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Minimal variance hedging for fractional Brownian motion. (English) Zbl 1056.60033

A stochastic (pathwise) integral was defined with respect to the fractional Brownian motion in case of Hurst parameter \(H>\frac 12\) by T. E. Duncan, Y. Hu and B. Pasik-Duncan [SIAM J. Control Optimization 38, No. 2, 582–612 (2000; Zbl 0947.60061)], or R. J. Eliott and J. van der Hoek [Math. Finance 13, No. 2, 301–330 (2003; Zbl 1069.91047)] or Y. Hu and B. Øksendal [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6, No. 1, 1–32 (2003; Zbl 1045.60072)] thanks to Wick product. Here the aim of the paper is to extend this stochastic integral to a \(d\)-dimensional fractional Brownian motion, the components of which being independent. The isometry property is preserved. A multi-dimensional Itô formula and an integration by parts formula are provided. Finally, there is an application to the problem of minimal variance hedging in a possible incomplete market driven by a \(d\)-dimensional fractional Brownian motion. We have to stress that the limit of these technics is the condition that any Hurst parameter is to be \(>\frac 12\).

MSC:

60G18 Self-similar stochastic processes
60H05 Stochastic integrals
91B28 Finance etc. (MSC2000)
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