Prediction of non zero mean ARMA

Starting from an $ARMA$ process $y(t)=W(z)e(t)=\frac{C(z)}{A(z)}e(t)$, wehere $e(t) \sim WN(\mu,\lambda^2)$ that is canonical and minimum phase we need to compute the unbiased processes: $$ \tilde{y}=y(t)-M_y\newline \tilde{e}(t)=e(t)-\mu $$ The resulting arma process $\tilde{y}(t)=\frac{C(z)}{A(z)}\tilde{e}(t)$ is still canonical and minimum phase.

We can compute its predictor $\hat{\tilde{y}}(t+k|t)$ and then: $$\hat{y}(t+k|t)=\hat{\tilde{y}}(t+k|t)+m_y=\frac{F(z)}{C(z)}y(t)+(1-\frac{F(1)}{C(1)})m_y$$