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Extended conditional G-expectations and related stopping times

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)).

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  • 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}} $.

    Mathematics Subject Classification: 60H10, 60H30.

    Citation:

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