On some expectation and derivative operators related to integral   representations of random variables with respect to a PII process
  Given a process with independent increments $X$ (not necessarily a martingale) and a large class of square integrable r.v. $H=f(X_T)$, $f$ being the Fourier transform of a finite measure $\mu$, we provide explicit Kunita-Watanabe and F\"ollmer-Schweizer decompositions. The representation is expressed by means of two significant maps: the expectation and derivative operators related to the characteristics of $X$. We also provide an explicit expression for the variance optimal error when hedging the claim $H$ with underlying process $X$. Those questions are motivated by finding the solution of the celebrated problem of global and local quadratic risk minimization in mathematical finance. 
St\'ephane Goutte (LPMA), Nadia Oudjane (LAGA), Francesco Russo (ENSTA   ParisTech, INRIA Rocquencourt, UMA)
math.PR