On the two-boundary first-passage-time problem for Gauss-Markov processes

The first-passage time problem through two time-dependent boundaries for one-dimensional Gauss-Markov processes is considered, both for fixed and for random initial states. The first passage time probability density functions are proved to satisfy a system of continuous-kernel integral equations that can be numerically solved by an accurate and computationally simple algorithm. A condition on the boundaries of the process is given such that this system reduces to a single non-singular integral equation. Closed-form results are also obtained for classes of double boundaries that are intimately related to certain symmetry properties of the considered processes. Finally, the double-sided problem is considered.