Stopping spikes, continuation bays and other features of optimal stopping with finite-time horizon

TL;DR

This paper analyzes optimal stopping problems with finite horizons and state-dependent discounting, revealing new regularity properties of the stopping boundary and geometric features like continuation bays and stopping spikes.

Contribution

It proves continuity and strict monotonicity of the optimal stopping boundary using probabilistic methods and introduces geometric features caused by atoms in the gain function's second derivative.

Findings

Optimal stopping boundary is continuous and strictly monotone.

Atoms in the second derivative induce geometric features in the stopping set.

Value function is continuously differentiable in time.

Abstract

We consider optimal stopping problems with finite-time horizon and state-dependent discounting. The underlying process is a one-dimensional linear diffusion and the gain function is time-homogeneous and difference of two convex functions. Under mild technical assumptions with local nature we prove fine regularity properties of the optimal stopping boundary including its continuity and strict monotonicity. The latter was never proven with probabilistic arguments. We also show that atoms in the signed measure associated with the second order spatial derivative of the gain function induce geometric properties of the continuation/stopping set that cannot be observed with smoother gain functions (we call them \emph{continuation bays} and \emph{stopping spikes}). The value function is continuously differentiable in time without any requirement on the smoothness of the gain function.