I was reading Zhang and Wang 2023 and I have some doubts regarding it. The realized Stochastic Volatility Model is expressed as follows:
$$\begin{matrix} y_t = \exp \big( \frac{h_t}{2} \big) \epsilon_t \quad \quad \quad & & & \epsilon_t\sim \text{N}(0,1), \\[12pt] z_t = \varsigma+h_t+u_t \quad \quad \ \ \ & & & \ \ u_t\sim N(0,\sigma^2_u), \\[12pt] h_{t+1} = \mu+\phi(h_t-\mu)+\eta_t \ & & & \ \ \ \eta_t\sim N(0,\sigma^2_\eta), \\[12pt] h_1 = \mu+\epsilon_0 \quad \quad \quad \quad \quad & & & \ \ \ \ \ \ \epsilon_0 \sim N(0,\frac{\sigma^2_\eta}{1-\phi^2}). \\[12pt] \end{matrix}$$
To ensure the strict stationarity and iterative nature of the stochastic process, the persistence parameter $|\phi| < 1$ is assumed in the logarithmic volatility equation $h_t$ and set $h_1$
My question: Why do we have the expectation: $$\mathbb{E}(z_t) = \varsigma+ \frac{\mu}{1-\phi}?$$
I cannot see why the expectation of $h_t$ should be $\frac{\mu}{1-\phi}$. $h_t$ should follow an AR(1) process and the expected value of an AR(1) process with drift is $\frac{\mu}{1-\phi}$ but in this case we have an additional term given by $-\phi \mu$. Am I missing something? Can someone explain why the $\mathbb{E}(z_t)$ is given by the above result?
my attempt was: since the AR(1) process is stationary given that $|\phi|<1$ the mean is constant over time and [\begin{align*} \mathbb{E}(h) - \phi\mathbb{E}(h) &= \mu - \phi\mu \\ (1-\phi)\mathbb{E}(h) &= \mu(1-\phi). \end{align*}] which gives $\mathbb{E}(h) = \mu$ why they got $\frac{\mu}{1-\phi}$?
