Paper: arXiv 2312.00613

Authors: Andrea Bovo, Tiziano De Angelis, Jan Palczewski

Abstract

We study zero-sum stochastic games between a singular controller and a stopper when the (state-dependent) diffusion matrix of the underlying controlled diffusion process is degenerate. In particular, we show the existence of a value for the game and determine an optimal strategy for the stopper. The degeneracy of the dynamics prevents the use of analytical methods based on solution in Sobolev spaces of suitable variational problems. Therefore we adopt a probabilistic approach based on a perturbation of the underlying diffusion modulated by a parameter $γ>0$. For each $γ>0$ the approximating game is non-degenerate and admits a value $u^γ$ and an optimal strategy $τ^γ_$ for the stopper. Letting $γ\to 0$ we prove convergence of $u^γ$ to a function $v$, which identifies the value of the original game. We also construct explicitly optimal stopping times $θ^γ_$ for $u^γ$, related but not equal to $τ^γ_$, which converge almost surely to an optimal stopping time $θ_$ for the game with degenerate dynamics.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly sophisticated theoretical analysis of zero-sum stochastic games involving degenerate diffusions, singular control, and optimal stopping. While mathematically rigorous and novel in its probabilistic perturbation approach, it is purely theoretical with zero empirical validation or financial data backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Solve Zero-Sum Game with Degenerate Diffusion Dynamics"] --> B["Inputs: Controlled State Process & Objective Function"]
    B --> C["Method: Introduce Non-Degenerate Approximation Parameter γ > 0"]
    C --> D["Compute Approximate Game Value u^γ & Stopping Strategy τ^γ_*"]
    D --> E["Construct Explicit Optimal Stopping Times θ^γ_* for Approximating Game"]
    E --> F["Pass to Limit: γ → 0 (Probabilistic Approach)"]
    F --> G["Key Outcomes: Prove Value Convergence (u^γ → v) & A.S. Convergence to Optimal Stopping Time θ_*"]