Modeling Continuous Time Series Driven by Fractional Gaussian Noise
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.
KeywordsBrownian Motion Stochastic Differential Equation Cauchy Sequence Synthetic Aperture Radar Fractional Brownian Motion
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