# Mean first-passage time for a marked Poisson process

Given a marked Poisson process in one dimension $$Y(t)=\sum_{\{t_i,a_i\}}g(t-t_i,a_i)$$

so that $$Y(t)$$ is a sum of impulses arriving as a Poisson process and the impulses $$g$$ belong to a family of curves governed by an additional random variable $$a$$, what is the expected first-passage time of $$Y$$ over a given threshold $$Y_{th}$$ from a given starting point $$Y_0$$? This seems like it should be a fairly straightforward problem as impulse times are uncorrelated and impulse shapes are independently identically distributed, but I am unable to find a comprehensive solution.