$(\omega,t,x)$, usually the argument $\omega \in \Omega$ will be omitted.

**Preliminary results** {#section 1}
=======================

First we introduce a few elementary facts on the fractional derivatives. The reader can find further details in [@EIK] and references therein. Recall that $\beta\in (0,1)$ and $$k_{\beta}(t):=t^{\beta-1}\Gamma(\beta)^{-1}, \quad t>0.$$ Let $T>0$. If $f$ is absolutely continuous on $[0, T]$ with $f(0)=0$ then $$\label{2014.1.23.2}
\frac{d}{dt} (k_{\beta}*f)=k_{\beta}* \frac{d}{dt}f, \quad t \in [0, T].$$ For functions $\varphi\in L_1([0,T])$, the Riemann-Liouville fractional integral of the order $\beta\in (0,1)$ is defined by $$I^{\beta} \varphi (t)=k_{\beta}*\varphi (t)=\frac{1}{\Gamma(\beta)}\int^t_0 (t-s)^{\beta-1}\varphi(s)ds.$$ Note that by Jensen’s inequality $$|I^\beta \varphi (t)|^p \leq \frac{1}{(\Gamma (\beta))^p} \left(\frac{t^\beta}{\beta}\right)^{p-1}
\int_0^t (t-s)^{\beta-1} |\varphi (s)|^p ds.$$ Thus it follows that for any $p\in [1,\infty]$, $$\label{eqn 7.03.1}
\|I^{\beta}\varphi\|_{L_p([0,T])}\leq N(T,\beta) \|\varphi\|_{L_p([0,T])}.$$ Consequently, if $\varphi_n \to \varphi$ in $L_p([0,T])$ then $I^{\beta}\varphi_n$ also converges to $I^{\beta} \varphi$ in $L_p([0,T])$. Also one can prove that if $f_n(\omega,t)$ converges in probability uniformly in $[0,T]$ then so does $I^{\beta}f_n$.

If $I^{1-\beta}\varphi$ is absolutely continuous, then Riemann-Liouville derivative of order $\beta$ is defined by $$\label{Riemann-Liouville}
D^{\beta}_t\varphi (t)=\frac{d}{dt} (I^{1-\beta}\varphi)(t).$$ If $\varphi$ is continuous and $I^{1-\beta}\varphi$ is absolutely continuous, the generalized functional derivative (or the Caputo derivative) of order $\beta$ is given by $$\begin{aligned}
          \label{caputo}
\partial^{\beta}_t\varphi (t)&=&D^{\beta}_t(\varphi-\varphi(0))
= D^{\beta}_t\varphi (t)-\frac{\varphi(0)}{t^{\beta}\Gamma(1-\beta)}.\end{aligned}$$ It 