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

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