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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.

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