Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linear-Quadratic Zero-Sum Stochastic Differential Game with Partial Observation

Zhiyong Yu, Wanying Yue

TL;DR

The paper addresses the problem of finding saddle points in a linear-quadratic zero-sum stochastic differential game under partial observation. It extends explicit and implicit feedback law concepts to incomplete information by introducing a conditional mean-field SDE (CMF-SDE) framework, employing Kalman-Bucy filtering and a separation principle, and applying a single-step completion-of-squares with a Riccati equation for $P(\cdot)$ and an auxiliary ODE for $p(\cdot)$ to obtain feedback laws. The main contributions include establishing the well-posedness of CMF-SDEs, deriving explicit saddle-point feedback laws under two structural conditions (I) and (II) (with coincidence when both hold), and applying the theory to a two-firm duopoly with partial observation, supported by numerical experiments. The results provide a practically implementable, feedback-based equilibrium construction for partially observed stochastic games and offer a template for extending to more complex observation models and broader applications.

Abstract

This paper is concerned with a kind of linear-quadratic (LQ, for short) two-person zero-sum stochastic differential game problems with partial observation. We propose the notions of explicit and implicit feedback laws under partial observation. With the help of a class of conditional mean-field stochastic differential equations (CMF-SDEs, for short), the separation principle, filtering techniques, and the method of completion of squares, we construct a saddle point in the form of feedback laws for the two players. Finally, the theoretical results are applied to investigate a duopoly competition problem with partial observation.

Linear-Quadratic Zero-Sum Stochastic Differential Game with Partial Observation

TL;DR

The paper addresses the problem of finding saddle points in a linear-quadratic zero-sum stochastic differential game under partial observation. It extends explicit and implicit feedback law concepts to incomplete information by introducing a conditional mean-field SDE (CMF-SDE) framework, employing Kalman-Bucy filtering and a separation principle, and applying a single-step completion-of-squares with a Riccati equation for and an auxiliary ODE for to obtain feedback laws. The main contributions include establishing the well-posedness of CMF-SDEs, deriving explicit saddle-point feedback laws under two structural conditions (I) and (II) (with coincidence when both hold), and applying the theory to a two-firm duopoly with partial observation, supported by numerical experiments. The results provide a practically implementable, feedback-based equilibrium construction for partially observed stochastic games and offer a template for extending to more complex observation models and broader applications.

Abstract

This paper is concerned with a kind of linear-quadratic (LQ, for short) two-person zero-sum stochastic differential game problems with partial observation. We propose the notions of explicit and implicit feedback laws under partial observation. With the help of a class of conditional mean-field stochastic differential equations (CMF-SDEs, for short), the separation principle, filtering techniques, and the method of completion of squares, we construct a saddle point in the form of feedback laws for the two players. Finally, the theoretical results are applied to investigate a duopoly competition problem with partial observation.

Paper Structure

This paper contains 13 sections, 7 theorems, 89 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Spaces
  3. Problem formulation
  4. Filtering and separation
  5. Completion of squares and saddle points
  6. A saddle point in explicit-implicit feedback law form
  7. A saddle point in implicit-explicit feedback law form
  8. The case where Conditions (I) and (II) hold
  9. Application to a duopoly problem
  10. Formulation of duopoly problem
  11. Solution to duopoly problem
  12. Numerical simulation
  13. Conclusion and further work

Key Result

Lemma 3.2

Let $u(\cdot) \in \mathscr U[0,T]$. Then $\mathbb F^u = \mathbb F^0$.

Figures (5)

  • Figure 1: Numerical solution of $P(\cdot)$.
  • Figure 2: Numerical solution of $p(\cdot)$.
  • Figure 3: Numerical solution of $\Sigma(\cdot)$.
  • Figure 4: One trajectory of $\widehat{x}^*(\cdot)$.
  • Figure 5: The induced pair of controls $(u_1^*(\cdot), u_2^*(\cdot))$.

Theorems & Definitions (10)

  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 4.1
  • Lemma 5.1
  • Theorem 5.2
  • Lemma 5.3
  • Theorem 5.4
  • Corollary 5.5