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A Doob-Meyer decomposition under model ambiguity: the case of compactness. (English) Zbl 1465.60036

Summary: We consider families of equivalent probability measures \(\mathcal{Q}\) with a property related to concepts known in the literature under different names such as rectangularity or multiplicative stability. For the problems considered in this paper such a property yields dynamical consistency. We prove under a weak-compactness assumption with general filtrations and continuous processes that all semimartingales have an additive decomposition as the sum of a predictable non-decreasing process and a universal local supermartingale, by this concept we mean a process that is a local supermartingale with respect to each element of \(\mathcal{Q}\). We also show that processes having a supermartingale property with respect to a superadditive nonlinear conditional expectation associated to the family \(\mathcal{Q}\) are always semimartingales under weak-compactness. These results are relevant in stochastic optimization problems including optimal stopping under model ambiguity.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
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