Weak equilibria for time‐inconsistent control: With applications to investment‐withdrawal decisions
研究了需要同时做出控制和停止策略的时间不一致问题，提出了均衡的正式定义，并通过扩展HJB系统验证均衡解，提供了数学金融和最优停止的应用实例。
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 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.