Modeling Continuous Time Series Driven by Fractional Gaussian Noise

  • Winston C. Chow
  • Edward J. Wegman
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 139)


We consider the stochastic differential equations, dX(t) = θX(t)dt + dB H (t); t > 0, and dX(t) = θ(t)X(t)dt + dB H (t); t > 0 where B H (t) is fractional Brownian motion. We find solutions for these differential equations and show the existence of the integrals related to these solutions. We then show that B H (t) is not a martingale. This implies that several conventional methods for defining integrals on fractional Brownian motion are inadequate. We demonstrate the existence of an estimator for θ which depends on the existence of integrals of certain integrals with respect to fractional Brownian motion. We conclude by showing the existence and Riemann sum approximations for these integrals.


Brownian Motion Stochastic Differential Equation Cauchy Sequence Synthetic Aperture Radar Fractional Brownian Motion 
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  1. Barton R.J. and Poor V.H.(1988), “Signal Detection in Fractional Gaussian Noise,” IEEE Transactions on Information Theory, 34: 943–959.MathSciNetCrossRefGoogle Scholar
  2. Christopeit N. (1986), “Quasi-Least-Squares Estimation in Semimartingale Regression Models,” Stochastics, 16: 255–278.MathSciNetMATHCrossRefGoogle Scholar
  3. Cramer H. and Leadbetter M.R.(1967), Stationary and Related Stochastic Processes, John Wiley and Sons, Inc.: New York.MATHGoogle Scholar
  4. Dobrushin R. (1979), “Gaussian and their subordinated generalized fields,” Annals of Probability, 7: 1–28.MathSciNetMATHCrossRefGoogle Scholar
  5. Gregotski M.E., Jensen O., and Arkani-Hamed J.(1991), “Fractal Stochastic Modeling of Aeromagnetic Data,” Geophysics, 56(11): 1706–1715.CrossRefGoogle Scholar
  6. Major P. (1981), Multiple Wiener-Ito Integrals, Lecture Notes in Mathematics, Springer-Verlag: New York.Google Scholar
  7. Mandelbrot B.B. (1983), The Fractal Geometry of Nature, W.H. Freeman and Company: New York.Google Scholar
  8. Shiryayev A.N. (1984), Probability, Springer-Verlag: New York.MATHGoogle Scholar
  9. Soong T.T. (1973), Random Differential Equations in Science and Engineering, Academic Press, Inc.: New York.MATHGoogle Scholar
  10. Stewart C.V., Moghaddam B., Hintz K.J., and Novak L.M.(1993), Fractional Brownian Motion Models for Synthetic Aperture Radar Imagery Scene Segmentation, Proceedings of the IEEE, 81(10): 1511–1522.Google Scholar
  11. Wegman E.J. and Habib M.K. (1992), “Stochastic Methods for Neural Systems,” J. Statistical Planning and Inference, 33: 5–26.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, LLC 2004

Authors and Affiliations

  • Winston C. Chow
  • Edward J. Wegman

There are no affiliations available

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