On the decompositions of T-quasi-martingales on Riesz spaces

The concept of a quasi-martingale is generalised to the Riesz space setting. Here we show that a quasi-martingale can be decomposed into the sum of a martingale and a quasi-potential. If, in addition, the quasi-martingale and its filtration are right continuous we show that the quasi-martingale can decomposed into the sum of a right continuous martingale and the difference of two positive right continuous potentials. The approach is measure-free and relies entirely on the order structure of Riesz spaces.