On $L^2$-projections on a space of stochastic integrals

Abstract

Let $X$ be an $\mathbb{R}^d$-valued continuous semimartingale, $T$ a fixed time horizon and $\Theta$ the space of all $\mathbb{R}^d$ -valued predictable $X$ -integrable processes such that the stochastic integral $G(\vartheta)=\int\vartheta dX$ is a square-integrable semimartingale. A recent paper gives necessary and sufficient conditions on $X$ for $G_T(\Theta)$ to be closed in $L^2(P)$. In this paper, we describe the structure of the $L^2$-projection mapping an $\mathscr{F}_T$-measurable random variable $H \in L^2(P)$ on $G_T(\theta)$ and provide the resulting integrand $\vartheta^H \in \Theta$ feedback form. This is related to variance-optimal hedging strategies in financial mathematics and generalizes previous results imposing very restrictive assumptions on $X$. Our proofs use the variance-optimal martingale measure $\tilda{P}$ for $X$ and weighted norm inequalities relating $\tilda{P}$ to the original measure $P$.