedge s}K(t,r)K(s,r)\,\mathrm{d}r.\end{aligned}$$

(i) Standard fractional Brownian motion $B^{H}$, with Hurst parameter $H\in(0,1)$, is the centered Gaussian process with covariance function $$\begin{aligned}
     \label{covfbm}
    R\left(  s,t\right)  =\frac{1}{2}\left(  s^{2H}+t^{2H}-\left\vert t-s \right\vert^{2H}\right).\end{aligned}$$ It is well-known that $B^{H}$ has a Volterra representation of the form where the kernel can be expressed as a particular hypergeometric function; cf. [@du1999].

(ii) The Riemann-Liouville process with Hurst parameter $H\in\left( 0, 1 \right) $ is determined by the kernel $K(t,s) := C_{H}(t-s)^{H-\frac{1}{2}} \mathds{1}_{[0, t)}(s)$. Like the fractional Brownian motion, it is a self-similar process with variance $t^{2H}$; however, it does not have stationary increments.

We will need the following condition on the kernel $K$.

\[amnCond\] There exists constants $C < \infty$ and $\alpha \in \left[ 0, \frac{1}{4} \right)$ such that

(i) $\left|  K(t,s) \right|  \leq C s^{-\alpha} (t - s)^{-\alpha}$ for all $0 < s < t \leq T$.

(ii) $\frac{\partial K(t,s)}{\partial t}$ exists for all $0 < s < t \leq T$ and 