Skip to main content Accessibility help
×
×
Home

Student processes

  • C. C. Heyde (a1) and N. N. Leonenko (a2)

Abstract

Stochastic processes with Student marginals and various types of dependence structure, allowing for both short- and long-range dependence, are discussed in this paper. A particular motivation is the modelling of risky asset time series.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Student processes
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Student processes
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Student processes
      Available formats
      ×

Copyright

Corresponding author

Postal address: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia. Email address: chris@maths.anu.edu.au
∗∗ Postal address: Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK. Email address: leonenkon@cardiff.ac.uk

References

Hide All
Ait-Sahalia, Y. (1996). Nonparametric pricing of interest rate derivative securities. Econometrica 64, 527560.
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.
Anh, V. V., Knopova, V. P. and Leonenko, N. N. (2004). Continuous-time stochastic processes with cyclical long-range dependence. Austral. N. Z. J. Statist. 46, 275296.
Barndorff-Nielsen, O. E. (1977). Exponentially decreasing distributions for logarithm of particle size. Proc. R. Soc. London A 353, 401419.
Barndorff-Nielsen, O. E. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Statist. 5, 151157.
Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 4168.
Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein–Uhlenbeck type processes. Theory Prob. Appl. 45, 175194.
Barndorff-Nielsen, O. E. and Halgreen, C. (1977). Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Z. Wahrscheinlichkeitsth. 38, 309312.
Barndorff-Nielsen, O. E. and Leonenko, N. N. (2005). Spectral properties of superpositions of Ornstein–Uhlenbeck type processes. To appear in Method. Comput. Appl. Prob.
Barndorff-Nielsen, O. E. and Pérez-Abreu, V. (1999). Stationary and self-similar processes driven by Lévy processes. Stoch. Process. Appl. 84, 357369.
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial econometrics. J. R. Statist. Soc. B 63, 167241.
Barndorff-Nielsen, O. E., Jensen, J. L. and Sørensen, M. (1998). Some stationary processes in discrete and continuous time. Adv. Appl. Prob. 30, 9891007.
Barndorff-Nielsen, O. E., Nicalato, E. and Shephard, N. (2002). Some recent developments in stochastic volatility modelling. Quant. Finance 2, 1123.
Bateman, H. and Erdélyi, A. (1953). Higher Transcendental Functions, Vol. 2. McGraw-Hill, New York.
Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth and Brooks, Pacific Grove, CA.
Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.
Bibby, B. M. and Sørensen, M. (1997). A hyperbolic diffusion model for stock prices. Finance Stoch. 1, 2541.
Bibby, B. M., Skovgaard, I. M. and Sørensen, M. (2003). Diffusion-type models with given marginal and autocorrelation function. Bernoulli 11, 191220.
Bingham, N. H. and Kiesel, R. (2002). Semi-parametric modelling in finance: theoretical foundations. Quant. Finance 2, 241250.
Bingham, N. H., Goldie, C. M. and Teugels, T. L. (1987). Regular Variation. Cambridge University Press.
Borland, L. (2002). A theory of non-Gaussian option pricing. Quant. Finance 2, 415431.
Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman and Hall, Boca Raton, FL.
Costa, J., Hero, A. and Vignat, C. (2003). On solutions to multivariate maximum α-entropy problems. In Proc. Energy Minimization Methods in Computer Vision and Pattern Recognition (Lisbon, July 2003; Lecture Notes Comput. Sci. 2683), Springer, Berlin, pp. 211228.
Courant, R. and Hilbert, D. (1953). Methods of Mathematical Physics. Interscience, New York.
Cramér, A. J. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. John Wiley, New York.
Dreier, I. and Kotz, S. (2002). A note on the characteristic function of the t-distribution. Statist. Prob. Lett. 57, 221224.
Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli 1, 281299.
Grosswald, E. (1976). The Student t-distribution of any degree of freedom is infinitely divisible. Z. Wahrscheinlichkeitsth. 36, 103109.
