Extended conditional G-expectations and related stopping times

Abstract

In this paper, we extend the definition of conditional G-expectation to a larger space on which the dynamical consistency still holds. We can consistently define, by taking the limit, the conditional G-expectation for each random variable X, which is the downward limit (respectively, upward limit) of a monotone sequence {Xi } in L1G(Ω) . 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-expectation .