Doob's martingale inequality

The Doob Maximalungleichung is one of the key inequalities in the stochastics. In addition to the Burkholder 's inequality it is one of the most common methods for calculating the (stochastic ) order of ( steady ) martingales.

Formulation

It should be a martingale or nonnegative submartingale and. In the case of continuous time is right-continuous. Then for all:

It denotes the Lp- norm. Note that the conjugate to real number, i.e. it is. According to the main proof step is the application of Hölder 's inequality.