Expected Value of an ARMA-GARCH Model An ARMA(p,q) model is given by
$ \qquad \qquad Y_t = c + \sum\limits_{i=1}^{p}\varphi_iY_{t-i}+\sum\limits_{i=1}^{q}\theta_i\varepsilon_{t-i} + \varepsilon$
with $\varepsilon_t \sim N(0,\sigma^2)$.
Let's say our model is simply an ARMA(1,1) model. The expected value for tomorrow's forecast then is
$\qquad \qquad E[Y_{t+1}] = E[c+\varphi_1Y_t+\theta_1\varepsilon_t + \varepsilon_{t+1}]$
$\qquad \qquad \qquad \qquad  = c+\varphi_1Y_t+\theta_1\varepsilon_t + E[\varepsilon_{t+1}]$
$\qquad \qquad \qquad \qquad = c+\varphi_1Y_t+\theta_1\varepsilon_t$
with $Y_t$ being our time series. Correct so far?
Now, a GARCH(p,q) model is given by
$\qquad \qquad X_t = \sigma_t\varepsilon_t$
$\qquad \qquad \sigma_t^2=\omega + \sum\limits_{i=1}^{q}\alpha_iX_{t-i}^2+\sum\limits_{i=1}^{p}\beta_i\sigma_{t-i}^2$
with $\varepsilon_t \sim N(0,\sigma^2)$ again and with $X_t$ being our time series here right?
Now my questions arise when looking at an ARMA(n,m)-GHARCH(p,q) model:
$\qquad \qquad Y_t = c + \sum\limits_{i=1}^{n}\varphi_iY_{t-i}+\sum\limits_{j=1}^{m}\theta_jX_{t-j}+X_t$.
I understand that $X_t$ ist the underlying time series of the garch model, but isn't that simply the time series we are looking at and trying to just fit to a GARCH-Model?
Next, when looking at tomorrow's expected value from the ARMA(1,1)-GARCH(1,1) model, it is
$\qquad \qquad E[Y_{t+1}] = ... = c+\varphi_1Y_t+\theta_1X_t + E[X_{t+1}]$
$\qquad \qquad \qquad \qquad = c+\varphi_1Y_t+\theta_1X_t + E[\sigma_{t+1}\varepsilon_{t+1}]$
$\qquad \qquad \qquad \qquad = c+\varphi_1Y_t+\theta_1X_t + E[\sigma_{t+1}]E[\varepsilon_{t+1}]$
$\qquad \qquad \qquad \qquad = c+\varphi_1Y_t+\theta_1X_t + E[\sigma_{t+1}]*0$
$\qquad \qquad \qquad \qquad = c+\varphi_1Y_t+\theta_1X_t $
Aren't $X_t$ and $Y_t$ basically the same time series were we tried to apply these models to? And where is $\sigma^2$ now, I thought our goal was to use tomorrow's forecasted variance in our combined model.
Thanks in advance for reading and taking your time.
 A: Consider a dependent variable $Y_t$ which is an original time series or a transformation (such as log-returns) thereof.

*

*ARMA models its conditional mean as time varying in a particular way.
When the cond. mean of $Y_t$ follows ARMA, the cond. variance of $Y_t$ can be constant, follow GARCH or whatever.
(See this answer for details.)

*GARCH models its conditional variance as time varying in a particular way.
When the cond. variance of $Y_t$ follows GARCH, the cond. mean of $Y_t$ can be constant, follow ARMA or whatever.
(See this answer for details.)

Consider a basic example of ARMA(1,1)-GARCH(1,1):
\begin{aligned}
y_t &= c+\varphi_1 y_{t-1}+\varepsilon_t+\theta_1\varepsilon_{t-1}, \\
\varepsilon_t &= \sigma_t z_t, \\
\sigma^2_t &= \omega+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2, \\
z_t &\sim i.i.d(0,1).
\end{aligned}
Given the estimated model, a one-step-ahead estimate/forecast of the conditional mean and variance of $Y_t$ are the following:
\begin{aligned}
\mathbb{E_t}(y_{t+1}) &= c+\hat\varphi_1 y_{t}+\hat\theta_1\hat\varepsilon_{t}, \\
\text{Var}_t(y_{t+1}) &= c+\hat\alpha_1\hat\varepsilon_{t}^2+\hat\beta_1\hat\sigma_{t}^2.
\end{aligned}
