Abstract: In this talk I will discuss the definition and the basic properties of integrals over the real line when the integrand is predictable and the integrator e.g. could be a two-sided Brownian motion (i.e. it is indexed by the real line). The first problem one encounters is that the two-sided Brownian motion is not a martingale in any filtration. Thus, it turns out that the right framework for the integrator is an increment martingale, or more generally an increment semimartingale, rather than a martingale or a semimartingale. The second problem is that sometimes the choice of filtration is a delicate matter. I discuss this in relation to representation of generalized Ornstein-Uhlenbeck processes.
This talk is based on joint works with Andreas Basse-O'Connor and Svend-Erik Graversen.