Student processes
C. C. Heyde and N. N. Leonenko
Source: Adv. in Appl. Probab. Volume 37, Number 2 (2005), 342-365.
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.
Primary Subjects: 60E99, 60G10, 60G35, 62M10
Keywords: Student distribution; Ornstein-Uhlenbeck-type process; self-decomposability; long-range dependence
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Mathematical Reviews number (MathSciNet):
MR2144557
References
Ait-Sahalia, Y. (1996). Nonparametric pricing of interest rate derivative securities. Econometrica 64, 527--560.
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.
Mathematical Reviews (MathSciNet):
MR1938167
Digital Object Identifier: doi:10.1239/jap/1037816015
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, 275--296.
Mathematical Reviews (MathSciNet):
MR2076396
Barndorff-Nielsen, O. E. (1977). Exponentially decreasing distributions for logarithm of particle size. Proc. R. Soc. London A 353, 401--419.
Barndorff-Nielsen, O. E. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Statist. 5, 151--157.
Mathematical Reviews (MathSciNet):
MR509451
Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41--68.
Mathematical Reviews (MathSciNet):
MR1804664
Digital Object Identifier: doi:10.1007/s007800050032
Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein--Uhlenbeck type processes. Theory Prob. Appl. 45, 175--194.
Mathematical Reviews (MathSciNet):
MR1967758
Barndorff-Nielsen, O. E. and Halgreen, C. (1977). Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Z. Wahrscheinlichkeitsth. 38, 309--312.
Mathematical Reviews (MathSciNet):
MR436260
Digital Object Identifier: doi:10.1007/BF00533162
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.
Mathematical Reviews (MathSciNet):
MR1967758
Barndorff-Nielsen, O. E. and Pérez-Abreu, V. (1999). Stationary and self-similar processes driven by Lévy processes. Stoch. Process. Appl. 84, 357--369.
Mathematical Reviews (MathSciNet):
MR1719337
Digital Object Identifier: doi:10.1016/S0304-4149(99)00061-7
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, 167--241.
Mathematical Reviews (MathSciNet):
MR1841412
Digital Object Identifier: doi:10.1111/1467-9868.00282
Barndorff-Nielsen, O. E., Jensen, J. L. and S\o rensen, M. (1998). Some stationary processes in discrete and continuous time. Adv. Appl. Prob. 30, 989--1007.
Mathematical Reviews (MathSciNet):
MR1671092
Digital Object Identifier: doi:10.1239/aap/1035228204
Barndorff-Nielsen, O. E., Nicalato, E. and Shephard, N. (2002). Some recent developments in stochastic volatility modelling. Quant. Finance 2, 11--23.
Mathematical Reviews (MathSciNet):
MR1919582
Digital Object Identifier: doi:10.1088/1469-7688/2/1/301
Bateman, H. and Erdélyi, A. (1953). Higher Transcendental Functions, Vol. 2. McGraw-Hill, New York.
Mathematical Reviews (MathSciNet):
MR58756
Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth and Brooks, Pacific Grove, CA.
Mathematical Reviews (MathSciNet):
MR1126464
Zentralblatt MATH:
0809.60046
Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.
Mathematical Reviews (MathSciNet):
MR1406564
Bibby, B. M. and S\o rensen, M. (1997). A hyperbolic diffusion model for stock prices. Finance Stoch. 1, 25--41.
Bibby, B. M., Skovgaard, I. M. and S\o rensen, M. (2003). Diffusion-type models with given marginal and autocorrelation function. Bernoulli 11, 191--220.
Mathematical Reviews (MathSciNet):
MR2132002
Project Euclid: euclid.bj/1116340291
Bingham, N. H. and Kiesel, R. (2002). Semi-parametric modelling in finance: theoretical foundations. Quant. Finance 2, 241--250.
