$\mathcal{L}^p$-PROJECTIONS OF RANDOM VARIABLES AND ITS APPLICATION TO FINANCE

The aim of this paper is to give an extension of the mean-variance hedging problem to the $\mathcal{L}^p$-setting, where 1 < p < ∞. Remark that the mean-variance hedging is corresponding to the case where p = 2. Firstly, we prove that the unique existence of the optimal hedging strategy in the $\mathcal{L}^p$-sense, which is the $\mathcal{L}^p$-projection of the underlying contingent claim onto a suitable space of stochastic integrations. Next, we obtain its feedback representation under some additional assumptions. Moreover, the valuation problem induced by the $\mathcal{L}^p$-projections naturally is discussed.