Deviation inequalities for continuous martingales

Abstract

We consider a broad class of continuous martingales whose local modulus of continuity is in some sense deterministic. We show that such martingales have Gaussian probability tails, provided we appropriately normalize them by their quadratic variation. As other applications of our methods, we provide energy inequalities and prove a new sufficient condition for the joint continuity of continuous additive functionals of Brownian motion indexed by their Revuz measures.