Extend the square-root boundary asymptotic to general one-dimensional diffusions

Establish the square-root asymptotic for the delivery boundary of a one-dimensional diffusion satisfying suitable coefficient regularity, integrability, and one-sided boundary assumptions, when the running reward vanishes at a point and has nonzero derivative there.

Background

The paper proves a square-root near-expiry expansion for the delivery boundary in the geometric Brownian-motion model. It then observes that the same Brownian scaling formally predicts an analogous expansion for a general one-dimensional diffusion dX=μ(X)dt+ν(X)dW when the running reward vanishes at a distinguished point and has nonzero derivative there.

The authors explicitly leave the generalization unresolved because the geometric Brownian-motion proof relies on an explicit solution and model-specific stopping-region properties. A successful extension would require appropriate regularity of the coefficients, integrability estimates, and one-sided assumptions on the boundary.