Stochastic Integrals and Two Filtrations

Abstract

In the definition of the stochastic integral, apart from the integrand and the integrator, there is an underlying filtration that plays a role. Thus, it is natural to ask: {\it Does the stochastic integral depend upon the filtration?} In other words, if we have two filtrations, $({\mathcal F}_\centerdot)$ and $({\mathcal G}_\centerdot)$, a process $X$ that is semimartingale under both the filtrations and a process $f$ that is predictable for both the filtrations, then are the two stochastic integrals - $Y=\int f\,dX$, with filtration $({\mathcal F}_\centerdot)$ and $Z=\int f\,dX$, with filtration $({\mathcal G}_\centerdot)$ the same? When $f$ is left continuous with right limits, then the answer is yes. When one filtration is an enlargement of the other, the two integrals are equal if $f$ is bounded but this may not be the case when $f$ is unbounded. We discuss this and give sufficient conditions under which the two integrals are equal.