A PROBABILISTIC APPROACH TO CONTINUOUS DIFFERENTIABILITY OF OPTIMAL STOPPING BOUNDARIES
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摘要
A probabilistic proof of the continuous differentiability of time-dependent free boundaries in time-inhomogeneous optimal stopping problems is established for a one-dimensional diffusion X with time-dependent gain and discounting, allowing finite or infinite horizons. The approach combines Lamperti transform, Girsanov change of measure, and Pitman’s theorem for a 3-dimensional Bessel process, yielding a local analysis that extends beyond PDE-based requirements and providing a probabilistic link between the value function and the Stefan problem. The results include a constructive representation of the boundary derivative and its continuity under mild regularity and Lipschitz conditions. The work also proves a probabilistic connection between v˙ and the Stefan problem via a time-derivative expansion near the boundary. [page::1][page::5]
