Penalty method for portfolio selection with capital gains tax

Many finance problems can be formulated as a singular stochastic control problem, where the associated Hamilton-Jacobi-Bellman (HJB) equation takes the form of variational inequality, and its penalty approximation equation is linked to a regular control problem. The penalty method, combined with a finite difference scheme, has been widely used to numerically solve singular control problems, and its convergence analysis in literature relies on the uniqueness of solution to the original HJB equation problem. We consider a singular stochastic control problem arising from continuous-time portfolio selection with capital gains tax, where the associated HJB equation problem admits infinitely many solutions. We show that the penalty method still works and converges to the value function, which is the minimal (viscosity) solution of the HJB equation problem. The key step is to prove that any admissible singular control can be approximated by a sequence of regular controls related to the corresponding penalized equation problem that admits a unique solution. Numerical results are presented to demonstrate the efficiency of the penalty method and to better understand optimal investment strategy in the presence of capital gains tax. Our approach sheds light on the robustness of the penalty method for general singular stochastic control problems.