Discrete stochastic integration in Riesz spaces

In this work we continue the developments of Kuo et al. (Indag Math 15:435-451, 2004; J Math Anal Appl 303:509-521, 2005) with the construction of the martingale transform or discrete stochastic integral in a Riesz space (measure-free) setting. The discrete stochastic integral is considered both in terms of a weighted sum of differences and via bilinear vector-valued forms. For this, analogues of the spaces L (2) and Mart(2) on Riesz spaces with a conditional expectation operator and a weak order unit are constructed using the f-algebra structure of the universal completion of the Riesz space and properties of the extension of the conditional expectation to its natural domain.