On some estimates for bounded submartingales and the shift inequality

Details On some estimates for bounded submartingales and the shift inequality On some estimates for bounded submartingales and the shift inequality

2010-08-03

arxiv.org/abs/1008.0573v1

Mathematics

Probability

12 pages

Scientific paper

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 $% X=Y+M$ can be unbounded. In this paper for some classes of increasing convex functions $f$ we will find the upper bounds for $\lim_n\sup_XEf(Y_n)$, where the supremum is taken over all submartingales $(X_n),0\leq X_n\leq 1,n=0,1,...$. We apply the stochastic control theory to prove these results.

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