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

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Stéphane Goutte
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Francesco Russo

#### 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.

#### Domains

Mathematics [math] Probability [math.PR]

### Dates and versions

hal-00665852 , version 1 (02-02-2012)

### Identifiers

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

### Cite

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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