On some estimates for bounded submartingales and the shift inequality

TL;DR

This paper investigates bounds on the expected values of increasing predictable processes derived from bounded submartingales, utilizing stochastic control theory to establish these estimates for specific classes of convex functions.

Contribution

It introduces new upper bounds for the limit supremum of expected values of the predictable component in the Doob decomposition for certain convex functions, expanding understanding of submartingale behavior.

Findings

Established upper bounds for $ ext{lim}_n ext{sup}_X E f(Y_n)$ for specific convex functions.

Applied stochastic control theory to derive these bounds.

Provided insights into the unboundedness of the predictable process in the Doob decomposition.

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.