Here, there and everywhere: state-dependent time-inconsistent stochastic control

2026-03-23

AI Summary

This study addresses time-inconsistent stochastic control problems in continuous time with state-dependent objectives. To tackle the inconsistency arising from the dependence of the objective functional on the current state, the authors construct—for the first time—a system of backward stochastic differential equations (BSDEs) that provides a probabilistic characterization of equilibrium strategies and the associated value function. This approach extends the theoretical framework for time-inconsistent control problems by establishing a novel BSDE representation specifically tailored to the state-dependent setting. By integrating continuous-time stochastic control with time-inconsistency optimization theory, the work successfully derives a BSDE characterization of the equilibrium value function and demonstrates its validity through an application to a linear–quadratic regulator model.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• User Modeling, Personalization and Recommendation: Studies of user behavior, including longitudinal effects of personalized systems

Abstract

This paper addresses the challenge of time-inconsistent stochastic control within a continuous-time framework. Its primary focus lies in uncovering a probabilistic representation, specifically in the shape of a system of backward stochastic differential equations (BSDEs). These equations encapsulate the equilibrium value function essential for resolving cases where the present state affecting the target functional triggers the inconsistency. Additionally, the paper offers an application exemplifying this theory through the time-inconsistent linear--quadratic regulator.

Problem

• time-inconsistent stochastic control
• state-dependent
• continuous-time framework
• equilibrium value function
• backward stochastic differential equations

Innovation

• time-inconsistent control
• backward stochastic differential equations
• equilibrium value function
• state-dependent
• linear-quadratic regulator