Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-20T16:31:01.298Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Jiti Gao
Jiti Gao*
Affiliation:
University of Western Australia
*
∗ Postal address: School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Continuous-time modeldiffusion processlong-range dependenceparameter estimationstochastic volatility

MSC classification

Primary: 60G17: Sample path properties
Secondary: 60G10: Stationary processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 467 - 482
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anh, V., Angulo, J., and Ruiz-Medina, M. (1999). Possible long-range dependence in fractional random fields. J. Statist. Planning Infer. 80, 95–110.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, 730–747.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, 637–654.Google 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, 101–149.Google Scholar
Comte, F., and Renault, E. (1998). Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291–323.Google Scholar
Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17, 1749–1766.CrossRefGoogle Scholar
Ding, Z., and Granger, C. W. J. (1996). Modelling volatility persistence of speculative returns: a new approach. J. Econometrics 73, 185–215.Google 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, 83–105.Google 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, 517–532.Google Scholar
Frisch, U. (1995). Turbulence. Cambridge University Press.Google Scholar
Gao, J., Anh, V., and Heyde, C. (2002). Statistical estimation of nonstationary Gaussian processes with long-range dependence and intermittency. Stoch. Process. Appl. 99, 295–321.Google 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, 517–535.Google Scholar
Heath, D., and Platen, E. (2002). A variance reduction technique based on integral representations. Quant. Finance 2, 362–369.Google Scholar
Heyde, C. (1999). A risky asset model with strong dependence through fractal activity time. J. Appl. Prob. 36, 1234–1239.Google 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, 169–182.Google Scholar
Hurst, S. R., Platen, E., and Rachev, S. R. (1997). Subordinated Markov index models: a comparison. Financial Eng. Japanese Markets 4, 97–124.Google Scholar
Kleptsyna, M., Kloeden, P., and Anh, V. (1998). Existence and uniqueness theorems for fBm stochastic differential equations. Problems Inf. Transmission 34, 51–61.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, 1057–1062.CrossRefGoogle Scholar
Mandelbrot, B., and van Ness, J. (1968). Fractional Brownian motion, fractional noises and applications. SIAM Rev. 10, 422–437.Google 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, 247–257.Google Scholar
Merton, R. C. (1973). The theory of rational option pricing. Bell J. Econom. 4, 141–183.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, 197–246.Google 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. 47–96.Google Scholar
Robinson, P. (1995). Gaussian semiparametric estimation of long-range dependence. Ann. Statist. 23, 1630–1661.Google Scholar
Robinson, P. (1999). The memory of stochastic volatility models. J. Econometrics 101, 195–218.Google Scholar
Rockafeller, R. T. (1970). Convex Analysis. Princeton University Press.Google Scholar
Shiryaev, A. N. (1999). Essentials of Stochastic Finance. World Scientific, Singapore.Google 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, 1569–1622.Google Scholar
Vasicek, O. (1977). An equilibrium characterization of the term structure. J. Financial Econom. 5, 177–188.Google Scholar