Extended Conditional G-Expectations and Related Stopping Times

Abstract

In this paper we extend the definition of time conditional G-expectations ${\hat{\mathbb{E}}}_t[\cdot ]$ to a larger domain on which the dynamical consistency still holds. In fact we can consistently define, by taking the limit, the time conditional expectations for each random variable $X$ which is the downward limit (resp. upward limit) of a monotone sequence $\{X_i\}$ in $L_G^1(\Omega )$. To accomplish this procedure, some careful analysis is needed. Moreover, we give 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-expectations.

28 pages

References in corpus (4)

• A theoretical framework for the pricing of contingent claims in the presence of model uncertainty
• A New Central Limit Theorem under Sublinear Expectations
• G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty
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• G-Doob-Meyer Decomposition and its Application in Bid-Ask Pricing for American Contingent Claim Under Knightian Uncertainty
• Pseudo-independence, independence and related limit theorems under sublinear expectations
• On the exit times of SDEs driven by $G$-Brownian motion