I wonder if this type of equation fits in the formalism of SDE
$$dX_{t}=(\mu_0+\eta_t)dt+\sigma dW_{t}$$
where $\eta_t$ is uncorrelated Gaussian noise.
First I wrote this, but I don't know if it makes sense.
$$dX_{t}=(\mu_0+dW_{1t})dt+\sigma dW_{2t}$$
Wiener Process 1 & 2 are uncorrelated.
Then I though of making a system of two equations,
$$dX_{t}=(\mu_0+\eta_t)dt+\sigma dW_{2t}\\ d\eta_{t}=dW_{1t}$$
But $\eta$ is not uncorrelated, is exactly the Wiener Process.
How should I proceed?