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Verification theorem related to a zero sum stochastic differential game, based on a chain rule for non-smooth functions

Carlo Ciccarella, Francesco Russo

TL;DR

The paper develops a PDE-based verification framework for zero-sum stochastic differential games by employing stochastic calculus via regularization and a chain rule for $C^{0,1}$-type quasi-strong solutions to Bellman-Isaacs equations. Under Isaacs' condition and the existence of a quasi-strong BI-solution, the authors establish that a saddle-point pair of feedback controls yields a Nash equilibrium and that the game value coincides with the BI-solution, with the result extending to degenerate diffusion cases. A fundamental lemma, derived from the Itô chain rule, connects the payoff to BI Hamiltonians, ensuring the value function governs the payoff under optimal feedbacks. The work also strengthens a verification theorem in stochastic control and demonstrates how the approach encompasses cases with non-smooth value functions, thereby broadening the applicability to regime-switching and non-classical regularity settings.

Abstract

In the framework of stochastic zero-sum differential games, we establish a verification theorem, inspired by those existing in stochastic control, to provide sufficient conditions for a pair of feedback controls to form a Nash equilibrium. Suppose the validity of the classical Isaacs' condition and the existence of a (what is termed) quasi-strong solution to the Bellman-Isaacs (BI) equations. If the diffusion coefficient of the state equation is non-degenerate, we are able to show the existence of a saddle point constituted by a couple of feedback controls that achieve the value of the game: moreover, the latter is equal to the (necessarily unique) solution of the BI equations. A suitable generalization is available when the diffusion is possibly degenerate. Similarly we have also improved a well-known verification theorem in stochastic control theory. The techniques of stochastic calculus via regularization we use, in particular specific chain rules, are borrowed from a companion paper of the authors.

Verification theorem related to a zero sum stochastic differential game, based on a chain rule for non-smooth functions

TL;DR

The paper develops a PDE-based verification framework for zero-sum stochastic differential games by employing stochastic calculus via regularization and a chain rule for -type quasi-strong solutions to Bellman-Isaacs equations. Under Isaacs' condition and the existence of a quasi-strong BI-solution, the authors establish that a saddle-point pair of feedback controls yields a Nash equilibrium and that the game value coincides with the BI-solution, with the result extending to degenerate diffusion cases. A fundamental lemma, derived from the Itô chain rule, connects the payoff to BI Hamiltonians, ensuring the value function governs the payoff under optimal feedbacks. The work also strengthens a verification theorem in stochastic control and demonstrates how the approach encompasses cases with non-smooth value functions, thereby broadening the applicability to regime-switching and non-classical regularity settings.

Abstract

In the framework of stochastic zero-sum differential games, we establish a verification theorem, inspired by those existing in stochastic control, to provide sufficient conditions for a pair of feedback controls to form a Nash equilibrium. Suppose the validity of the classical Isaacs' condition and the existence of a (what is termed) quasi-strong solution to the Bellman-Isaacs (BI) equations. If the diffusion coefficient of the state equation is non-degenerate, we are able to show the existence of a saddle point constituted by a couple of feedback controls that achieve the value of the game: moreover, the latter is equal to the (necessarily unique) solution of the BI equations. A suitable generalization is available when the diffusion is possibly degenerate. Similarly we have also improved a well-known verification theorem in stochastic control theory. The techniques of stochastic calculus via regularization we use, in particular specific chain rules, are borrowed from a companion paper of the authors.
Paper Structure (10 sections, 12 theorems, 73 equations)

This paper contains 10 sections, 12 theorems, 73 equations.

Table of Contents

  1. Introduction and problem formulation
  2. Preliminaries and Stochastic calculus
  3. Preliminaries
  4. Concept of PDE solution
  5. The Itô chain rule formula
  6. Application to game theory
  7. Basic setting
  8. The fundamental lemma
  9. Verification theorem and value of the game
  10. The case of control theory

Key Result

Theorem 2.7

Let $\sigma : [0,T]\times \mathbb R^d \rightarrow L(\mathbb R^m , \mathbb R^d)$ be a Borel locally bounded function and $F_0: \Omega \times [0,T] \rightarrow \mathbb R^d$ be an a.s. locally bounded progressively measurable field. Let $(S_s)_{s\in [t,T]}$ be a Itô process such that and Let $h:[t,T] \times \mathbb R^d \rightarrow \mathbb R$ and $g: \mathbb R^d \rightarrow \mathbb R$ as in the lin

Theorems & Definitions (46)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 36 more