On some expectation and derivative operators related to integral representations of random variables with respect to a PII process.
january, 2013
Publication type:
Paper in peer-reviewed journals
Journal:
Stochastic Analysis and Applications, vol. 31, pp. 108--141
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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. }, doi={10.1080/07362994.2013.741395 }, journal={Stochastic Analysis and Applications }, year={2013 }, month={1}, volume={31 }, pages={108--141}, }