A probabilistic approach to continuous differentiability of optimal stopping boundaries

Paper: arXiv 2405.16636

Authors: Tiziano De Angelis, Damien Lamberton

Abstract

We obtain the first probabilistic proof of continuous differentiability of time-dependent optimal boundaries in optimal stopping problems. The underlying stochastic dynamics is a one-dimensional, time-inhomogeneous diffusion. The gain function is also time-inhomogeneous and not necessarily smooth. Moreover, we include state-dependent discount rate and the time-horizon can be either finite or infinite. Our arguments of proof are of a local nature that allows us to obtain the result under more general conditions than those used in the PDE literature. As a byproduct of our main result we also obtain the first probabilistic proof of the link between the value function of an optimal stopping problem and the solution of the Stefan’s problem.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a groundbreaking theoretical result, offering the first probabilistic proof of continuous differentiability of optimal stopping boundaries using advanced stochastic calculus tools like Bessel processes and Girsanov’s theorem. However, as a purely theoretical probability paper, it contains no empirical analysis, backtesting, or financial datasets.

Research Flowchart

  flowchart TD
    A[Research Goal: Probabilistic proof of C1 differentiability of optimal stopping boundaries] --> B[Model Inputs: 1D time-inhomogeneous diffusion, non-smooth gain, state-dependent discount]
    B --> C[Methodology: Local probabilistic arguments & stochastic analysis]
    C --> D[Computational & Analytical Process: Analyzing local behavior of value function near boundary]
    D --> E[Outcome 1: Proof of continuous differentiability for time-dependent boundaries]
    D --> F[Outcome 2: Probabilistic proof linking optimal stopping to Stefan's problem]