Variance of ARMA process?

Question

Problem
Given $ARMA(1,1)$ stationary process $$x_t = 0.7 x_{t-1} + u_t + 0.2 u_{t-1} $$ where $u_t$ is white noise, with standard deviation $\sigma(u_t) = 4$

Note, stationarity of $x_t$ implies that $$ \Bbb E(x_t) = const = \mu$$ $$ \Bbb Cov(x_t, x_{t+k}) = \gamma_k$$

Find $ \Bbb Var(x_{t+1}|x_t, u_t)$

My ideas
We could substitute the expression of $x_t$ into the variance (I can do that), but I do not really understand what the fact that it is $ARMA(1,1)$ process gives us?

Answer 1

Actually, the fact that it is $ARMA(1,1)$ process does not change the procedure much. $$ Var( x_{t+1}| x_y, u_t) = Var(0.7x_t + u_{t+1} + 0.2u_t |x_t, u_t) = Var(u_{t+1}) = 16$$ as $ 0.7x_t$ and $ 0.2u_t$ are considered as constants so disappear from the variance and $u_{t+1}$ is independent of $x_t, u_t$ so it is usual non-conditional variance.