Admissible Trading Strategies Under Transaction Costs

Abstract

A well known result in stochastic analysis reads as follows: for an \(\mathbb{R}\)-valued super-martingale X = (X t )0 ≤ t ≤ T such that the terminal value X T is non-negative, we have that the entire process X is non-negative. An analogous result holds true in the no arbitrage theory of mathematical finance: under the assumption of no arbitrage, an admissible portfolio process x + (H ⋅ S) verifying x + (H ⋅ S) T ≥ 0 also satisfies x + (H ⋅ S) t ≥ 0, for all 0 ≤ t ≤ T. In the present paper we derive an analogous result in the presence of transaction costs. In fact, we give two versions: one with a numeraire-based, and one with a numeraire-free notion of admissibility. It turns out that this distinction on the primal side perfectly corresponds to the difference between local martingales and true martingales on the dual side. A counter-example reveals that the consideration of transaction costs makes things more delicate than in the frictionless setting.