## A general fractional white noise theory and applications to finance.(English)Zbl 1069.91047

In this paper, the authors present a new framework for a fractional Browinian motion in which processes with all Hurst indices in $$(0,1)$$ can be considered on the space of tempered distributions under the same probability measure. A key result is the generalized Itô isometry for stochastic integrals with respect to the fractional Brownian motion. The paper extends the contribution by Hu and Øksendal (2000) with the Hurst index in $$(1/2,1)$$. Option pricing formulas in a multiple fractional Brownian Black-Scholes market are derived.

### MSC:

 91B28 Finance etc. (MSC2000) 91B70 Stochastic models in economics 60H40 White noise theory
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### References:

 [1] DOI: 10.1007/s007800050006 · Zbl 0940.91023 [2] Anderson T., J. Finance 52 pp 975– (1997) [3] DOI: 10.1088/1469-7688/2/3/302 [4] DOI: 10.1023/A:1008634027843 · Zbl 0924.60034 [5] Duncan T. E., SIAM J. Control Optim 38 pp 582– (2000) [6] Hida T., White Noise Analysis (1993) · Zbl 0791.60029 [7] Hille E., Ann. Math 27 (2) pp 427– (1958) [8] Holden H., Stochastic Partial Differential Equations (1996) [9] Y.Hu, and B.Oksendal(2000 ): Fractional White Noise Analysis and Applications to Finance . Preprint, University of Oslo. [10] Ito K., J. Math. Soc. Japan 3 pp 157– (1951) [11] Kuo H.-H., White Noise Distribution (1996) [12] Lindstom T., Bull. London Math. Soc 25 pp 83– (1993) [13] Los C., Wavelet Multiresolution Analysis of High Frequency Asian Exchange Rates [14] Muller W. A., A Practical Guide to Heavy Tails: Statistical Techniques and Applications pp 55– (1998) [15] Rogers L. C. G., Math. Finance 7 pp 95– (1997) [16] Samorodnitsky G., Stable Non-Gaussian Random Processes (1994) · Zbl 0925.60027 [17] Samko S. G., Fractional Integrals and Derivatives: Theory and Applications (1987) [18] I.Simonsen, and K.Sneppen(2001 ): Anti-correlations in the Nordic Electricity Spot Market . Preprint, Norwegian University of Science and Technology. [19] Thangavelu S., Lectures on Hermite and Laguerre Expansions (1993) · Zbl 0791.41030
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