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Convex concentration inequalities and forward-backward stochastic calculus. (English) Zbl 1112.60034

Summary: Given \((M_t)_{t \in R_+}\) and \((M^*_t)_{t \in R_+}\), respectively, a forward and a backward martingale with jumps and continuous parts, we prove that \(E[\varphi (M_t+M^\ast_t)]\) is non-increasing in \(t\) when \(\varphi\) is a convex function, provided the local characteristics of \((M_t)_{t \in R_+}\) and \((M^\ast_t)_{t \in R_+}\) satisfy some comparison inequalities. We deduce convex concentration inequalities and deviation bounds for random variables admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component.

MSC:

60G44 Martingales with continuous parameter
39B62 Functional inequalities, including subadditivity, convexity, etc.
60E15 Inequalities; stochastic orderings
60F10 Large deviations
60H07 Stochastic calculus of variations and the Malliavin calculus
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