  Consider a set of discounted optimal stopping problems for a one-parameter
family of objective functions and a fixed diffusion process, started at a fixed
point. A standard problem in stochastic control/optimal stopping is to solve
for the problem value in this setting. In this article we consider an inverse
problem; given the set of problem values for a family of objective functions,
we aim to recover the diffusion. Under a natural assumption on the family of
objective functions we can characterise existence and uniqueness of a diffusion
for which the optimal stopping problems have the specified values. The solution
of the problem relies on techniques from generalised convexity theory
