数学系 Seminar 第1966期
“60周年”系庆系列报告
报告主题：Optimal Consumption with Reference to Past Spending Maximum
报告人：李迅 教授 （香港理工大学）
报告时间：2020年7月8日（周三） 11:00-13:00
参会方式：腾讯 会议
会议ID：834 300 427
会议密码：123456
邀请人：刘见礼
主办部门：理学院数学系
报告摘要：This work studies an infinite-time horizon optimal consumption problem under exponential utility, together with non-negativity constraint on consumption rate and a reference point to the past consumption peak. The performance is measured by the distance between the consumption rate and a fraction $0\leq\lambda\leq 1$ of the historical consumption maximum. To overcome its path-dependent nature, the consumption running maximum process is chosen as an auxiliary state process that renders the value function two dimensional depending on the wealth variable $x$ and the reference variable $h$. The associated Hamilton-Jacobi-Bellman (HJB) equation is expressed in different forms across three regions to take into account all constraints. By employing the dual transform and smooth-fit principle, the classical solution of the HJB equation is obtained in an analytical form, which in turn provides the feedback optimal investment and consumption. For $0<\lambda<1$, we are able to find four free boundary curves $x_1(h)$, $\breve{x}(h)$, $x_2(h)$ and $x_3(h)$ for the wealth level $x$ that are nonlinear functions of $h$ such that the feedback optimal consumption satisfies: (i) $c^*(x,h)=0$ when $x\leq x_1(h)$; (ii) $0<c^*(x,h)<\lambda h$ when $x_1(h)<x<\breve{x}(h)$; (iii) $\lambda h\leq c^*(x,h)<h$ when $\breve{x}(h)\leq x<x_2(h)$; (iv) $c^*(x,h)=h$ but the running maximum process remains flat when $x_2(h)\leq x<x_3(h)$; (v) $c^*(x,h)=h$ and the running maximum process increases when $x=x_3(h)$. Similar conclusions can be made in a simpler fashion for two extreme cases when $\lambda=0$ and $\lambda=1$. Numerical examples are also presented to illustrate some theoretical conclusions and financial insights.
报告人简介：李迅教授于1992年在上海科学技术大学数学系获学士学位，1995年在上海大学数学系获硕士学位，2000年在香港中文大学系统工程和工程管理系获博士学位。2000年-2001 年在香港中文大学系统工程和工程管理系开展博士后研究。2001 年-2003年, 在卡尔加里大学数学和计算金融实验室开展博士后研究。2003年-2007年, 在新加坡国立大学数学系任Visiting Fellow. 2007加入香港理工大学应用数学系，现为该系教授。他的主要研究兴趣为应用概率、随机控制论及其金融应用。其论文主要发表在SIAM Journal on Control and Optimization, Annals of Applied Probability, IEEE Transactions on Automatic Control, Automatica, Mathematical Finance and Quantitative Finance等国际著名期刊。
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