Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Zero-sum stopper vs. singular-controller games with constrained control directions

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.

Zero-sum stopper vs. singular-controller games with constrained control directions

Abstract

We consider a class of zero-sum stopper vs. singular-controller games in which the controller can only act on a subset of the 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 -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.
Paper Structure (10 sections, 16 theorems, 168 equations)

This paper contains 10 sections, 16 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Approximation procedure
  4. Challenges in the constrained setup
  5. Notation
  6. Properties of the approximating problems and stability estimates
  7. Some stability estimates
  8. Convergence of the approximating problems
  9. Limits as γ->0
  10. Relaxing Assumption \ref{['ass:gen3']} into Assumption \ref{['ass:gen2']}

Key Result

Theorem 2.3

Under Assumptions ass:gen1 and ass:gen2, the game described above admits a value $v$ (i.e., eq:valuegame_1 holds) with the following properties: Moreover, for any given $(t,x)\in\mathbb{R}^{d+1}_{0,T}$ and any $(n,\nu)\in \mathcal{A}^{d_0}$, the stopping time $\theta_*=\theta_*(t,x;n,\nu)\in\mathcal{T}_t$ is optimal for the stopper, where $\theta_*=\tau_*\wedge\sigma_*$ and $\mathsf P_x$-a.s.

Theorems & Definitions (37)

  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 3.1
  • proof
  • Theorem 3.3
  • Remark 3.4
  • Lemma 3.5
  • ...and 27 more