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Time-Inconsistent Stochastic Linear-quadratic Differential Game

Qinglong Zhou, Gaofeng Zong

Abstract

We consider a general time-inconsistent stochastic linear-quadratic differential game. The time-inconsistency arises from the presence of quadratic terms of the expected state as well as state-dependent term in the objective functionals. We define an equilibrium strategy, which is different from the classical one, and derived a sufficient conditions for equilibrium strategies via a system of forward-backward stochastic differential equations. When the state is one-dimensional and the coefficients are all deterministic, we find an explicit equilibrium strategy. The uniqueness of such equilibrium strategy is also given.

Time-Inconsistent Stochastic Linear-quadratic Differential Game

Abstract

We consider a general time-inconsistent stochastic linear-quadratic differential game. The time-inconsistency arises from the presence of quadratic terms of the expected state as well as state-dependent term in the objective functionals. We define an equilibrium strategy, which is different from the classical one, and derived a sufficient conditions for equilibrium strategies via a system of forward-backward stochastic differential equations. When the state is one-dimensional and the coefficients are all deterministic, we find an explicit equilibrium strategy. The uniqueness of such equilibrium strategy is also given.

Paper Structure

This paper contains 6 sections, 10 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Sufficient conditions
  4. Existence and uniqueness of equilibrium strategy when coefficients are deterministic
  5. An intuitional idea and the uniqueness of the equilibrium strategy
  6. Existence of the equilibrium strategies

Key Result

Proposition 3.1

For any $t\in[0,T),\epsilon>0$, and $v_1,v_2\in L_{\mathcal{F}_t}^2(\Omega,\mathbb{R}^l)$, define $u_i^{t,\epsilon,v_i}, i=1,2$ by (spiked.control). Then where $\Lambda_i(s;t)=B_{i,s}p_i(s;t)+\sum_{j=1}^d(D_{i,s}^j)'k_i^j(s;t)+R_{i,s}u_{i,s}^*$ and $H_i(s;t)=R_{i,s}+\sum_{j=1}^d(D_{i,s}^j)'P_i(s;t)D_{i,s}^j$ for $i=1,2$.

Theorems & Definitions (11)

  • Definition 2.1
  • Proposition 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 4.1
  • Proposition 4.2
  • Theorem 4.3
  • Theorem 4.4
  • ...and 1 more