Uniform Integrability of Square Integrable Martingales

Abstract

Let $(M_t, \mathscr{F}_t)_{t \geqq 0}$ be a continuous square integrable martingale and let $A_t$ be the natural increasing process in the Doob decomposition of $M_t^2$. Extending a result of Burgess Davis we show that there exist constants $C_1$ and $C_2$ such that $C_1 E\lbrack A_t^{\frac{1}{2}}\rbrack \leqq E\lbrack \sup_{s \leqq t} |M_s|\rbrack \leqq C_2 E\lbrack A_t^{\frac{1}{2}}\rbrack$ for all $t > 0$. Now if $A_\infty = \lim_{t\rightarrow \infty} A_t$, we find moment conditions on $A_\infty$ which relate to uniform integrability of $M_t$. In particular, $E\lbrack A_\infty^{\frac{1}{2}}\rbrack < \infty$ implies $M_t$ is uniformly integrable which implies $E\lbrack A_\infty^{1/\delta}\rbrack < \infty$ for all $\delta > 4$.