GLMs Driven by Latent Processes

$E(v_{t+l}|v_t)=\exp ({\rho }^l{\sigma }^2(1-{\rho }^l)/2+{\rho }^l\ln v_t)$
Proof: $v_t\sim \text{log-normal}AR(1)$, $a_t=\ln v_t\sim \text{normal}AR(1{)}_{N(-{\sigma }^2/2,{\sigma }^2)}$, $E_v=E(\ln v_t)=-\frac{{\sigma }^2}{2}$
$a_{t+l}-E_v=\rho (a_{t+l-1}-E_v)+{\epsilon }_{t+l-1}$ $={\rho }^2(a_{t+l-2}-E_v)+\rho {\epsilon }_{t+l-2}+{\epsilon }_{t+l-1}$ $=\dots$ $={\rho }^l(a_t-E_v)+\underset{E(\mathcal{E})=0}{\underbrace{\mathcal{E}}}$, $\therefore a_{t+l}={\rho }^l{a}_t+E_v(1-{\rho }^l)+\mathcal{E}$
$\therefore E(a_{t+l}|a_t)={\rho }^l{a}_t-\frac{{\sigma }^2}{2}(1-{\rho }^l)$
$\because v_t=e^{a_t},a_t=\ln v_t$, $\therefore E(v_{t+l}|v_t)=E(e^{a_{t+l}}|a_t)$

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${\gamma }_v(l)$ for log-normal AR(1)
$v_t=exp(Z_t)$, while $Z_t$ is $AR(1)$, $Z_t\sim N(-\frac{{\sigma }^2}{2},{\sigma }^2)$, $Corr(Z_{t+l},Z_t)={\rho }^l$.