Impulse Control and Optimal Portfolio Selection with General Transaction Cost

In this paper we study an optimal portfolio selection problem under general transaction cost. We consider a simplified financial market that consists of a risk free asset and a risky asset, but the admissible portfolios are only allowed to have piecewise constant paths, reflecting a more practical perspective. The problem is then reduced to an impulse control problem with subadditive transaction costs. We prove the existence of the optimal strategy for a fairly large class of cost functionals, and we show that the number of trading times is a random variable with finite expectation. Our result covers the cost functionals such as the commonly used fixed cost case, and more importantly, those subadditive cost functionals possibly without fixed cost components. Unlike the commonly seen results that characterize the value function via variational inequalities, we attack the optimal strategy directly.