First Passage through a Continuous Barrier: Pathwise Decomposition, Random-Time Structure, and Compensators

Let t be the first-passage time of a continuous barrier by a càdlàg adapted process. We show that t admits a canonical fourfold pathwise decomposition into continuous contact, contact from the left followed by an upward jump, exact hit by jump, and strict overshoot by jump from below. This refinement is more informative than the classical contact-versus-overshoot dichotomy for random-time purposes, because it separates modes with different predictability properties. In particular, the left-contact component always defines an accessible stopping time and becomes predictable under a no-premature-left-contact condition, which we prove to be both sufficient and necessary for the canonical running-supremum announcing sequence to work. On the gap side, under a structural exclusion of predictable gap-crossings, the corresponding restricted time is totally inaccessible. In the semimartingale setting, we obtain a sharp compensator criterion for the predictable-side condition, explicit compensator formulas for the jump-driven crossing modes, and a decomposition of the compensator of the default indicator into its predictable jump part and continuous part. As an application, for a mean-reverting affine jump-diffusion with upward exponential jumps, we derive the boundary-value problem governing the overshoot mode, prove that the differentiated third-order ODE is equivalent to the original problem only when a boundary compatibility condition is retained, and establish verification and uniqueness for the discounted problem. This yields an explicit Green-Volterra representation, a first-order small-q expansion expansion, and, in the undiscounted case, closed formulas for the overshoot and creeping probabilities