 $(S_0, Z)\sim \mathcal{N}(\mu_{S_0Z}, K_{S_0Z})$ with mean $\mu_{S_0Z} = (0,0)$ and covariance $$K_{S_0 Z} = \begin{bmatrix} \mathbb{E}[S_0^0] & \mathbb{E}[S_0 Z] \\ \mathbb{E}[Z S_0]  & \mathbb{E}[Z^0]  \end{bmatrix}