Weak equilibria for time‐inconsistent control: With applications to investment‐withdrawal decisions

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 time‐inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair false(trueû,Cfalse)$(\hat{u},C)$ of control‐stopping 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 nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant 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.

Binary options trading: A deep dive into user perspective and satisfaction

“…Meanwhile, based on Liang and Yuan [24] , on their research propose an investment-withdrawal decision model argued where the time-inconsistent decision makers are provided with both the opportunity to choose portfolios and the right to stop discretionarily. Most of binary options traders often have a mistrust of the trading platform and the brokers, which further exacerbates the withdrawal problem [23] .…”

Equilibria of time‐inconsistent stopping for one‐dimensional diffusion processes

“…Another approach to address the time‐inconsistency is to look for a subgame perfect Nash equilibrium; given the future selves follow the equilibrium strategy, the current self has no incentive to deviate from it. For equilibrium strategies, we refer to the works (Björk et al., 2021; Ekeland & Lazrak, 2010; Ekeland & Pirvu, 2008; He & Jiang, 2021; Ekeland & Lazrak, 2006; Hernández & Possamaï, 2020; Hamaguchi, 2021; Huang & Zhou, 2021; Wang & Yong, 2021; Wei et al., 2017) among others for time‐inconsistent control, and Christensen and Lindensjö (2020a, 2020b); Ebert and Strack (2018); He and Zhou (2022); Huang and Nguyen‐Huu (2018); Liang and Yuan (2021); Tan et al. (2021); Bodnariu et al.…”

“…The weak equilibrium concept for time inconsistent stopping is proposed in Christensen and Lindensjö (2018), and further studied in Christensen and Lindensjö (2020a); Liang and Yuan (2021); Tan et al. (2021).…”