Generalization of an Inequality of Birnbaum and Marshall, with Applications to Growth Rates for Submartingales

摘要

The well-known submartingale maximal inequality of Birmbaum and marshall (1961) is generalized to provide upper tail inequalities for suprema of processes which are products of a submartingale by a nonincreasing nonnegative predictable process. The new inequalities are proved by applying an inequality of Lenglart (1977), and are then used to provide best-possible universal growth-rates for a general submartingale in terms of the predictable compensator of its positive part. Applications of these growth rates include strong asymptotic upper bounds on solutions to certain stochastic differential equations, and strong asymptotic lower bounds on Brownian-motion occupational-times. (Author)