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Linear-Quadratic Non-zero Sum Differential Game with Asymmetric Delayed Information

Yuxin Ye, Jingtao Shi

TL;DR

This work tackles a linear-quadratic non-zero-sum differential game with asymmetric time delays, where the two players access different delayed information. The authors apply the stochastic maximum principle to derive a delayed forward-backward SDE (DFBSDE) framework and then employ a time discretization with backward iteration to decouple and solve the system; a continuous-time solution emerges as the limit, characterized by Riccati-type equations. The main contribution is a constructive, state-estimate feedback Nash equilibrium under invertibility conditions, enabling explicit computation of equilibrium controls $u_i(t)$ in terms of conditional state estimates $E_{t-h_i}[x(t)]$. This advances the theory of stochastic differential games with asymmetric information and delays and has potential applications in finance, networked control, and multi-agent systems operating under partial information.

Abstract

This paper is concerned with a linear-quadratic non-zero sum differential game with asymmetric delayed information. To be specific, two players exist time delays simultaneously which are different, leading the dynamical system being an asymmetric information structure. By virtue of stochastic maximum principle, the stochastic Hamiltonian system is given which is a delayed forward-backward stochastic differential equation. Utilizing discretisation approach and backward iteration technique, we establish the relationship between forward and backward processes under asymmetric delayed information structure and obtain the state-estimate feedback Nash equilibrium of our problem.

Linear-Quadratic Non-zero Sum Differential Game with Asymmetric Delayed Information

TL;DR

This work tackles a linear-quadratic non-zero-sum differential game with asymmetric time delays, where the two players access different delayed information. The authors apply the stochastic maximum principle to derive a delayed forward-backward SDE (DFBSDE) framework and then employ a time discretization with backward iteration to decouple and solve the system; a continuous-time solution emerges as the limit, characterized by Riccati-type equations. The main contribution is a constructive, state-estimate feedback Nash equilibrium under invertibility conditions, enabling explicit computation of equilibrium controls in terms of conditional state estimates . This advances the theory of stochastic differential games with asymmetric information and delays and has potential applications in finance, networked control, and multi-agent systems operating under partial information.

Abstract

This paper is concerned with a linear-quadratic non-zero sum differential game with asymmetric delayed information. To be specific, two players exist time delays simultaneously which are different, leading the dynamical system being an asymmetric information structure. By virtue of stochastic maximum principle, the stochastic Hamiltonian system is given which is a delayed forward-backward stochastic differential equation. Utilizing discretisation approach and backward iteration technique, we establish the relationship between forward and backward processes under asymmetric delayed information structure and obtain the state-estimate feedback Nash equilibrium of our problem.
Paper Structure (4 sections, 6 theorems, 75 equations)

This paper contains 4 sections, 6 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Main results
  4. Concluding remarks

Key Result

Lemma 3.1

If $(u_1(\cdot),u_2(\cdot))\in \mathcal{U}_{ad}^1\times\mathcal{U}_{ad}^2$ is a Nash equilibrium for Problem SDG-ADI, then it satisfies the following equalities: where $(p_i(\cdot),q_i(\cdot)), i=1,2$ satisfies: and now $x(\cdot)\equiv x(\cdot;u_1(\cdot),u_2(\cdot))$ denoted the solution of (linear SDE) corresponding to $(u_1(\cdot),u_2(\cdot))$.

Theorems & Definitions (9)

  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Remark 3.1
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.2