Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T14:00:51.795Z Has data issue: false hasContentIssue false
  • English
  • Français

Aspects of prediction

Part of: Probability theory and stochastic processes Statistics

Published online by Cambridge University Press:  30 March 2016

N. H. Bingham  and
Badr Missaoui
N. H. Bingham
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK. Email address: [email protected].
Badr Missaoui
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK. Email address: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.

Keywords

Stationary processKolmogorov isomorphism theoremtime seriesstochastic volatilityvolatility clusteringBanach spacecovariance operatorlocally convexreproducing-kernel Hilbert spacesampling theorem

MSC classification

Primary: 60-02: Research exposition (monographs, survey articles)
Secondary: 62-02: Research exposition (monographs, survey articles)
Type
Part 5. Finance and econometrics
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 189 - 201
Copyright
Copyright © Applied Probability Trust 2014 

References

Albiac, F., and Kalton, N. J. (2006). Topics in Banach Space Theory (Graduate Texts Math. 233). Springer, New York.Google Scholar
Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Springer, New York.Google Scholar
Alpay, D., Timoshenko, O., and Volok, D. (2009). Carathéodory-Fejér interpolation in locally convex topological vector spaces. Linear Algebra Appl. 431, 1257–1266.Google Scholar
Antoniadis, A., and Sapatinas, T. (2003). Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes. J. Multivariate Anal. 87, 133–158.Google Scholar
Antoniadis, A., Paparoditis, E., and Sapatinas, T. (2006). A functional wavelet-kernel approach for time series prediction. J. R. Statist. Soc. B 68, 837–857.Google Scholar
Applebaum, D., and Riedle, M. (2010). Cylindrical Lévy processes in Banach spaces. Proc. London Math. Soc. 101, 697–726.Google Scholar
Aronszajn, N. (1950). Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404.Google Scholar
Belyaev, Y. K. (1959). Analytic random processes. Theory Prob. Appl. 4, 402–409.Google Scholar
Bergstrom, A. R. (1990). Continuous time Econometric Modelling. Oxford University Press.Google Scholar
Bingham, N. H. (2012). Szegö's theorem and its probabilistic descendants. Prob. Surveys 9, 287–324.Google Scholar
Bingham, N. H. (2012). Multivariate prediction and matrix Szegö theory. Prob. Surveys 9, 325–339.Google Scholar
Bingham, N. H. (2014). Modelling and prediction of financial time series. Commun. Statist. Theory Meth. 43, 1351–1361.Google Scholar
Bingham, N. H., Fry, J. M., and Kiesel, R. (2010). Multivariate elliptic processes. Statist. Neerlandica 64, 352–366.Google Scholar
Boas, R. P. Jr. (1972). Summation formulas and band-limited signals. Tohoku Math. J. 24, 121–125.Google Scholar
Bosq, D. (2000). Linear Processes in Function Spaces. Theory and Applications (Lecture Notes Statist. 149). Springer, New York.Google Scholar
Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (2008). Time Series Analysis. Forecasting and Control, 4th edn. John Wiley, Hoboken, NJ.Google Scholar
Brockwell, P. J. (2001). Continuous-time ARMA processes. In Stochastic Processes: Theory and Methods (Handbook Statist. 19), Elsevier, London, pp. 249–276.Google Scholar
Brockwell, P. J. (2001). Lévy-driven CARMA processes. Ann. Inst. Statist. Math. 53, 113–124.Google Scholar
Brockwell, P. J., and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Brockwell, P. J., Chadraa, E., and Lindner, A. (2006). Continuous-time GARCH processes. Ann. Appl. Prob. 16, 790–826.Google Scholar
Butzer, P. L. et al. (2011). The sampling theorem, Poisson's summation formula, general Parseval formula, reproducing kernel formula and the Paley-Wiener theorem for bandlimited signals—their interconnections. Appl. Anal. 90, 431–461.Google Scholar
Chobanyan, S. A., and Vakhania, N. N. (1983). The linear prediction and approximation of weak second order random elements. In Prediction Theory and Harmonic Analysis: The Pesi Masani Volume, eds Mandrekar, V. and Saleti, H., North-Holland, Amsterdam, pp. 37–60.Google Scholar
Cramér, H. (1942). On harmonic analysis in certain function spaces. Ark. Mat. Astr. Fys. 28B, 7 pp.Google Scholar
Cramér, H., and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. John Wiley, New York.Google Scholar
