Free-boundary problems and smooth fit

For a Markov diffusion and discounted reward, the value dominates the immediate gain because stopping now is allowed. In a continuation region, local waiting gives the equation $V_t+\mathcal LV-rV=0$. In a stopping region, the value equals the gain and waiting must not improve its expected discounted value. These statements combine into a variational inequality, with a maximum for this reward convention. The unknown interface between the two regions is the free boundary.

For a perpetual time-homogeneous problem, the time derivative disappears, but behavior at infinity and admissibility of the proposed hitting time still matter. Solving the continuation ordinary differential equation introduces constants that the growth condition and boundary matching can determine. A candidate must dominate the gain everywhere and have the correct supermartingale inequality in the stopping region. Matching conditions alone can select a false candidate that fails either of these global tests.

Value matching requires the limiting continuation value to equal the stopping gain at the boundary. Smooth fit additionally equates the relevant first derivatives. For ordinary one-dimensional Brownian motion and a smooth gain near a regular threshold, this matching can be justified by the path behavior and the associated stopping verification. It does not require equality of second derivatives: a valid value function can be continuously differentiable and only piecewise twice differentiable.

Smooth fit is not a universal consequence of continuity of paths. Peskir constructs regular diffusions and differentiable gains for which ordinary spatial smooth fit fails, and explains the role of differentiability of the scale function. Jump processes, irregular boundaries, or nonsmooth gains can require different matching principles. Here the threshold lies away from the gain's kink, and Brownian scale is smooth. We will verify the resulting piecewise function directly, rather than extend its boundary rule to all stopping problems.

Let $X=x+\sigma W$ on the real line, use positive discount r, and gain $(x-K)^+$. For a threshold b above K, the decaying continuation solution is proportional to an exponential with rate $k=\sqrt{2r}/\sigma$. Value matching and smooth fit yield $b=K+1/k$. The candidate exceeds the gain below b by the elementary exponential tangent inequality, and above b its generator-minus-discount expression is nonpositive because the gain is positive and affine there.

The candidate is continuously differentiable at b, so the local-time contribution associated with a first-derivative jump vanishes in the generalized Itô formula. Its bounded first derivative gives a true discounted martingale term. Discounted Brownian moment bounds justify the terminal tail limit, and the threshold hitting-time Laplace transform attains the candidate below b. These checks certify the value among stopping times, taking a zero payoff on an infinite stopping time if needed.