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Given a process with independent increments $X$ (not necessarily amartingale) and a large class of square integrable r.v. $H=f(X_T)$, $f$ beingthe Fourier transform of a finite measure $\mu$, we provide explicitKunita-Watanabe and F\"ollmer-Schweizer decompositions. The representation isexpressed by means of two significant maps: the expectation and derivativeoperators related to the characteristics of $X$. We also provide an explicitexpression for the variance optimal error when hedging the claim $H$ withunderlying process $X$. Those questions are motivated by finding the solutionof the celebrated problem of global and local quadratic risk minimization inmathematical finance.

On some expectation and derivative operators related to integral representations of random variables with respect to a PII process