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

Abstract

The aim of this paper is to give an extension of the mean-variance hedging problem to the [Formula: see text]-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 [Formula: see text]-sense, which is the [Formula: see text]-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 [Formula: see text]-projections naturally is discussed.