This paper determines first-passage time distributions with a twofold emphasis on the dynamics of the state variables and interest rate uncertainty. Underlyings follow two-dimensional geometric Brownian motions, Ornstein-Uhlenbeck processes or Poisson jump-diffusion processes, and boundaries are either fixed or indexed on risk-free bonds. Forward-neutral changes of numeraire enable one to derive generic valuation expressions, while changing time allows one to determine closed-form solutions for geometric Brownian motions and moving barriers. In turn, the latter formulas are used to reduce the variance of Monte Carlo simulations in the case of jump-diffusion processes, by means of the control variate method.