Krein's spectral theory and the Paley-Wiener expansion for fractional Brownian motion

In this paper we develop the spectral theory of the fractional Brownian motion (fBm) using the ideas of Krein's work on continuous analogous of orthogonal polynomials on the unit circle. We exhibit the functions which are orthogonal with respect to the spectral measure of the fBm and obtain an explicit reproducing kernel in the frequency domain. We use these results to derive an extension of the classical Paley-Wiener expansion of the ordinary Brownian motion to the fractional case.

[1]  Harry Hochstadt,et al.  The functions of mathematical physics , 1972 .

[2]  M. Talagrand,et al.  Probability in Banach spaces , 1991 .

[3]  A. S. ÜstünelÉcole.  Stochastic Analysis of the Fractional Brownian Motion , 1996 .

[4]  Kacha Dzhaparidze,et al.  A series expansion of fractional Brownian motion , 2002 .

[5]  Henrik Hult,et al.  Approximating some volterra type stochastic integrals with applications to parameter estimation , 2003 .

[6]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[7]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

[8]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[9]  H. Vincent Poor,et al.  Linear estimation of self-similar processes via Lamperti's transformation , 2000 .

[10]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[11]  M. Morf,et al.  Inverses of Toeplitz operators, innovations, and orthogonal polynomials , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[12]  Q. Shao,et al.  Gaussian processes: Inequalities, small ball probabilities and applications , 2001 .

[13]  M. Ismail.  Polynomials Orthogonal on the Unit Circle , 2005 .

[14]  J. H. Zanten,et al.  Optimality of an explicit series expansion of the fractional Brownian sheet , 2005 .

[15]  Werner Linde,et al.  Approximation, metric entropy and small ball estimates for Gaussian measures , 1999 .

[16]  A. Yaglom.  Correlation Theory of Stationary and Related Random Functions I: Basic Results , 1987 .

[17]  Murad S. Taqqu,et al.  Rate Optimality of Wavelet Series Approximations of Fractional Brownian Motion , 2003 .

[18]  N. Wiener,et al.  Fourier Transforms in the Complex Domain , 1934 .

[19]  P. Levy.  Processus stochastiques et mouvement brownien , 1948 .

[20]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[21]  M. Taqqu,et al.  Are classes of deterministic integrands for fractional Brownian motion on an interval complete , 2001 .

[22]  G. Watson.  Bessel Functions. (Scientific Books: A Treatise on the Theory of Bessel Functions) , 1923 .

[23]  Jan Abraham Ferreira,et al.  A frequency domain approach to some results on fractional Brownian motion , 2002 .

[24]  W. Rudin.  Real and complex analysis , 1968 .

[25]  G. Gripenberg,et al.  On the prediction of fractional Brownian motion , 1996, Journal of Applied Probability.

[26]  Thomas Kühn,et al.  Optimal series representation of fractional Brownian sheets , 2002 .

[27]  Helly.  Fourier transforms in the complex domain , 1936 .

[28]  I. Norros,et al.  An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions , 1999 .

[29]  G. Molchan.  Linear Problems for a Fractional Brownian Motion: Group Approach , 2003 .

[30]  K. O. Dzhaparidze,et al.  A series expansion of fractional Brownian motion with Hurst index exceeding 1/2 , 2002 .