Mathematical Reviews (MathSciNet):
MR1922404
Digital Object Identifier: doi:10.1088/1469-7688/2/4/201
Bingham, N. H., Goldie, C. M. and Teugels, T. L. (1987). Regular Variation. Cambridge University Press.
Mathematical Reviews (MathSciNet):
MR898871
Zentralblatt MATH:
0617.26001
Borland, L. (2002). A theory of non-Gaussian option pricing. Quant. Finance 2, 415--431.
Mathematical Reviews (MathSciNet):
MR1948281
Digital Object Identifier: doi:10.1088/1469-7688/2/6/301
Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman and Hall, Boca Raton, FL.
Mathematical Reviews (MathSciNet):
MR2042661
Zentralblatt MATH:
1052.91043
Costa, J., Hero, A. and Vignat, C. (2003). On solutions to multivariate maximum $\alpha $-entropy problems. In Proc. Energy Minimization Methods in Computer Vision and Pattern Recognition (Lisbon, July 2003; Lecture Notes Comput. Sci. 2683), Springer, Berlin, pp. 211--228.
Courant, R. and Hilbert, D. (1953). Methods of Mathematical Physics. Interscience, New York.
Mathematical Reviews (MathSciNet):
MR65391
Cramér, A. J. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. John Wiley, New York.
Mathematical Reviews (MathSciNet):
MR217860
Dreier, I. and Kotz, S. (2002). A note on the characteristic function of the $t$-distribution. Statist. Prob. Lett. 57, 221--224.
Mathematical Reviews (MathSciNet):
MR1912079
Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli 1, 281--299.
Grosswald, E. (1976). The Student $t$-distribution of any degree of freedom is infinitely divisible. Z. Wahrscheinlichkeitsth. 36, 103--109.
Mathematical Reviews (MathSciNet):
MR426091
Digital Object Identifier: doi:10.1007/BF00533993
Halgreen, C. (1979). Self-decomposability of generalized inverse Gaussian and hyperbolic distributions. Z. Wahrscheinlichkeitsth. 47, 13--17.
Mathematical Reviews (MathSciNet):
MR521527
Havdra, M. and Charvát, F. (1967). Quantification method of classification processes: concept of structural $\alpha $-entropy. Kybernetika 3, 30--35.
Mathematical Reviews (MathSciNet):
MR209067
Zentralblatt MATH:
0178.22401
Heyde, C. C. (1999). A risky asset model with strong dependence through fractal activity time. J. Appl. Prob. 36, 1234--1239.
Mathematical Reviews (MathSciNet):
MR1746407
Digital Object Identifier: doi:10.1239/jap/1032374769
Heyde, C. C. and Gay, R. (2002). Fractals and contingent claims. Preprint, Australian National University.
Heyde, C. C. and Liu, S. (2001). Empirical realities for a minimal description risky asset model. The need for fractal features. J. Korean Math. Soc. 38, 1047--1059.
Mathematical Reviews (MathSciNet):
MR1849338
Heyde, C. C. and Yang, Y. (1997). On defining long-range dependence. J. Appl. Prob. 34, 939--944.
Mathematical Reviews (MathSciNet):
MR1484026
Hurst, S. R. and Platen, E. (1997). The marginal distribution of returns and volatility. In L$_1$-Statistical Procedures and Related Topics (IMS Lecture Notes Monogr. Ser. 31), ed. Y. Dodge, IMS, Hayward, CA, pp. 301--314.
Mathematical Reviews (MathSciNet):
MR1833594
Zentralblatt MATH:
0937.62107
Hurst, S. R., Platen, E. and Rachev, S. R. (1997). Subordinated Markov models: a comparison. Finan. Eng. Japanese Markets 4, 97--124.
Jurek, Z. J. (2001). Remarks on the self-decomposability and new examples. Demonstr. Math. 34, 241--250.
Mathematical Reviews (MathSciNet):
MR1833180
Jurek, Z. J. and Mason, J. D. (1993). Operator-Limit Distributions in Probability Theory. John Wiley, New York.
