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

6 MATHRISK - Mathematical Risk handling
Inria Paris-Rocquencourt, UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech
Abstract : 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 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.
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Preprints, Working Papers, ...
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https://hal-ensta-paris.archives-ouvertes.fr//hal-00665852
Contributor : Francesco Russo <>
Submitted on : Thursday, February 2, 2012 - 8:30:17 PM
Last modification on : Thursday, April 29, 2021 - 3:20:26 AM
Long-term archiving on: : Thursday, May 3, 2012 - 3:10:46 AM

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• HAL Id : hal-00665852, version 1
• ARXIV : 1202.0619

### Citation

Stéphane Goutte, Nadia Oudjane, Francesco Russo. On some expectation and derivative operators related to integral representations of random variables with respect to a PII process. 2012. ⟨hal-00665852⟩

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