A summability criterion for stochastic integration

In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes $X$ with values in a Banach space $E$; namely, $X$ is of class (LD), and the stochastic measure $I_{X}$ is bounded and strongly additive in $L_{E}^{p}$ (in particular, if $I_{X}$ is bounded in $L_{E}^{p}$ and $c_{0}\nsubseteq E$) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with $p$-integrable semivariation or $p$-integrable variation, the conditions of $I_{X}$ to be bounded and have bounded semivariation are superfluous. For processes with $1$-integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.