Summable processes versus semimartingales

The classical stochastic integral $\int H dX$ is defined for real-valued semimartingales $X$. For processes with values in a Banach space $E$, the stochastic integral $\int H dX$ is defined for locally summable processes $X$, using a measure-theoretical approach.

We investigate the relationship between semimartingales and locally summable processes.

A real-valued, locally summable process is a special semimartingale. We prove that in infinite-dimensional Banach spaces, a locally summable process is not necessarily a semimartingale.