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arXiv · 1008.0573

On some estimates for bounded submartingales and the shift inequality

Leonid Galtchouk·Isaac Sonin
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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 $% X=Y+M$ 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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Leonid Galtchouk, Isaac Sonin. 2010-08-03. On some estimates for bounded submartingales and the shift inequality. https://arxiv.org/abs/1008.0573

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