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On some expectation and derivative operators related to integral representations of random variables with respect to a PII process. (English) Zbl 1288.60058

The objective of this paper is to provide explicit expressions for the Kunita-Watanabe and the Föllmer-Schweizer decompositions, as well as for the mean-variance hedging strategy, in case the driving process has independent increments. These decompositions are valid under any probability measure, and not just under a martingale measure.
The authors consider a random variable / payoff of the form \(f(X_T)\), where \(X\) is the driving process and the function \(f\) can be represented as the Fourier transform of a finite measure (see also [F. Hubalek et al., Ann. Appl. Probab. 16, No. 2, 853–885 (2006; Zbl 1189.91206)]). They provide explicit expressions for the Kunita-Watanabe and Föllmer-Schweizer decompositions, and for the mean-variance hedging strategy, using the properties of the log-characteristic function of \(X_t\) as well as results from Fourier analysis. They also relate their representations with analogous results derived using Malliavin calculus for processes with jumps.

MSC:

60G51 Processes with independent increments; Lévy processes
60H05 Stochastic integrals
91G20 Derivative securities (option pricing, hedging, etc.)

Citations:

Zbl 1189.91206
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