Finite-Horizon Optimal Investment with Transaction Costs: Construction of the Optimal Strategies

Abstract

We consider the problem of utility maximization in a Black-Scholes market in the presence of proportional transaction costs. While it is well-known that the value function of this problem is the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation (cf. Belak et al. (2013)), and that the HJB equation admits a classical solution on a reduced state space (cf. Dai and Yi (2009)), it has been an open problem to construct the optimal strategies. We provide a simple method to prove the existence of the candidate optimal strategies, verify their optimality, study the regularity in detail and show that value function coincides with the classical solution of the HJB on the reduced state space.