This article is concerned with a forward problem for the following sub-diffusion equation driven by standard Brownian motion
\[
({}^{C}D_{t}^{\gamma} + A)u(t) = f(t) + B(t)\dot{W}(t), \quad t \in J := (0, T),
\]
where ${}^{C}D_{t}^{\gamma}$ is the conformable derivative, $\gamma \in \left(\tfrac{1}{2}, 1\right]$. Under some flexible assumptions on $f$, $B$ and the initial data, we investigate the existence, regularity, and continuity of the solution on two spaces $L^{r}(J; L^{2}(\Omega, H^{\alpha}))$ and $C^{\alpha}(J; L^{2}(\Omega, H))$ separately.