A Generalization of Stochastic Integration with Respect to Semimartingales

Abstract

On the real line, there exist $\sigma$-finite measures which are not Radon measures, but are nevertheless defined on all bounded intervals $\big(\text{e.g.} \frac{1}{x} \sin \frac{1}{x} dx, \text{or} \sum_n\frac{(-1)^n}{n} \delta_{1/n}\big).$ Similarly, in stochastic calculus, there exist processes that, though not semimartingales, can be obtained as stochastic integrals of predictable processes with respect to semimartingales. This paper deals with such processes.