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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.

MSC:

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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