I have an issue with calculating the expected value of the exponential of a normally distributed variable times a discrete random variable. Let $\xi_{i} \overset{\text{iid}}{\sim} \mathcal{N}(0, \Delta t)$, $\sigma_{i}$ a time-homogeneous Markov Chain with three possible states $\sigma_{\text{low}}, \sigma_{\text{med}}$ and $\sigma_{\text{high}}$, $\alpha > 0$ and $x_{i} \in [0,10]$ some constants and $i = 0, 1, \dots, 10$. Furthermore, we assume that $\xi_{i}$ and $\sigma_{i}$ are independent for all $i$. The transition matrix for $\sigma_{i}$ is \begin{align*} p^{\sigma} = \begin{pmatrix} 0.93 & 0.04 & 0.03 \\ 0.72 & 0.22 & 0.06 \\ 0.44 & 0.48 & 0.08 \end{pmatrix} \end{align*} I want to find an expression for \begin{align*} \mathbb{E} \left[\exp(-\alpha \sum_{j = t+1}^{T} x_{j}\sigma_{j}\xi_{j}) \mid \sigma_{t} = v\right], \end{align*} where $v$ could be any of the three states. My approach is \begin{align*} \mathbb{E} \left[\exp(-\alpha \sum_{j = n+1}^{T} x_{j}\sigma_{j}\xi_{j}) \mid \sigma_{t} = v\right] &= \mathbb{E} \left[\prod_{j = n+1}^{T} \exp(-\alpha x_{j}\sigma_{j}\xi_{j}) \mid \sigma_{t} = v \right] \\ &= \prod_{j = n+1}^{T} \mathbb{E} \left[\exp(-\alpha x_{j}\sigma_{j}\xi_{j}) \mid \sigma_{t} = v \right] \end{align*} However, I am unsure how to continue as I don't know how to handle the product of $\sigma_{j}$ and $\xi_{j}$.
Thanks for any tips and help in advance.