Phase-type or hyperexponential jump laws and higher-order local systems

Derive and analyze the higher‑order local system that arises when the upward exponential jump‑size distribution in the mean‑reverting affine jump‑diffusion is replaced by a phase‑type or hyperexponential distribution via state‑space augmentation, and establish the corresponding mode‑separated boundary‑value problems and solutions.

Background

In the presented affine model, the exponential jump law enables factorization in the compensation argument and leads to tractable OIDE/ODE formulations and explicit representations. The conclusion suggests generalizing the jump law to richer classes such as phase‑type or hyperexponential distributions.

Such generalizations typically require augmenting the state space to obtain Markovian dynamics, which is expected to produce a higher‑order local system. Characterizing and solving this system for the mode‑separated first‑passage transforms remains an open problem.

References

Several directions remain open. One may sharpen the discounted analysis of Fq, Gq, and Hq, treat nonconstant continuous barriers in affine models, or replace the exponential jump law by phase- type or hyperexponential distributions, where one expects a higher-order local system after augmenting the state space.

— First Passage through a Continuous Barrier: Pathwise Decomposition, Random-Time Structure, and Compensators  (2604.03125 - Guillaume, 3 Apr 2026) in Section 7, Conclusion