An alternative proof of a result of Takaoka

In Karatzas and Kardaras's paper on semimartingale financial models, it is proved that the NUPBR condition is a property of the local characteristic of the asset process alone. In Takaoka's paper on NUPBR, it is proved that the NUPBR condition is equivalent to the existence of a simga-martingale deflator. However, Takaoka's paper founds its proof on Delbaen and Schachermayer's fundamental asset pricing theorem, i.e. the NFLVR condition, which is not a pure property of the local characteristic of the asset process. In this paper we give an alternative proof of the result of Takaoka, which makes use only the properties of the local characteristic of the asset process.