Optimal Consumption in a Brownian Model with Absorption and Finite Time Horizon

We construct ϵ-optimal strategies for the following control problem: Maximize \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {E}[ \int_{[0,\tau)}e^{-\beta s}\,dC_{s}+e^{-\beta\tau}X_{\tau}]$\end{document}, where Xt=x+μt+σWt−Ct, τ≡inf{t>0|Xt=0}∧T, T>0 is a fixed finite time horizon, Wt is standard Brownian motion, μ, σ are constants, and Ct describes accumulated consumption until time t. It is shown that ϵ-optimal strategies are given by barrier strategies with time-dependent barriers.