On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries

Making use of the integral equations given earlier, the asymptotic behaviour of the first-passage time (FPT) p.d.f.’s through certain time-varying boundaries, including periodic boundaries, is determined for a class of one-dimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Ornstein-Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.