I am trying to solve the following problem. Let the time series $S_t$, $t\in \{1,...N\}$ and consider its corresponding return $R_t$ defined as \begin{equation} R_t=\log(\frac{S_t}{S_{t-1}})=\mu_t+\sigma_t \end{equation} Let $\Psi=\{\Psi_t\}$ be a $\sigma$-algebra where $\Psi_t=\sigma(\{S1,..,S_t\})=\sigma(\{R_1,..,R_t\})$.
The model is described as follow: \begin{equation} \mu_t=c+\theta_1R_{t-1}+\theta_2R_{t-2} \end{equation} \begin{equation} \sigma_t=\sigma_{t,1}+\sigma_{t,2} \end{equation} where \begin{equation}\sigma_{t,1}|\Psi_{t-1}\sim N(0,h) \;\;\;\;\;\;\;\sigma_{t,2}=\sum_{k=1}^{N_t}V_{t,k}-\lambda\phi \end{equation}
with $N_t \sim Poiss(\lambda)$ and $V_{t,k} \sim N(\phi,\theta^2)$ i.i.d. for $k=1,2...$. $\;\;$So $E[\sigma_{t,2}|\Psi_{t-1}]=0$.
The model parameters are $\{c,\theta_1,\theta_2,h,\lambda,\phi,\theta\}$ and I have already estimated them by maximizing the log-likelihood function.
Now, I would like to compute a forecast for $\hat{S_t}=\hat{S_{t-1}}e^{\hat{R_t}}$ , $t\in\{N+1,...,N'\}$ with $N'>N$, but I don't know how to do it.
I have already tried taking conditional expectation $E[S_t|\Psi_{t-1}]$ but the result is a straight line and I think it is a too rough solution.
Any kind of help would be appreciated.