  Tied-down renewal processes are generalisations of the Brownian bridge, where
an event (or a zero crossing) occurs both at the origin of time and at the
final observation time $t$. We give an analytical derivation of the two-time
correlation function for such processes in the Laplace space of all temporal
variables. This yields the exact asymptotic expression of the correlation in
the Porod regime of short separations between the two times and in the
persistence regime of large separations. We also investigate other quantities,
such as the backward and forward recurrence times, as well as the occupation
time of the process. The latter has a broad distribution which is determined
exactly. Physical implications of these results for the Poland Scheraga and
related models are given. These results also give exact answers to questions
posed in the past in the context of stochastically evolving surfaces.
