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 It\^o integrals $\left(\int_{0}^{\cdot}YdX^{n}\right)_{n=1}^{\infty}$ for a fixed integrand $Y$. Under a boundedness condition, we find convex combinations $(\widetilde{X}^{n})_{n=1}^{\infty}$ of $(X^{n})_{n=1}^{\infty}$ with Fatou limit $\widetilde{X}$, such that $\left(\int_{0}^{\cdot}Yd\widetilde{X}^{n}\right)_{n=1}^{\infty}$ converges in a Fatou-like sense to $\int_{0}^{\cdot}Yd\widetilde{X}$ 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.