For a stochastic process $G(t)$, one technique of making the process continuous is to define a process $G^m(t) := \int_0^t m e^{m (s-t)} G(s) ds$. I would like to understand the intuition behind such a definition and verify that it is in fact continuous almost surely. Furthermore, it is claimed that $G^m$ converges to $G$ in the $L_2$ sense as $m \rightarrow \infty$.
Are there more natural methods to smooth out stochastic processes?