# On Some Expectation and Derivative Operators Related to Integral Representations of Random Variables with Respect to a PII process

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RR-FiME-12-05

### Stéphane GOUTTE, Nadia OUDJANE, Francesco RUSSO

Given a process with independent increments X (not necessarily a martingale) and a large class of square integrable r.v. H = f(XT ), f being the Fourier transform of a finite measure μ, we provide explicit Kunita-Watanabe and Föllmer-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.