Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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A Note on the Wick Integral with Respect to Fractional Brownian Sheet

  • Rhee, Joon-Hee (Department of Business and Administration, Soong-Sil University) ;
  • Kim, Yoon-Tae (Department of Statistics, Hallym University)
  • Published : 2010.01.31
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Abstract

By using the white noise theory for fractional Brownian sheet, we give new representations of the Wick integrals of various types with respect to fractional Brownian sheet with Hurst parameters $H_1,H_2{\in}$(0, 1).

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References

  1. Biagini, F., Oksendal, B., Sulem, A. and Wallner, N. (2004). An introduction to white noise theory and Malliavin calculus for fractional Brownian motion, Proceedings: Mathematical, Physical and Engineering Sciences, 460, 347-372. https://doi.org/10.1098/rspa.2003.1246
  2. Cairoli, R. and Walsh, J. B. (1975). Stochastic integrals in the plane, Acta Mathematica, 134, 111-183. https://doi.org/10.1007/BF02392100
  3. Elliott, R. J. and van der Hoek, J. (2003). A general fractional white noise theory and applications to finance, Mathematical Finance, 13, 301-339. https://doi.org/10.1111/1467-9965.00018
  4. Holden, H., Oksendal, B., Uboe, J. and Zhang, T. (1996). Stochastic Partial Differential Equations, Birkhauser, Boston.
  5. Hu, Y., Oksendal, B. and Zhang, T. (2004). General fractional multiparameter white noise theory and stochastic partial differential equations, Communications in Partial Differential Equations, 29, 1-23. https://doi.org/10.1081/PDE-120028841
  6. Kim, Y. T. (2006). An Ito formula of generalized functionals and local time for fractional Brownian sheet, Stochastic Analysis and Applications, 24, 973-997. https://doi.org/10.1080/07362990600870082
  7. Kim, Y. T. (2009). A note on the differentiation formula in Stratonovich type for a fractional Brownian sheet, Journal of the Korean Statistical Society, 38, 259-265. https://doi.org/10.1016/j.jkss.2008.11.005
  8. Kim, Y. T. and Jeon, J. W. (2006). An Ito formula for a fractional Brownian sheet with arbitrary Hurst parameters, Proceedings of the American Mathematical Society, 134, 3677-3683. https://doi.org/10.1090/S0002-9939-06-08466-8
  9. Kim, Y. T., Jeon, J. W. and Park, H. S. (2008). Various types of stochastic integrals with respect to fractional Brownian sheet and their applications, Journal of Mathematical Analysis and Applications, 341, 1382-1398. https://doi.org/10.1016/j.jmaa.2007.10.071
  10. Kim, Y. T., Jeon, J. W. and Park, H. S. (2009). Differentiation formula in Stratonovich version for fractional Brownian sheet, Journal of Mathematical Analysis and Applications, 359, 106-125. https://doi.org/10.1016/j.jmaa.2009.05.037
  11. Kim, Y. T. and Park, H. S. (2009). Stratonovich calculus with respect to fractional Brownian sheet, Stochastic Analysis and Applications, 27, 1-22. https://doi.org/10.1080/07362990802558238
  12. Kim, Y. T. and Rhee, J. (2008). A note on Ito formula for a fractional Brownian sheet with Hurst parameters $H_1,\;H_2\;{\epsilon}$ (0, 1), Journal of the Korean Statistical Society, 37, 349-354. https://doi.org/10.1016/j.jkss.2008.02.005
  13. Tudor, C. A. and Viens, F. G. (2003). Ito Formula and Local Time for the Fractional Brownian Sheet, Electronic Journal of Probability, 8, 1-31.