I'm reading the following paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2543719 . On page 15, the author states that in a jump-diffusion process, the log jump sizes, $Y_k$, follow a binomial jump diffusion process on the interval $[-2a, a]$, where $a>0$, i.e. log jump sizes has a density
$$f(y)=\begin{cases} -2a, & \text{with probability } p\\ -a, & \text{with probability} 1-p \end{cases}$$
Then the author goes on to state that
$$\mathbb{E}[e^{Y}] = pe^{-2a} + (1-p)e^{-a}.$$
I have two questions: i) I'm fairly new to the topic of Levy proceses and jump-diffusion models. Is the aforementioned distribution known in literature as a binomial jump diffusion process? Since a simple Google search doesn't seem to come up with anything like that.
ii) I am unable to see how the author derives the equation for $\mathbb{E}[e^{Y}]$. If someone can point me in the direction for how to do go about deriving it I'd be greatful.