Let $X_t$ denote the position of this stochastic oscillator, and $V_t$ denote its velocity. A meaningful interpretation of the quoted differential equation is $ X_t = x_0 + \int_0^t V_s \mathrm{d} s, \quad V_t = v_0 - \omega^2 \int_0^t X_s \mathrm{d} s + \sigma W_t $ where $W_t$ denotes the standard Wiener process. The deterministic case corresponds to $\sigma = 0$.
In the differential form this SDE reads: $ \mathrm{d} \begin{pmatrix} X_t \\ V_t \end{pmatrix} = \hat{B}.\begin{pmatrix} X_t \\ V_t \end{pmatrix} \mathrm{d} t + \begin{pmatrix} 0 \cr \sigma \end{pmatrix} \mathrm{d} W_t $ where $\hat{B} = \begin{pmatrix} 0 & 1 \\ -\omega^2 & 0 \end{pmatrix}$. This is an exactly solvable system, with $(X_t, V_t)$ being a Guassian process. It is solved using Ito lemma: $ \mathrm{d} \left( \mathrm{e}^{-\hat{B} t}\cdot \begin{pmatrix} X_t \\ V_t \end{pmatrix} \right) = \mathrm{e}^{-\hat{B} t}\cdot \begin{pmatrix} 0 \cr \sigma \end{pmatrix} \mathrm{d} W_t $ Which implies $ \begin{pmatrix} X_t \\ V_t \end{pmatrix} = \mathrm{e}^{\hat{B} t} \cdot \begin{pmatrix} x_0 \\ v_0 \end{pmatrix} + \mathrm{e}^{\hat{B} t} \cdot \int_0^t \mathrm{e}^{-\hat{B} s} \begin{pmatrix} 0 \cr \sigma \end{pmatrix} \mathrm{d} W_s $ Using $ \mathrm{e}^{-\hat{B} t} = \begin{pmatrix} \cos(\omega t) & - \frac{\sin(\omega t)}{\omega} \cr \omega \sin(\omega t) & \cos(\omega t) \end{pmatrix} $ we arrive at the solution: $ \begin{eqnarray} X_t &=& x_0 \cos(\omega t) + \frac{v_0}{\omega} \sin(\omega t) + \frac{\sigma}{\omega} \int_0^t \sin((t-s) \omega) \mathrm{d} W_s \\ V_t &=& v_0 \cos(\omega t) - x_0 \omega \sin(\omega t) + \sigma \int_0^t \cos((t-s) \omega) \mathrm{d} W_s \end{eqnarray} $ Since $(X_t, V_t)$ is Gaussian, value of the process at any $t$ is a multinormal random vector with mean and covariance matrix found by using Ito isometry: $ \begin{eqnarray} \mathbb{E}(X_t) &=& x_0 \cos(\omega t) + \frac{v_0}{\omega} \sin(\omega t) \\ \mathbb{E}(V_t) &=& v_0 \cos(\omega t) - x_0 \omega \sin(\omega t) \\ \mathbb{Var}(X_t) &=& \frac{\sigma^2}{\omega^2} \int_0^t \sin^2(\omega (t-s)) \mathrm{d} s = \frac{\sigma^2}{\omega^2} \left( \frac{t}{2} - \frac{\sin(2 \omega t)}{4 \omega} \right) \\ \mathbb{Var}(V_t) &=& \sigma^2 \int_0^t \cos^2(\omega (t-s)) \mathrm{d} s = \sigma^2 \left( \frac{t}{2} + \frac{\sin(2 \omega t)}{4 \omega} \right) \\ \mathbb{Cov}(X_t,V_t) &=& \frac{\sigma^2}{\omega} \int_0^t \sin(\omega (t-s)) \cos(\omega (t-s)) \mathrm{d}s = \sigma^2 \frac{ \sin^2(\omega t)}{2 \omega^2} \end{eqnarray} $