In the present paper we study moving average processes $X_t=\int \phi(t-s)-\psi(-s)\, dW_s,$ where $\phi$ and $\psi$ are deterministic functions and $(W_t)_{t\in\mathbb{R}}$ is a Wiener process. Necessary and sufficient condition on $(\phi,\psi)$ are provided for $(X_t)_{t\geq 0}$ to be an $(\mathcal{F}^{X,\infty}_t)_{t\geq 0}$-semimartingale, where $\mathcal{F}^{X,\infty}_t:=\sigma(X_s:s\in (-\infty,t])$. Our results are constructive - meaning that they provide a simple method to obtain $\phi$ and $\psi$ for which $(X_t)_{t\geq 0}$ is an $(\mathcal{F}^{X,\infty}_t)_{t\geq 0}$-semimartingale or an $(\mathcal{F}^{X,\infty}_t)_{t\geq 0}$-Wiener process. Several examples are considered.

In the last part of the paper we study general Gaussian processes with stationary increments, $(X_t)_{t\in\mathbb{R}}$. We provide necessary and sufficient conditions on spectral measure for $(X_t)_{t\geq 0}$ to be an $(\mathcal{F}^{X,\infty}_t)_{t\geq 0}$-semimartingale.