  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.
