A model of optimal consumption under liquidity risk with random trading times * Huyˆ en PHAM Laboratoire de Probabilit´ es et Mod` eles Al´ eatoires CNRS, UMR 7599 Universit´ e Paris 7 e-mail: pham@math.jussieu.fr and Institut Universitaire de France Peter TANKOV Laboratoire de Probabilit´ es et Mod` eles Al´ eatoires CNRS, UMR 7599 Universit´ e Paris 7 e-mail: tankov@math.jussieu.fr September 2006 This version : April 2007 Abstract We consider a portfolio/consumption choice problem in a market model with liqui- dity risk. The main feature is that the investor can trade and observe stock prices only at exogenous Poisson arrival times. He may also consume continuously from his cash holdings, and his goal is to maximize his expected utility from consumption. This is a mixed discrete/continuous stochastic control problem, nonstandard in the literature. The dynamic programming principle leads to a coupled system of Integro-Differential Equations (IDE), and we provide a convergent numerical algorithm for the resolution to this coupled system of IDE. Several numerical experiments illustrate the impact of the restricted liquidity trading opportunities, and we measure in particular the utility loss with respect to the classical Merton consumption problem. Key words : liquidity, random trading times, portfolio/consumption problem, integrodif- ferential equations, cost of liquidity. MSC Classification (2000) : 93E20, 49K22, 91B28. JEL Classification : G11. * The authors are grateful to Fr´ ed´ eric Bonnans for fruitful discussions, and the two anonymous referees for valuable suggestions. 1