摘要：
We consider 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 multi-dimensional controlled diffusion model and propose a formal definition of their equilibriums. We show that an admissible pair $(\hat{u},C)$ of control-stopping policy is equilibrium if and only if the auxiliary function associated to it solves the extended HJB system, providing 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 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 hyperbolic discount.

报告人简介：
梁宗霞：清华大学数学科学系长聘教授, 博士生导师. 主要从事精算科学, 金融数学, 概率论，随机控制与优化等理论方面的研究. 有数十篇论文发表在上述领域里高水平学术期刊上如Mathematical Finance,   Insurance: Mathematics and Economics,  Scandinavian Actuarial Journal,  North American Actuarial Journal,  Stochastic Processes and their Applications,  Ann. Inst. Henri Poincare Probab. Statist.,  Journal of Functional Analysis,  SIAM Journal on Control and Optimization，EJOR.