We consider the problem of portfolio optimisation with general càdlàg price processes in the presence of proportional transaction costs. In this context, we develop a general duality theory. In particular, we prove the existence of a dual optimiser as well as a shadow price process in an appropriate generalised sense. This shadow price is defined by means of a “sandwiched” process consisting of a predictable and an optional strong supermartingale, and pertains to all strategies that remain solvent under transaction costs. We provide examples showing that, in the general setting we study, the shadow price processes have to be of such a generalised form.
"Duality theory for portfolio optimisation under transaction costs." Ann. Appl. Probab. 26 (3) 1888 - 1941, June 2016. https://doi.org/10.1214/15-AAP1136