Deviation inequalities for continuous martingales (Q1382466)

A class of continuous martingales \(M_t\) is considered that have a locally deterministic modulus in certain sense. It is shown that if they are appropriately normalized by their quadratic variation, then their probabilistic tails are Gaussian. Besides this main result, other consequences of the basic assumption are examined and energy inequalities and results on the smoothness of the sample functions of \(t\mapsto M_t\) are provided. They are used to obtain estimates for the smoothness of continuous additive functionals of multidimensional Brownian motion viewed as functions of their Revuz measures.