NON-CONCAVE PORTFOLIO OPTIMIZATION WITHOUT THE CONCAVIFICATION PRINCIPLE

QIAN SHUAIJIE

Publication

NON-CONCAVE PORTFOLIO OPTIMIZATION WITHOUT THE CONCAVIFICATION PRINCIPLE

QIAN SHUAIJIE

Abstract

Non-concave portfolio optimization problems appear in many areas of finance and economics. Existing literature solves some of these problems using the concavification principle, and derives equivalent, concave optimization problems whose value functions are still concave. We study non-concave portfolio optimization, where the concavification principle may not hold. In particular, we focus on two problems. First, we consider a non-concave utility maximization problem with portfolio constraints. We find that adding bounded portfolio constraints can significantly affect economic insights in the existing literature. As the resulting value function is likely discontinuous, we introduce a new definition of viscosity solution and show the convergence of monotone, stable, and consistent finite difference schemes via comparison principle. Second, we study a portfolio selection model with capital gains tax formulated by Ben Tahar, Soner and Touzi (2010), where the average tax basis is employed as an approximation. The associated HJB equation problem turns out to admit 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. Our approach sheds light on the robustness of the penalty method for general singular stochastic control problems.