I have SDE of the form,
$$
da(t)=(y(t)-c(t))dt - \gamma (c(t))dN(t),
$$ where $dN(t)$ is a jump process in Poisson form, $dN(t) \sim \operatorname{Poi}(\lambda)$.
Now, when there is an event, so that $dN(t)=1$, we get $\tilde{a}(t) = a(t)-\gamma (c(t)) $. Here, $\gamma( c(t))$ is differentiable. How would I derive the SDE for $\dot{\tilde{a}}(t)$?
If $c(t)$ is a continuous and differentiable function over time, without any stochastic element, then can I write, $$ d\tilde{a(t)} = (y(t)-c(t))dt - \gamma(c(t))dN(t) - \gamma '(c(t)) dc(t).$$ Then, if I need to find $df(\tilde{a}(t))$, how would I treat the term, $\gamma '(c(t)) dc(t)$?