Sharpen discounted analysis of Fq, Gq, and Hq

Strengthen the discounted analysis of the transforms Fq(x) = Hq(x) − Gq(x), Gq(x) = E_x[e^{-q T_a} 1_{J+}], and Hq(x) = E_x[e^{-q T_a} 1_{T_a<∞}] for first passage of the constant barrier a by the mean-reverting affine jump‑diffusion dX_t = (a + B X_t) dt + σ dW_t + dJ_t with upward exponential jumps (compound Poisson with intensity λ and Exp(ν) sizes). Develop sharper results beyond the current Green–Volterra representation and first‑order small‑q expansion, such as improved representations, bounds, or asymptotics for these mode‑separated discounted transforms.

Background

The paper develops a pathwise fourfold decomposition of first passage and analyzes the associated random-time structure. In the affine jump‑diffusion with upward exponential jumps, it derives an integro‑differential equation (OIDE) for the overshoot mode, proves that a third‑order ODE is equivalent only with a boundary compatibility condition, and establishes verification and uniqueness for the discounted problem.

In the mean‑reverting case, the authors further obtain a Green–Volterra representation for the derivative Wq = Gq′ and a first‑order small‑q expansion identifying the overshoot‑conditioned first‑passage‑time moment. The conclusion notes that further sharpening of the discounted analysis of Fq, Gq, and Hq remains open.