Optimal Stopping and Singular Stochastic Control

When should a random process be stopped, and how hard should it be pushed? The two questions have separate literatures, separate equations and separate textbooks. This book shows that they are one question.

Start from a single Hamilton–Jacobi–Bellman equation and let the control grow without bound. Cap the rate at which the process may be killed, and the penalized equation appears; as the cap grows, optimal stopping is what remains. Cap the rate at which it may be pushed instead, and the limit is singular control, with a gradient constraint and a reflecting barrier. Differentiate the second limit and the first returns: the barrier is the free boundary, one derivative apart.

Every theorem is proved in full on one running model, and every claim is checked by computation: two solvers that share nothing but a grid agree to the third decimal across the whole plane of time and state.

Nikolaos Vardakis is a probabilist who works on optimal stopping and stochastic control.