Weighted exponential inequality for differentially subordinate martingales

摘要

The paper contains a study of weighted exponential inequalities for differentially subordinate martingales, under the assumption that the underlying weight satisfies Muckenhoupt's condition A(infinity). The proof exploits certain functions enjoying appropriate size conditions and concavity. The martingales are adapted, uniformly integrable, and cadlag we do not assume any path-continuity restrictions. Because of this generality, we need to handle jump parts of processes which forces us to construct a Bellman function satisfying a stronger condition than local concavity. As a corollary, we will establish some new weighted L-p estimates for differential subordinates of bounded martingales.