Discrete stochastic integration in Riesz spaces

Abstract

In this work we continue the developments of W.-C. Kuo, C. C. A. Labuschagne, B. A. Watson, Discrete time stochastic processes on Riesz spaces, Indag. Math., 15 (2004), 435-451, and W.-C. Kuo, C. C. A. Labuschagne, B. A. Watson, Conditional Expectations on Riesz Spaces, J. Math. Anal. Appl., 303 (2005), 509-521, 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.