Positivity (2014) 18:425–437
On the decompositions of T -quasi-martingales
on Riesz spaces
Jessica J. Vardy · Bruce A. Watson
Received: 17 October 2012 / Accepted: 26 August 2013 / Published online: 4 September 2013
© Springer Basel 2013
Abstract 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.
Keywords Riesz space · Conditional expectation · Quasi-martingale
Mathematics Subject Classiﬁcation (2000) 46A40 · 47B60 · 60G20 · 60G48
Quasi-martingales were ﬁrst introduced by H. Rubin in an invited lecture at the Institute
of Mathematical Statistics in 1956. In  quasi-martingales were formally introduced
and deﬁned by Fisk. Fisk gave necessary and sufﬁcient conditions under which a
quasi-martingale with continuous sample paths could be decomposed into the sum of
a martingale and a process having almost every sample path of bounded variation.
Orey, in , generalised Fisk’s results to right continuous processes (or F-processes,
in Orey’s terminology). Finally, in , Rao gave an elegant and greatly simpliﬁed
B. A. Watson was supported in part by NRF Grant IFR2011032400120 and by the Centre for Applicable
Analysis and Number Theory.
J. J. Vardy · B. A. Watson (
School of Mathematics, University of the Witwatersrand,
Private Bag X3, P.O. WITS, Johannesburg 2050, South Africa