Some Norm Estimates for Semimartingales--- Under Linear and Nonlinear Expectations

Details Some Norm Estimates for Semimartingales--- Under Linear and Nonlinear Expectations Some Norm Estimates for Semimartingales--- Under Linear and Nonlinear Expectations

2011-07-20

arxiv.org/abs/1107.4020v1

Mathematics

Probability

27 pages

Scientific paper

In this paper we introduce two types of norms for semimartingales, under both linear and nonlinear expectations. The first norm is motivated by quasimartingales, and characterizes square integrable semimartingales. The second norm characterizes the absolute continuity of the finite variation part of the semimartingale with respect to the Lebesgue measure. One typical example of nonlinear expectation is the $G$-expectation introduced by Peng \cite{Peng2}. By applying our estimates, we prove a Doob-Meyer type decomposition for $G$-submartingales, and obtain the component $\Gamma$ in the $G$-martingale representation theorem, which improves the result of Soner-Touzi-Zhang \cite{STZa}.

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