Stochastic Integrals and Two Filtrations

TL;DR

This paper investigates whether stochastic integrals depend on the underlying filtration, providing conditions under which integrals with different filtrations are equal, especially focusing on cases with bounded and unbounded integrands.

Contribution

It offers new insights and sufficient conditions for the equality of stochastic integrals across different filtrations, extending understanding beyond the case of bounded integrands.

Findings

When the integrand is left continuous with right limits, integrals are equal across filtrations.

For filtration enlargements, integrals are equal if the integrand is bounded.

Unbounded integrands may lead to differences in the stochastic integrals across 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.