Convex Concentration Inequalities and Forward-Backward Stochastic Calculus

Abstract

Given $(M_t)_{t\in \mathbb{R}_+}$ and $(M^*_t)_{t\in \mathbb{R}_+}$ respectively a forward and a backward martingale with jumps and continuous parts, we prove that $E[\phi (M_t+M^*_t)]$ is non-increasing in $t$ when $\phi$ is a convex function, provided the local characteristics of $(M_t)_{t\in \mathbb{R}_+}$ and $(M^*_t)_{t\in \mathbb{R}_+}$ 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