Multidimensional Breeden-Litzenberger representation for state price densities and static hedging

In this article, we consider European options of type $h(X^1_T, X^2_T,\ldots, X^n_T)$ depending on several underlying assets. We study how such options can be valued in terms of simple vanilla options in non-specified market models. We consider different approaches related to static hedging and derive several pricing formulas for a wide class of payoff functions $h:\R_+^n\rightarrow \R$. We also give new relations between prices of different options both in one dimensional and multidimensional case.