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

Stéphane Goutte, Nadia Oudjane and Francesco Russo
january, 2013
Publication type:
Paper in peer-reviewed journals
Journal:
Stochastic Analysis and Applications, vol. 31, pp. 108–--141
arXiv:
images/icons/icon_arxiv.png 1202.0619
Keywords :
Föllmer-Schweizer decomposition, Kunita-Watanabe decomposition, Lévy processes, Characteristic functions, Processes with independent increments, Global and local quadratic risk minimization, Expectation and derivative operators.
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.
BibTeX:
@article{Gou-Oud-Rus-2013,
    author={Stéphane Goutte and Nadia Oudjane and Francesco Russo },
    title={On some expectation and derivative operators related to 
           integral representations of random variables with respect to a 
           PII process. },
    journal={Stochastic Analysis and Applications },
    year={2013 },
    month={1},
    volume={31 },
    pages={108–--141},
}