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.