Some norm estimates for semimartingales. (English) Zbl 1288.60054

The first goal of the paper is to introduce a norm which characterizes square integrable semimartingales, under a fixed (linear) probability measure. The main feature is that the norm involves only the semimartingale itself, without involving directly its Doob-Meyer decomposition. It is proved that a progressively measurable process is a square integrable semimartingale if and only if it has finite norm. Then the norm is extended to semimartingales under nonlinear expectations, in particular, the \(G\)-expectation. It is proved that any progressively measurable process with finite norm under \(G\)-expectation has to be a semimartingale under each probability measure. Using a similar idea, a new norm for the barriers of doubly reflected backward stochastic differential equations (BSDEs) is introduced and some a priori estimates for the solutions are established. The introduced norm provides an alternative but more tractable characterization for the standard Mokobodzki’s condition.


60G48 Generalizations of martingales
60G46 Martingales and classical analysis
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
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