Properly discounted asset prices are semimartingales

We study general undiscounted asset price processes, which are only assumed to be non- negative, adapted and RCLL (but not a priori semimartingales). Traders are allowed to use simple (piecewise constant) strategies. We prove that under a discounting-invariant condition of absence of arbitrage, the original prices discounted by the value process of any simple strategy with positive wealth must follow semimartingales. We also establish a corresponding version of the fundamental theorem of asset pricing that involves supermartingale discounters with an additional strict positivity property