In this paper asymptotic results in a long-term growth rate portfolio optimization model under both fixed and proportional transaction costs are obtained. More precisely, the convergence of the model when the fixed costs tend to 0 is investigated. A suitable limit model with purely proportional costs is introduced and the convergence of optimal boundaries, asymptotic growth rates, and optimal risky fraction processes is rigorously proved. The results are based on an in-depth analysis of the convergence of the solutions to the corresponding Hamilton–Jacobi–Bellman equations.

We consider a class of impulse control problems for general underlying strong Markov processes on the real line, which allows for an explicit solution. The optimal impulse times are shown to be of a threshold type and the optimal threshold is characterised as a solution of a (typically nonlinear) equation. The main ingredient we use is a representation result for excessive functions in terms of expected suprema.

A martingale approach to a dynamic program with general state and action spaces is taken. Several necessary and sufficient conditions are given for a policy to be optimal. The results comprehend and modify different criteria of optimality given for dynamic programming problems. Finally, two applications are stated.