Robust Time-Inconsistent Stochastic Control Problems
20 Pages Posted: 13 Sep 2017 Last revised: 23 Mar 2018
Date Written: September 11, 2017
This paper establishes a general analytical framework for continuous-time stochastic control problems for an ambiguity-averse agent (AAA) with time-inconsistent preference, where the control problems do not satisfy Bellman's principle of optimality. The AAA is concerned about model uncertainty in the sense that she is not completely confident in the reference model of the controlled Markov state process and rather considers some similar alternative models. The problems of interest are studied within a set of dominated models and the AAA seeks for an optimal decision that is robust with respect to model risks. We adopt a game-theoretic framework and the concept of subgame perfect Nash equilibrium to derive an extended dynamic programming equation and extended Hamilton -- Jacobi -- Bellman -- Isaacs equations for characterizing the robust dynamically optimal control of the problem. We also prove a verification theorem to theoretically support our construction of robust control. To illustrate the tractability of the proposed framework, we study an example of robust dynamic mean-variance portfolio selection under two cases: 1. constant risk aversion; and 2. state-dependent risk aversion.
Keywords: Robust Time-Consistent Stochastic Control, Time-Inconsistent Preference, Extended Dynamic Programming Approach, Hamilton--Jacobi--Bellman--Isaacs Equations, Robust Dynamic Mean-Variance Portfolio, State-Dependent Risk and Ambiguity Aversion, Abel's Differential Equations
JEL Classification: C61, C68, C72, D81, D92, G11
Suggested Citation: Suggested Citation