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

Dylan Possamaï, Mateo Rodriguez Polo

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.

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

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.
Paper Structure (26 sections, 19 theorems, 178 equations, 2 figures)

This paper contains 26 sections, 19 theorems, 178 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Time-inconsistent stochastic control
  3. Probabilistic setting
  4. Target functional
  5. Game formulation
  6. Functional spaces
  7. Auxiliary weighted functional spaces and norms
  8. Main results
  9. An informal derivation of the BSDE system
  10. Assumptions
  11. The extended dynamic programming principle
  12. A necessity result
  13. Verification theorem
  14. Well-posedness of the solution
  15. An example: the linear--quadratic time-inconsistent regulator
  16. ...and 11 more sections

Key Result

Theorem 2.1

Consider a stochastic kernel $(\mathbb{Q}_\omega)_{\omega\in\Omega}$, and let $\tau\in\mathcal{T}_{0,T}(\mathbb{F})$. Suppose the map $\omega \longmapsto \mathbb{Q}_\omega$ is $\mathcal{F}_\tau$-measurable and $\mathbb{Q}_\omega[\Omega_\tau^\omega]=1$ for all $\omega\in \Omega$. Given $\mathbb{P} \i

Figures (2)

  • Figure 1: Comparison of state trajectories (left) and control effort (right). The equilibrium strategy (blue) maintains the state slightly near the target $x_0=1.0$ with moderate effort. The naive strategy (red dashed) applies more control initially.
  • Figure 2: Sensitivity analysis. The total expected cost $J(0, x_0)$ is plotted against the inconsistency parameter $\Gamma$. The equilibrium strategy consistently outperforms the naive strategy as the penalty parameter increases.

Theorems & Definitions (55)

  • Remark 1
  • Remark 2
  • Theorem 2.1: Concatenated measure
  • Definition 1: Equilibrium control
  • Remark 3
  • Remark 4
  • Definition 2: Locally uniform random fields
  • Remark 5: General growth conditions
  • Proposition 1: Banach structure
  • proof
  • ...and 45 more