The finite-horizon consumption-investment and retirement problem with borrowing constraint
报告人简介：杨舟，华南师范大学数学科学学院，教授，博士导师。主要从事金融数学和随机控制方面的研究，主要研究方向为：美式衍生产品定价、最优投资组合、最优停时问题、金融中的自由边界问题。部分研究成果发表于MATH OPER RES、SIAM J CONTROL OPTIM、SIAM J MATH ANAL、J DIFFER EQUATIONS等期刊。曾主持五项国家基金和多项省部级基金。
报告内容介绍：In this paper, we study the optimization problem of an economic agent who chooses the best time for retirement as well as consumption and investment in the presence of a mandatory retirement date. Moreover, the agent faces the borrowing constraint which is constrained in the ability to borrow against future income during working. By utilizing the dual-martingale method for the borrowing constraint, we derive a dual two-person zero-sum game between a singular-controller and a stopper over finite-time horizon. The value of the game satisfies a min-max type of parabolic variational inequality involving both obstacle and gradient constraints, which gives rise to two time-varying free boundaries that correspond to the optimal retirement and the wealth binding, respectively. Using partial differential equation (PDE) techniques, including many technical and non-standard arguments, we establish the uniqueness and existence of a strong solution to the variational inequality, as well as the monotonicity and smoothness of the two free boundaries. Furthermore, the value of game is shown to be the solution to the variational inequality, and we establish a duality theorem to characterize the optimal strategy. To the best our knowledge, this paper is the first to study the zero-sum games between a singular-controller and a stopper over finite-time horizon in the mathematical finance literature.
