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Robust Time-inconsistent Linear-Quadratic Stochastic Controls: A Stochastic Differential Game Approach

Bingyan Han, Chi Seng Pun, Hoi Ying Wong

TL;DR

The paper develops a robust time-inconsistent linear-quadratic stochastic control framework by formulating a stochastic differential game between a decision-maker and an adversary who distorts the probability measure. Using spike variation and an augmented state that includes the Radon-Nikodým derivative, it derives sufficient conditions for Nash-equilibrium (time-consistent) policies without requiring a preference ordering, and provides semi-analytic examples with state- and control-dependent ambiguity aversion. The main contributions are (i) a tractable SDG approach to TIC robust controls, (ii) explicit equilibrium characterizations via first- and second-order adjoints, and (iii) numerical insights showing ambiguity aversion can alter effective risk more strongly than linear risk aversion, with practical implications for robust portfolio design under model misspecification.

Abstract

This paper studies robust time-inconsistent (TIC) linear-quadratic stochastic control problems, formulated by stochastic differential games. By a spike variation approach, we derive sufficient conditions for achieving the Nash equilibrium, which corresponds to a time-consistent (TC) robust policy, under mild technical assumptions. To illustrate our framework, we consider two scenarios of robust mean-variance analysis, namely with state- and control-dependent ambiguity aversion. We find numerically that with time inconsistency haunting the dynamic optimal controls, the ambiguity aversion enhances the effective risk aversion faster than the linear, implying that the ambiguity in the TIC cases is more impactful than that under the TC counterparts, e.g., expected utility maximization problems.

Robust Time-inconsistent Linear-Quadratic Stochastic Controls: A Stochastic Differential Game Approach

TL;DR

The paper develops a robust time-inconsistent linear-quadratic stochastic control framework by formulating a stochastic differential game between a decision-maker and an adversary who distorts the probability measure. Using spike variation and an augmented state that includes the Radon-Nikodým derivative, it derives sufficient conditions for Nash-equilibrium (time-consistent) policies without requiring a preference ordering, and provides semi-analytic examples with state- and control-dependent ambiguity aversion. The main contributions are (i) a tractable SDG approach to TIC robust controls, (ii) explicit equilibrium characterizations via first- and second-order adjoints, and (iii) numerical insights showing ambiguity aversion can alter effective risk more strongly than linear risk aversion, with practical implications for robust portfolio design under model misspecification.

Abstract

This paper studies robust time-inconsistent (TIC) linear-quadratic stochastic control problems, formulated by stochastic differential games. By a spike variation approach, we derive sufficient conditions for achieving the Nash equilibrium, which corresponds to a time-consistent (TC) robust policy, under mild technical assumptions. To illustrate our framework, we consider two scenarios of robust mean-variance analysis, namely with state- and control-dependent ambiguity aversion. We find numerically that with time inconsistency haunting the dynamic optimal controls, the ambiguity aversion enhances the effective risk aversion faster than the linear, implying that the ambiguity in the TIC cases is more impactful than that under the TC counterparts, e.g., expected utility maximization problems.
Paper Structure (14 sections, 9 theorems, 91 equations, 4 figures)

This paper contains 14 sections, 9 theorems, 91 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. The Models and the Controls
  4. Comparison between Different Equilibrium Controls
  5. Sufficient Conditions
  6. Examples and Well-posedness Results
  7. State-dependent ambiguity aversion
  8. Control-dependent ambiguity aversion
  9. Numerical Studies
  10. Varying Risk Aversion
  11. Varying Ambiguity Aversion
  12. Discussion: Relations between Risk and Ambiguity Aversions
  13. Conclusions
  14. Proofs

Key Result

Theorem 3.1

For $t\in [0,T)$ and $s \in [t, T]$, define If $h^*$ validates then inequality ineq:h* holds for $h^*$.

Figures (4)

  • Figure 1: Varying risk aversion
  • Figure 2: Varying ambiguity aversion
  • Figure 3: A linear relation between risk and ambiguity aversion
  • Figure 4: A quadratic relation between risk and ambiguity aversion

Theorems & Definitions (20)

  • Definition 2.2
  • Definition 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Lemma 4.1
  • Proposition 4.2
  • Lemma 4.3
  • Proposition 4.4
  • Corollary 5.1
  • ...and 10 more