  Stochastic bridges are commonly used to impute missing data with a lower
sampling rate to generate data with a higher sampling rate, while preserving
key properties of the dynamics involved in an unbiased way. While the
generation of Brownian bridges and Ornstein-Uhlenbeck bridges is well
understood, unbiased generation of such stochastic bridges subject to a given
extremum has been less explored in the literature. After a review of known
results, we compare two algorithms for generating Brownian bridges constrained
to a given extremum, one of which generalises to other diffusions. We further
apply this to generate unbiased Ornstein-Uhlenbeck bridges and unconstrained
processes, both constrained to a given extremum, along with more tractable
numerical approximations of these algorithms. Finally, we consider the case of
drift, and applications to geometric Brownian motions.
