On some estimates for bounded submartingales and the shift inequality

Abstract

It is well known that if a submartingale $X$ is bounded then the increasing predictable process $Y$ and the martingale $M$ from the Doob decomposition can be unbounded. In this paper for some classes of increasing convex functions $f$ we will find the upper bounds for ${\lim }_n{\sup }_{X}Ef(Y_n)$, where the supremum is taken over all submartingales $(X_n),0\le X_n\le 1,n=0,1,...$. We apply the stochastic control theory to prove these results.

12 pages