Halgreen, C. (1979). Self-decomposability of generalized inverse Gaussian and hyperbolic distributions. Z. Wahrscheinlichkeitsth. 47, 1317.
Havdra, M. and Charvát, F. (1967). Quantification method of classification processes: concept of structural α-entropy. Kybernetika 3, 3035.
Heyde, C. C. (1999). A risky asset model with strong dependence through fractal activity time. J. Appl. Prob. 36, 12341239.
Heyde, C. C. and Gay, R. (2002). Fractals and contingent claims. Preprint, Australian National University.
Heyde, C. C. and Liu, S. (2001). Empirical realities for a minimal description risky asset model. The need for fractal features. J. Korean Math. Soc. 38, 10471059.
Heyde, C. C. and Yang, Y. (1997). On defining long-range dependence. J. Appl. Prob. 34, 939944.
Hurst, S. R. and Platen, E. (1997). The marginal distribution of returns and volatility. In L1-Statistical Procedures and Related Topics (IMS Lecture Notes Monogr. Ser. 31), ed. Dodge, Y., IMS, Hayward, CA, pp. 301314.
Hurst, S. R., Platen, E. and Rachev, S. R. (1997). Subordinated Markov models: a comparison. Finan. Eng. Japanese Markets 4, 97124.
Jurek, Z. J. (2001). Remarks on the self-decomposability and new examples. Demonstr. Math. 34, 241250.
Jurek, Z. J. and Mason, J. D. (1993). Operator-Limit Distributions in Probability Theory. John Wiley, New York.
Kotz, S., Kozubowski, T. J. and Podgorski, K. (2001). The Laplace Distribution and Generalizations. Birkhäuser, Boston, MA.
Kwapien, S. and Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston, MA.
Lamperti, J. W. (1962). Semi-stable stochastic process. Trans. Amer. Math. Soc. 104, 6278.
Leonenko, N. N. (1999). Limit Theorems for Random Fields with Singular Spectrum. Kluwer, Dordrecht.
Madan, D. B. and Seneta, E. (1990). The variance gamma (V.G.) model for share market returns. J. Business 63, 511524.
Mandelbrot, B. B. (2001a). Scaling in financial prices. I. Tails and dependence. Quant. Finance 1, 113123.
Mandelbrot, B. B. (2001b). Scaling in financial prices. II. Multifractals and the star equation. Quant. Finance 1, 124130.
Rajput, B. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451487.
Rényi, A. (1961). On measures of entropy and application. In Proc. 4th Berkeley Symp. Math. Statist. Prob., Vol. 1, University of California Press, Berkeley, CA, pp. 547561.
Rosiński, J. (1991). On a class of infinitely divisible processes represented as mixtures of Gaussian processes. In Stable Processes and Related Topics, eds Cambanis, S., Samorodnitsky, G. and Taqqu, M. S., Birkhäuser, Basel, pp. 2741.
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
Schoutens, W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives. John Wiley, Chichester.
Seneta, E. (2004). Fitting the variance-gamma model to financial data. In Stochastix Methods and Their Applications (J. Appl. Prob. Spec. Vol. 41A), Applied Probability Trust, Sheffield, pp. 177187.
Sørensen, M. and Bibby, M. (2003). Hyperbolic processes in finance. In Handbook of Heavy Tailed Distributions in Finance, ed. Rachev, S. T., Elsevier, Amsterdam, pp. 211248.
Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287302.
Tarami, B. and Pourahmadi, M. (2003). Multi-variate t autoregressions: innovations, prediction variances and exact likelihood equations. J. Time Ser. Anal. 24, 739754.
Tsallis, C. and Bukman, D. J. (1996). Anomalous diffusion in the presence of external forces: exact time-dependent solutions and their thermostatistical basis. Phys. Rev. E 54, 21972200.
Tsallis, C., Levy, S. V. F., Souza, A. M. C. and Maynard, R. (1995). Statistical-mechanical foundation of the ubiquity of Lévy distributions in nature. Phys. Rev. Lett. 75, 35893593.
Vignat, C. and Bercher, J. F. (2003). Analysis of signals in the Fisher–Shannon information plane. Phys. Rev. A 312, 2733.
Watson, G. N. (1958). A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.
Witkovsky, V. (2002). Exact distribution of positive linear combinations of inverted chi-square random variables with odd degrees of freedom. Statist. Prob. Lett. 56, 4550.
Woyczyński, W. A. (1998). Burgers-KPZ Turbulence (Lecture Notes Math. 1700). Springer, Berlin.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed