Sharp Bounds for the Total Variance of Uniformly Bounded Semimartingales

Abstract

Let $S_n = f + X_1 + \cdots + X_n$ be an expectation-decreasing semimartingale with values in the unit interval, and let $V_n$ be the conditional variance of $X_n$ given the past. Then $E(\sum V_n)$ is less than $f(2 - f)$, and this bound is sharp. Sharper bounds are available if the process $S_0, S_1, \cdots$ satisfies suitable additional constraints.