Martingale conditions for discounted stock process under short‑rate models

Determine parameter conditions under which the discounted stock price process exp(−∫_0^t R_u du)·S_t is a true martingale under the risk‑neutral measure for the short‑rate models considered in the paper (including Vasicek, Ho–Lee, Hull–White, Black–Karasinski, and the extended models EV+, CIR++, and EEV+), so that the optimal stopping results for convertible bonds apply without assuming constant equity volatility.

Background

In the analysis of American-style convertible bonds, the paper notes that the results rely on the discounted stock process being a true martingale under the risk-neutral measure. Without appropriate parameter restrictions, the discounted stock process is only a local martingale, which invalidates certain optimal stopping arguments.

The authors reference sufficient conditions established in related stochastic volatility settings but state that analogous conditions for the specific short-rate models treated here (time-homogeneous and time-inhomogeneous, including extended models) have not been established in this work.