Nonconstant continuous barriers in affine models

Extend the analysis of first passage in affine jump‑diffusion models to nonconstant continuous barriers b(t), deriving and analyzing the corresponding mode‑separated discounted transforms (including overshoot and continuous‑contact components) and their governing boundary‑value problems.

Background

The paper’s detailed analytic treatment focuses on a constant barrier b(t) ≡ a in a mean‑reverting affine jump‑diffusion with upward exponential jumps, yielding explicit OIDE/ODE formulations, a Green–Volterra representation for the discounted overshoot transform, and closed‑form expressions in the undiscounted limit.

The authors indicate that treating time‑dependent continuous barriers within the same affine framework remains open, which would require formulating and solving the appropriate boundary‑value problems for the mode‑separated transforms when b varies with time.