Characterization of free boundaries in general optimal stopping problems

Determine the number of free boundaries and characterize their monotonicity properties and mutual relationships in general optimal stopping problems, including how these features depend on model parameters and affect continuation and stopping regions.

Background

The authors emphasize that directly approximating free boundaries is challenging because their structure is not universally known across models. In particular, the number of free boundaries, their monotonicity, and interrelationships can vary substantially with parameters, as evidenced by examples where continuation regions change under different regimes.

This uncertainty hampers methods that rely on explicit free-boundary parameterization and motivates approaches that avoid pre-specifying boundary geometry. Establishing general principles about the number and behavior of free boundaries would clarify when such parameterizations are feasible and reliable.