Mathematical Reviews (MathSciNet):
MR1243181
Zentralblatt MATH:
0850.60003
Kotz, S., Kozubowski, T. J. and Podgorski, K. (2001). The Laplace Distribution and Generalizations. Birkhäuser, Boston, MA.
Mathematical Reviews (MathSciNet):
MR1935481
Zentralblatt MATH:
0977.62003
Kwapien, S. and Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston, MA.
Mathematical Reviews (MathSciNet):
MR1167198
Zentralblatt MATH:
0751.60035
Lamperti, J. W. (1962). Semi-stable stochastic process. Trans. Amer. Math. Soc. 104, 62--78.
Mathematical Reviews (MathSciNet):
MR138128
Leonenko, N. N. (1999). Limit Theorems for Random Fields with Singular Spectrum. Kluwer, Dordrecht.
Mathematical Reviews (MathSciNet):
MR1687092
Zentralblatt MATH:
0963.60048
Madan, D. B. and Seneta, E. (1990). The variance gamma (V.G.) model for share market returns. J. Business 63, 511--524.
Mandelbrot, B. B. (2001a). Scaling in financial prices. I. Tails and dependence. Quant. Finance 1, 113--123.
Mathematical Reviews (MathSciNet):
MR1810018
Digital Object Identifier: doi:10.1088/1469-7688/1/1/308
Mandelbrot, B. B. (2001b). Scaling in financial prices. II. Multifractals and the star equation. Quant. Finance 1, 124--130.
Mathematical Reviews (MathSciNet):
MR1810019
Digital Object Identifier: doi:10.1088/1469-7688/1/1/309
Rajput, B. and Rosi\'nski, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451--487. %\vfill
Mathematical Reviews (MathSciNet):
MR1001524
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. 547--561.
Mathematical Reviews (MathSciNet):
MR132570
Rosi\'nski, J. (1991). On a class of infinitely divisible processes represented as mixtures of Gaussian processes. In Stable Processes and Related Topics, eds S. Cambanis, G. Samorodnitsky and M. S. Taqqu, Birkhäuser, Basel, pp. 27--41.
Mathematical Reviews (MathSciNet):
MR1119350
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
Mathematical Reviews (MathSciNet):
MR1280932
Zentralblatt MATH:
0925.60027
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
Mathematical Reviews (MathSciNet):
MR1739520
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. 177--187.
Mathematical Reviews (MathSciNet):
MR2057573
Digital Object Identifier: doi:10.1239/jap/1082552198
S\o rensen, M. and Bibby, M. (2003). Hyperbolic processes in finance. In Handbook of Heavy Tailed Distributions in Finance, ed. S. T. Rachev, Elsevier, Amsterdam, pp. 211--248.\vfill
Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287--302.
Mathematical Reviews (MathSciNet):
MR400329
Digital Object Identifier: doi:10.1007/BF00532868
Tarami, B. and Pourahmadi, M. (2003). Multi-variate $t$ autoregressions: innovations, prediction variances and exact likelihood equations. J. Time Ser. Anal. 24, 739--754.
Mathematical Reviews (MathSciNet):
MR2020918
Digital Object Identifier: doi:10.1111/j.1467-9892.2003.00332.x
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, 2197--2200.
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, 3589--3593.
Mathematical Reviews (MathSciNet):
MR1362033
Digital Object Identifier: doi:10.1103/PhysRevLett.75.3589
Vignat, C. and Bercher, J. F. (2003). Analysis of signals in the Fisher--Shannon information plane. Phys. Rev. A 312, 27--33.
Mathematical Reviews (MathSciNet):
MR1996910
Digital Object Identifier: doi:10.1016/S0375-9601(03)00570-X
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, 45--50.
Mathematical Reviews (MathSciNet):
MR1881529
Woyczyński, W. A. (1998). Burgers-KPZ Turbulence (Lecture Notes Math. 1700). Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR1732301
