The Itô integral and near-martingales in Riesz spaces

Abstract

A new class of the Itô integral for Brownian motion is defined and studied in the framework of Riesz spaces. The stochastic process with respect to this stochastic integral is non-adapted and it is a motivitation to construct near-martingales in Riesz spaces. Furthermore, we state Doob–Meyer decomposition theorem for near-submartingales in Riesz spaces.

Keywords:

• Itô integral
• near-martingale
• Brownian motion
• Riesz space
• instantly independent

MATHEMATICS SUBJECT CLASSIFICATION 2020:

• Primary 46A40
• Secondary 60G44
• 60G48

Acknowledgments

The authors would like to sincerely thank the referee for several valuable comments, which led to an improvement of this article.