A representation theorem for set-valued submartingales

Abstract

The integral representation theorem for martingales has been widely used in probability theory. In this work, we propose and prove a general representation theorem for a class of set-valued submartingales. We also extend the stochastic integral representation for non-trivial initial set-valued martingales. Moreover, we show that this result covers the existing ones in the literature for both degenerated and non-degenerated set-valued martingales.

Keywords:

• Set-valued integrals
• set-valued martingales
• random sets
• set-valued stochastic processes
• interval-Valued martingale
• Martingale representation theorem

Acknowledgments

The authors would like to thank the two anonymous referees for their valuable comments and useful suggestions that helped to improve the quality of the paper.

Disclosure statement

The authors declare that they have no conflict of interest.

Additional information

Funding

The first author, L.T. Tuyen, was supported by Institute of Information Technology, Vietnam Academy of Scinece and Technology, under the project no. CS.23.02.