Series talks to celebrate 60th anniversary of Math. Dept.
Title: Optimal Consumption with Reference to Past Spending Maximum
Speaker: Prof. Xun Li ( The Hong Kong Polytechnic University)
Time: 2020-7-8 (Wednesday) 10:00-13:00
Tencent Conference :
Conference ID：834 300 427
Password: 123456
Inviter: Jianli Liu
Abstract:
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 and $\lambda="1$." numerical examples are also presented to illustrate some theoretical conclusions and financial insights.