This paper is concerned with numerical solutions to a singular stochastic control problem arising from the continuous-time portfolio selection with proportional transaction costs. The associated value function is governed by a variational inequality with gradient constraints. We propose a penalty method to deal with the gradient constraints and employ a finite difference discretization. Convergence analysis is presented. We also show that the standard penalty method can be applied in the case

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... a single risky asset where the problem can be reduced to a standard variational inequality. Numerical results are given to demonstrate the efficiency of the methods and to examine the behaviour of the optimal trading strategy. . Dai would like to thank his students Joline A.V. Uichanco and Kaiyun Chong for parts of numerical implementation. He also thanks Yue-Kuen Kwok and Xingye Yue for helpful comments and an anonymous referee for valuable suggestions.