Optimal consumption and portfolio selection problem with downside consumption constraints Yong Hyun Shin * , Byung Hwa Lim, U Jin Choi Department of Mathematics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305701, Republic of Korea Abstract We study a general optimal consumption and portfolio selection problem of an infinitely-lived investor whose consump- tion rate process is subjected to downside constraint. That is, her consumption rate is greater than or equals to some posi- tive constant. We obtain the general optimal policies in an explicit form using martingale method and Feynman–Kac formula. We derive some numerical results of optimal consumption and portfolio in the special case of a constant relative risk aversion (CRRA) utility function. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Consumption; Portfolio selection; Utility maximization; Downside consumption constraint; Martingale method 1. Introduction We study a general optimal consumption and portfolio selection problem of an infinitely-lived investor whose consumption rate process is subjected to downside constraint. That is, her consumption rate is greater than or equals to some positive constant. We obtain the general optimal policies in explicit forms using mar- tingale method and Feynman–Kac formula. We derive properties of the optimal policies and some numerical results of the optimal consumption and portfolio in the special case of a constant relative risk aversion (CRRA) utility function. Historically, Merton [7,8] introduced the dynamic programming method in order to study the optimal con- sumption and portfolio selection problem in continuous-time. Karatzas et al. [3] extended this work to the general utility function. They presented an explicit solution of a general consumption-portfolio problem using the dynamic programming method. Cox and Huang [1] and Karatzas et al. [4] introduced the martingale method independently. Lakner and Nygren [6] solved the portfolio optimization problem with both consumption and terminal wealth downside constraints using the gradient operator and Clark–Ocone formula in Malliavin calculus on a finite horizon. Since we only consider an infinite horizon case in this paper, we need not consider the 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.11.053 * Corresponding author. E-mail addresses: yhshin@kaist.ac.kr (Y.H. Shin), st2211@kaist.ac.kr (B.H. Lim), ujinchoi@kaist.ac.kr (U.J. Choi). Applied Mathematics and Computation 188 (2007) 1801–1811 www.elsevier.com/locate/amc