On the first-passage time of an integrated Gauss-Markov process

It is considered the integrated process $X(t)=x+{\int }_0^{t}Y(s)ds,$ where $Y(t)$ is a Gauss-Markov process starting from $y\mathrm{.}$ The first-passage time (FPT) of $X$ through a constant boundary and the first-exit time of $X$ from an interval $(a,b)$ are investigated, generalizing some results on FPT of integrated Brownian motion. An essential role is played by a useful representation of $X,$ in terms of Brownian motion which allows to reduces the FPT of $X$ to that of a time-changed Brownian motion. Some explicit examples are reported; when theoretical calculation is not available, the quantities of interest are estimated by numerical computation.