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Parameter dependent optimal thresholds, indifference levels and inverse optimal stopping problems
Consider the classic infinite-horizon problem of stopping a one-dimensional
diffusion to optimise between running and terminal rewards and suppose we are
given a parametrised family of such problems. We provide a general theory of
parameter dependence in infinite-horizon stopping problems for which threshold
strategies are optimal. The crux of the approach is a supermodularity condition
which guarantees that the family of problems is indexable by a set valued map
which we call the indifference map. This map is a natural generalisation of the
allocation (Gittins) index, a classical quantity in the theory of dynamic
allocation. Importantly, the notion of indexability leads to a framework for
inverse optimal stopping problems
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