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

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

2012-02-03

arxiv.org/abs/1202.0619v1

Mathematics

Probability

29 pages

Scientific paper

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 F\"ollmer-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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