Weak equilibria for time-inconsistent control: with applications to investment-withdrawal decisions

TL;DR

This paper develops a framework for analyzing time-inconsistent stopping control problems, characterizing equilibria via extended HJB systems, with applications in finance and control theory.

Contribution

It introduces a formal definition of equilibria for stopping control problems and provides methods to verify or exclude such equilibria, including explicit solutions and existence results.

Findings

Explicit equilibrium for wealth-dependent reward functions

Non-existence of equilibrium for constant proportion strategies with non-exponential discounting

Existence of non-constant equilibria described by boundary value problems

Abstract

This paper considers time-inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the timeinconsistent stopping control problems under general multi-dimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair $(\hat{u},C)$ of controlstopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a non-exponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general non-constant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two-dimensional example with a hyperbolic discount.