Stochastic integration on the real line

Stochastic integration on the predictable $\sigma$-field with respect to increment semimartingales, and, more generally, $\sigma$-finite $L^0$-valued measures is studied. The latter are also known as formal semimartingales. In particular, the triplet of $\sigma$-finite measures is introduced and used to characterize the set of integrable processes. Special attention is given to Lévy processes indexed by the real line. Surprisingly, many of the basic properties break down in this situation compared to the usual $\mathbf{R}_+$ case. The results enable us to define, represent, and study different classes of stationary processes.