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Modelling long-range-dependent Gaussian processes with application in continuous-time financial models

Published online by Cambridge University Press:  14 July 2016

Jiti Gao*
University of Western Australia
Postal address: School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia. Email address:


This paper considers a class of continuous-time long-range-dependent Gaussian processes. The corresponding spectral density is assumed to have a general and flexible form, which covers some important and special cases. For example, the spectral density of a continuous-time fractional stochastic differential equation is included. A modelling procedure is then established through estimating the parameters involved in the spectral density by using an extended continuous-time version of the Gauss–Whittle objective function. The resulting estimates are shown to be strongly consistent and asymptotically normal. An application of the modelling procedure to the identification and modelling of a fractional stochastic volatility is discussed in some detail.

Research Papers
Copyright © Applied Probability Trust 2004 

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Anh, V., Angulo, J., and Ruiz-Medina, M. (1999). Possible long-range dependence in fractional random fields. J. Statist. Planning Infer. 80, 95110.CrossRefGoogle Scholar
Anh, V. V., Heyde, C. C., and Leonenko, N. N. (2002). Dynamic models of long-memory processes driven by Lévy noise. J. Appl. Prob. 39, 730747.CrossRefGoogle Scholar
Baillie, R. T., and King, M. L. (eds) (1996). Special issue of Journal of Econometrics. (Ann. Econom. 73.)Google Scholar
Black, F., and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Political Econom. 3, 637654.CrossRefGoogle Scholar
Brockwell, P., and Davis, R. (1990). Time Series Theory and Methods. Springer, New York.Google Scholar
Comte, F., and Renault, E. (1996). Long memory continuous-time models. J. Econometrics 73, 101149.CrossRefGoogle Scholar
Comte, F., and Renault, E. (1998). Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291323.CrossRefGoogle Scholar
Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17, 17491766.CrossRefGoogle Scholar
Ding, Z., and Granger, C. W. J. (1996). Modelling volatility persistence of speculative returns: a new approach. J. Econometrics 73, 185215.CrossRefGoogle Scholar
Ding, Z., Granger, C. W. J., and Engle, R. F. (1993). A long memory property of stock market returns and a new model. J. Empirical Finance 1, 83105.CrossRefGoogle Scholar
Dym, H., and McKean, H. (1976). Gaussian Process, Function Theory and the Inverse Spectral Problem. Academic Press, New York.Google Scholar
Fox, R., and Taqqu, M. S. (1986). Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14, 517532.CrossRefGoogle Scholar
Frisch, U. (1995). Turbulence. Cambridge University Press.Google ScholarPubMed
Gao, J., Anh, V., and Heyde, C. (2002). Statistical estimation of nonstationary Gaussian processes with long-range dependence and intermittency. Stoch. Process. Appl. 99, 295321.CrossRefGoogle Scholar
Gao, J., Anh, V., Heyde, C., and Tieng, Q. (2001). Parameter estimation of stochastic processes with long-range dependence and intermittency. J. Time Ser. Anal. 22, 517535.CrossRefGoogle Scholar
Heath, D., and Platen, E. (2002). A variance reduction technique based on integral representations. Quant. Finance 2, 362369.CrossRefGoogle Scholar
Heyde, C. (1999). A risky asset model with strong dependence through fractal activity time. J. Appl. Prob. 36, 12341239.CrossRefGoogle Scholar
Heyde, C., and Gay, R. (1993). Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence. Stoch. Process. Appl. 45, 169182.CrossRefGoogle Scholar
Hurst, S. R., Platen, E., and Rachev, S. R. (1997). Subordinated Markov index models: a comparison. Financial Eng. Japanese Markets 4, 97124.CrossRefGoogle Scholar
Kleptsyna, M., Kloeden, P., and Anh, V. (1998). Existence and uniqueness theorems for fBm stochastic differential equations. Problems Inf. Transmission 34, 5161.Google Scholar
Kloeden, P., and Platen, E. (1999). Numerical Solution of Stochastic Differential Equations (Appl. Math. 23). Springer, New York.Google Scholar
Mandelbrot, B., and Taylor, H. (1967). On the distribution of stock price differences. Operat. Res. 15, 10571062.CrossRefGoogle Scholar
Mandelbrot, B., and van Ness, J. (1968). Fractional Brownian motion, fractional noises and applications. SIAM Rev. 10, 422437.CrossRefGoogle Scholar
Mandelbrot, B., Fisher, A., and Calvet, L. (1997). A multifractal model of asset returns. Cowles Foundation Discussion Paper 1164.Google Scholar
Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous time case. Rev. Econom. Statist. 51, 247257.CrossRefGoogle Scholar
Merton, R. C. (1973). The theory of rational option pricing. Bell J. Econom. 4, 141183.CrossRefGoogle Scholar
Merton, R. C. (1990). Continuous-Time Finance. Blackwell, Oxford.Google Scholar
Platen, E. (1999). An introduction to numerical methods for stochastic differential equations. Acta Numerica 8, 197246.CrossRefGoogle Scholar
Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego, CA.Google Scholar
Prudnikov, A., Brychkov, Y., and Marichev, O. (1990). Integrals and Series, Vol. 3. Gordon and Breach, New York.Google Scholar
Robinson, P. (1994). Time series with strong dependence. In Advances In Econometrics, Sixth World Congress (Econom. Soc. Monogr. 23), Vol. 1, ed. Sims, C. A., Cambridge University Press, pp. 4796.CrossRefGoogle Scholar
Robinson, P. (1995). Gaussian semiparametric estimation of long-range dependence. Ann. Statist. 23, 16301661.CrossRefGoogle Scholar
Robinson, P. (1999). The memory of stochastic volatility models. J. Econometrics 101, 195218.CrossRefGoogle Scholar
Rockafeller, R. T. (1970). Convex Analysis. Princeton University Press.CrossRefGoogle Scholar
Shiryaev, A. N. (1999). Essentials of Stochastic Finance. World Scientific, Singapore.CrossRefGoogle Scholar
Sims, C. A. (1984). Martingale-like behavior of asset prices and interest rates. Discussion Paper 205, Department of Economics, University of Minnesota.Google Scholar
Sundaresan, S. (2001). Continuous-time methods in finance: a review and an assessment. J. Finance 55, 15691622.CrossRefGoogle Scholar
Vasicek, O. (1977). An equilibrium characterization of the term structure. J. Financial Econom. 5, 177188.CrossRefGoogle Scholar