在线学术报告

报告标题: Optimal Consumption with Reference to Past Spending Maximum

报告人：李迅(教授，香港理工大学)

报告摘要:  This paper studies an infinite 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 the piecewise manner across different regions to take into account constraints. By employing the dual transform and smooth-fit principle, the classical solution of the HJB equation is obtained in an ****ytical form, which in turn provides the feedback optimal investment and consumption. For $0<\lambda<1$, we are able to find four 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 $\lambda=0$ and $\lambda=1$. Numerical examples are also presented to illustrate some theoretical results and financial insights.

报告时间：2020年12月15日 (周二)下午15:00-16:00

报告地点：腾讯会议（会议ID：301992278）

报告人简介：Xun Li (李迅) received BSc from the Department of Mathematics at Shanghai University of Science and Technology in 1992, and obtained MSc in the Department of Mathematics at Shanghai University in 1995. He completed his PhD in the Department of Systems Engineering and Engineering Management at the Chinese University of Hong Kong in 2000, and he stayed with the same department as a postdoctoral research fellow until 2001. From 2001 to 2003, he was a postdoctoral fellow in the Mathematical and Computational Finance Laboratory at University of Calgary. From 2003 to 2007, he was a visiting fellow (research assistant professor)  in the Department of Mathematics at the National University of Singapore. He joined the Department of Applied Mathematics at the Hong Kong Polytechnic University as Assistant Professor in 2007, Associate Professor in 2013, and is currently Professor. His main research areas are stochastic control and applied probability with financial applications, and he has published in journals such as SIAM Journal on Control and Optimization, Annals of Applied Probability, Journal of Differential Equations, IEEE Transactions on Automatic Control, Automatica, Mathematical Finance, and Quantitative Finance.

邀请人：喻高航