# American Institute of Mathematical Sciences

December  2021, 6(4): 369-390. doi: 10.3934/puqr.2021018

## Extended conditional G-expectations and related stopping times

 1 Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, Shandong, China 2 School of Mathematics, Shandong University, Jinan 250100, Shandong, China

humingshang@sdu.edu.cn

Received  November 29, 2021 Accepted  December 05, 2021 Published  December 2021

Fund Project: Mingshang Hu is supported by National Key R&D Program of China (Grant No. 2018YFA0703900) and the National Natural Science Foundation of China (Grant No. 11671231). Shige Peng is supported by the Tian Yuan Projection of the National Natural Science Foundation of China (Grant Nos. 11526205 and 11626247) and the National Basic Research Program of China (973 Program) (Grant No. 2007CB814900 (Financial Risk)).

In this paper, we extend the definition of conditional $G{\text{-}}{\rm{expectation}}$ to a larger space on which the dynamical consistency still holds. We can consistently define, by taking the limit, the conditional $G{\text{-}}{\rm{expectation}}$ for each random variable $X$, which is the downward limit (respectively, upward limit) of a monotone sequence $\{X_{i}\}$ in $L_{G}^{1}(\Omega)$. To accomplish this procedure, some careful analysis is needed. Moreover, we present a suitable definition of stopping times and obtain the optional stopping theorem. We also provide some basic and interesting properties for the extended conditional $G{\text{-}}{\rm{expectation}}$.

Citation: Mingshang Hu, Shige Peng. Extended conditional G-expectations and related stopping times. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 369-390. doi: 10.3934/puqr.2021018
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