Paper: arXiv 2306.05113

Authors: Andrea Bovo, Tiziano De Angelis, Jan Palczewski

Abstract

We consider a class of zero-sum stopper vs. singular-controller games in which the controller can only act on a subset $d_0<d$ of the $d$ coordinates of a controlled diffusion. Due to the constraint on the control directions these games fall outside the framework of recently studied variational methods. In this paper we develop an approximation procedure, based on $L^1$-stability estimates for the controlled diffusion process and almost sure convergence of suitable stopping times. That allows us to prove existence of the game’s value and to obtain an optimal strategy for the stopper, under continuity and growth conditions on the payoff functions. This class of games is a natural extension of (single-agent) singular control problems, studied in the literature, with similar constraints on the admissible controls.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents advanced theoretical mathematical research on stochastic differential games, singular control, and optimal stopping with constrained control directions. It offers strong mathematical novelty and rigorous proofs, but lacks any empirical data, backtesting, or practical financial application.

Research Flowchart

  flowchart TD
    A[Research Goal: Value & Strategies for Stopper vs. Singular-Controller Games with Directional Constraints] --> B[Inputs: Controlled Diffusion in d-dimensions, Controls restricted to d0 Subspace, Payoffs]
    B --> C[L1-Stability Estimates for Controlled Diffusion]
    C --> D[Approximation Scheme & A.S. Convergence of Stopping Times]
    D --> E[Analysis of Continuity & Growth Conditions on Payoffs]
    E --> F[Key Outcome: Existence of Game's Value & Optimal Strategy for Stopper]