Fatou limits of stochastic integrals

Abstract

The convergence of stochastic integrals is essential to stochastic analysis, especially in applications to mathematical finance, where they model the gains associated with a self-financing strategy. However, Fatou convergence of $(X^n{)}_{n=1}^{\infty }$ $\unicode{x2014}$a notion introduced for its amenability to compactness principles$\unicode{x2014}$implies little about the sequence of Itô integrals ${\left({\int }_0^{\cdot }Yd{X}^n\right)}_{n=1}^{\infty }$ for a fixed integrand $Y$. Under a boundedness condition, we find convex combinations $({\tilde{X}}^n{)}_{n=1}^{\infty }$ of $(X^n{)}_{n=1}^{\infty }$ with Fatou limit $\tilde{X}$, such that ${\left({\int }_0^{\cdot }Yd{\tilde{X}}^n\right)}_{n=1}^{\infty }$ converges in a Fatou-like sense to ${\int }_0^{\cdot }Yd\tilde{X}$ for all continuous semimartingales $Y$. The result is sharp, in the sense that continuity of $Y$ cannot be relaxed to being the left limits process of a semimartingale.