Mean-variance hedging for continuous processes: New proofs and examples

Let $X$ be a special semimartingale of the form $X=X_0+M+\int d\langle M\rangle\,\widehat\lambda$ and denote by $\widehat K=\int \widehat\lambda^{\rm tr}\,d\langle M\rangle\,\widehat\lambda$ the mean-variance tradeoff process of $X$. Let $\Theta$ be the space of predictable processes $\theta$ for which the stochastic integral $G(\theta)=\int\theta\,dX$ is a square-integrable semimartingale. For a given constant $c\in{\Bbb R}$ and a given square-integrable random variable $H$, the mean-variance optimal hedging strategy $\xi^{(c)}$ by definition minimizes the distance in ${\cal L}^2(P)$ between $H-c$ and the space $G_T(\Theta)$. In financial terms, $\xi^{(c)}$ provides an approximation of the contingent claim $H$ by means of a self-financing trading strategy with minimal global risk. Assuming that $\widehat K$ is bounded and continuous, we first give a simple new proof of the closedness of $G_T(\Theta)$ in ${\cal L}^2(P)$ and of the existence of the FÃllmer-Schweizer decomposition. If moreover $X$ is continuous and satisfies an additional condition, we can describe the mean-variance optimal strategy in feedback form, and we provide several examples where it can be computed explicitly. The additional condition states that the minimal and the variance-optimal martingale measures for $X$ should coincide. We provide examples where this assumption is satisfied, but we also show that it will typically fail if $\widehat K_T$ is not deterministic and includes exogenous randomness which is not induced by $X$.