A complete representation theorem for G-martingales

AbstractIn this article, we establish a complete representation theorem for G-martingales. Unlike the existing results in the literature, we provide the existence and uniqueness of the second-order term, which corresponds to the second-order derivative in Markovian case. The main ingredient of the article is a new norm for that second-order term, which is based on an operator introduced by Song.Keywords:: G-expectationsG-martingalesmartingale representation theoremnonlinear expectationsAMS Subject Classification:: 60H1060H30 Notes1. Email: peng@sdu.edu.cn2. Email: jianfenz@usc.eduAdditional informationFundingThe first author was partially supported by NSF of China [grant number 10921101] and National Basic Research Program of China (973 Program) [grant number 2007CB814900]. The second author was supported by NCMIS; Youth Grant of NSF [grant number 11101406]; Key Project of NSF [grant number 11231005]; Key Lab of Random Complex Structures and Data Science, CAS [grant number 2008DP173182]. The third author was supported in part by NSF [grant number DMS 10-08873].