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

Abstract: Let t be the first-passage time of a continuous barrier by a c{à}dl{à}g 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

• The paper introduces a fourfold pathwise decomposition that distinguishes between continuous contact and jump-induced barrier crossings.
• The paper establishes sharp compensator criteria and explicit formulas for different crossing modes in semimartingale settings.
• The paper applies its decomposition to an affine jump-diffusion model, providing analytical solutions useful for credit risk modeling.

Detailed Analysis of "First Passage through a Continuous Barrier: Pathwise Decomposition, Random-Time Structure, and Compensators" (2604.03125)

Overview and Motivation

The paper addresses the rigorous analysis of the first-passage problem for general càdlàg adapted processes crossing a continuous (time-dependent) barrier. Unlike the standard dichotomy that classifies crossings as either continuous contact (creeping) or jump-overshoot, this work introduces a canonical fourfold pathwise decomposition of crossing modes. This decomposition enables a more granular understanding of random time structures associated with first passage, particularly in terms of their predictability, accessibility, and compensators.

The research is motivated by questions in fluctuation theory, mathematical finance (notably credit risk), and the theory of stochastic processes, where the predictability and randomness of first passage times fundamentally influence applications and modeling strategies.

Canonical Pathwise Decomposition

The core technical contribution is the pathwise decomposition of the first passage event into four mutually exclusive crossing modes:

• Continuous contact: Trajectory first touches the barrier continuously.
• Left contact, then jump: Path reaches the barrier from below, touches it, and jumps upward.
• Exact hit by jump: Process jumps from below and lands exactly on the barrier.
• Strict overshoot by jump: Path jumps from below and lands strictly above the barrier.

This fine classification separates crossing events according to the "geometry" of their realization and, crucially, their measurability and predictability with respect to the underlying filtration.

The decomposition is shown to produce a disjoint, exhaustive partition of the first-passage event. Notably, it decouples modes visible at the predictable stopping times (e.g., continuous contact and left-contact-then-jump) from those realized strictly by unpredictable jumps (e.g., exact hit by jump, strict gap overshoot).

Random-Time Structure: Predictability and Accessibility

The analysis reveals that the left-contact modes (continuous contact and left-contact-then-jump) always yield stopping times that are accessible. Moreover, under a no-premature-left-contact condition (the path does not visit the barrier before actual passage), the predictable structure is characterized: the filtering canonical sequence based on the running supremum of the process converges to the left-contact time, and the paper proves this condition is both necessary and sufficient.

Conversely, under the structural exclusion of predictable gap-crossings (specifically, when predictable stopping times cannot see a jump-over), the gap modes yield totally inaccessible stopping times, signifying that these events are maximally "unpredictable" and can only be realized by unannounced jumps.

Compensator Decomposition in the Semimartingale Setting

The study proceeds to a detailed stochastic calculus analysis in the semimartingale context. Using compensator theory, the author provides:

• Sharp compensator criteria for the predictability-oriented no-premature-left-contact condition.
• Explicit compensator formulas for the jump-driven crossing modes.
• A canonical decomposition of the default indicator's compensator into predictable jump and continuous components, providing a direct probabilistic bridge to hazard-process-based default modeling in credit risk.

Notably, for processes with no predictable jumps, the continuous contact time is predictable and the jump-over time is totally inaccessible—a distinction that is critical for modeling default in finance.

Analytic Solution for Affine Jump-Diffusion Models

The theoretical abstraction is grounded by specializing to an affine mean-reverting jump-diffusion with upward exponential jumps—a prototypical model in finance for asset and credit dynamics. The paper delivers:

• The integro-differential equation (OIDE) governing the Laplace transform of the overshoot mode.
• Reduction of this equation to a third-order ODE under boundary compatibility conditions, and demonstration of necessary and sufficient uniqueness and verification properties for bounded classical solutions in the discounted case.
• An exact Green-Volterra representation for the derivative of the discounted overshoot probability, featuring parabolic-cylinder functions and a Volterra integral kernel, thereby separating the "local" diffusive part from the nonlocal jump effect.
• A first-order small-$q$ expansion for Laplace transforms, revealing that the overshoot-time moment appears as the leading correction.
• In the undiscounted limit, the problem simplifies to an explicit closed-form parabolic-cylinder function formula, yielding both overshoot and creeping probabilities directly.

Contrasts with Existing Literature

The fourfold decomposition extends classical results (e.g., creeping versus overshoot in Lévy process fluctuation theory) by isolating the precise random-time and filtration-theoretic nature of first passage, in contrast to analyses focused solely on transform computation. Compared with the affine jump-diffusion literature (e.g., double-exponential cases), the paper addresses non-constant coefficients and boundary-value complications absent in previous work.

In the context of credit risk, the endogenous decomposition of the compensator based on the geometry of sample paths is a new contribution not captured in reduced-form models or incomplete information frameworks.

Implications and Future Directions

Practical Implications:

The results are directly relevant for modeling the timing and mathematical nature of rare events such as structural default in credit risk (predictable vs. surprise defaults), as well as for robust option pricing and path-dependent payout structures in financial mathematics.

Theoretical Implications:

The findings clarify the filtration-theoretic underpinnings of the randomness and predictability of first-passage times, providing a template for analyzing more complex models involving state-dependent or time-inhomogeneous barriers, broader jump distributions, and possibly higher-dimensional settings.

Future Developments:

Potential directions include:

• Extending the discounted analysis to more general jump distributions (e.g., phase-type, hyperexponential).
• Analyzing time-dependent and stochastic barriers in affine frameworks.
• Developing analogous decompositions and compensator structures in multi-dimensional processes and non-Markovian settings.
• Utilizing the pathwise decomposition for inverse problems, such as reconstructing barrier geometry or jump law characteristics from observed first-passage data.

Conclusion

The paper establishes a precise, filtration-aware pathwise decomposition of first passage through continuous barriers, delineates the corresponding random-time structure, and connects these to compensator analysis in semimartingale settings. The explicit analytic results for affine models reinforce the practical tractability of the approach. Collectively, the results offer foundational insights and technical tools for stochastic modeling wherever the geometry and predictability of barrier crossings are material.