Cugliari, J. (2011). Prévision non paramétrique de processusà valeurs fonctionnelles. Application à la consommation d'électricité. , Université Paris-Sud XI.Google Scholar
De Branges, L. (1968). Hilbert Spaces of Entire Functions. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Denisov, S. A. (2006). Continuous analogs of polynomials orthogonal on the unit circle and Krein systems. IMRS Int. Math. Res. Surv. 2006, 54517.Google Scholar
Doob, J. L. (1953). Stochastic Processes. John Wiley, New York.Google Scholar
Dym, H., and McKean, H. P. Jr. (1970). Application of de Branges spaces of integral functions to prediction of stationary Gaussian processes. Illinois J. Math. 14, 299–343.Google Scholar
Dym, H., and McKean, H. P. (1970). Extrapolation and interpolation of stationary Gaussian processes. Ann. Math. Statist. 41, 1817–1844.Google Scholar
Dym, H., and McKean, H. P. (1976). Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Academic Press, New York.Google Scholar
Fang, K. T., Kotz, S., and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions. Chapman and Hall, London.Google Scholar
Ferraty, F., and Romain, Y. (eds) (2010). The Oxford Handbook of Functional Data Analysis. Oxford University Press.Google Scholar
Finkenstädt, B. and Rootzén, H. (eds) (2004). Extreme Values in Finance, Telecommunications, and the Environment. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Gel'fand, I. M., and Vilenkin, N. Y. (1964). Generalized Functions, Vol. 4. Academic Press, New York.Google Scholar
Ginovyan, M. S., and Mikaelyan, L. V. (2010). Prediction error for continuous-time stationary processes with singular spectral densities. Acta Appl. Math. 110, 327–351.Google Scholar
Gorniak, J., and Weron, A. (1980/81). Aronszajn-Kolmogorov type theorems for positive definite kernels in locally convex spaces. Studia Math. 69, 235–246.Google Scholar
Gouriéroux, C. (1997). ARCH Models and Financial Applications. Springer, New York.Google Scholar
Grenander, U. and Szegö, G. (1958). Toeplitz Forms and Their Applications. University of California Press, Berkeley, CA.Google Scholar
Hajduk-Chmielewska, G. (1988). The Wold-Cramér concordance problem for Banach-space-valued stationary processes. Studia Math. 91, 31–43.Google Scholar
Haug, S., Klüppelberg, C., Lindner, A., and Zapp, M. (2007). Method of moment estimation in the COGARCH(1,1) model. Econometric J. 10, 320–341.Google Scholar
Higgins, J. R. (1977). Completeness and Basis Properties of Sets of Special Functions (Camb. Tracts Math. 72). Cambridge University Press.Google Scholar
Higgins, J. R. (1985). Five short stories about the cardinal series. Bull. Amer. Math. Soc. (N.S.) 12, 45–89.Google Scholar
Higgins, J. R. (1996). Sampling Theory in Fourier and Signal Analysis: Foundations. Clarendon Press, Oxford.Google Scholar
Higgins, J. R., and Stens, R. L. (eds) (1996). Sampling Theory in Fourier and Signal Analysis: Advanced Topics. Oxford University Press.Google Scholar
Inoue, A. (2008). AR and MA representations of partial autocorrelation functions, with applications. Prob. Theory Relat. Fields 140, 523–551.Google Scholar
Inoue, A., and Kasahara, Y. (2006). Explicit representation of finite predictor coefficients and its applications. Ann. Statist. 34, 973–993.Google Scholar
Ito, K. (1954). Stationary random distributions. Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 28, 209–223.Google Scholar
Janson, S. (1997). Gaussian Hilbert Spaces (Camb. Tracts Math. 129). Cambridge University Press.Google Scholar
Kabanov, Y., Liptser, R., and Stoyanov, J. (eds) (2006). From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift. Springer, Berlin.Google Scholar
Kakihara, Y. (1997). Multidimensional Second Order Stochastic Processes. World Scientific, River Edge, NJ.Google Scholar
Kakihara, Y. (1997). Dilations of Hilbert-Schmidt operator-valued measures and applications. In Stochastic Processes and Functional Analysis: In Celebration of M. M. Rao's 65th Birthday, eds Goldstein, J. A., Gretsky, N. E. and Uhl, J. J., Dekker, New York, pp. 123–135.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
Karhunen, K. (1950). Uber die Struktur stationärer zufälliger funktionen. Ark. Mat. 1, 141–160.CrossRefGoogle Scholar
Kasahara, Y., and Bingham, N. H. (2014). Verblunsky coefficients and Nehari sequences. Trans. Amer. Math. Soc. 366, 1363–1378.Google Scholar
Katznelson, Y. (2004). An Introduction to Harmonic Analysis, 3rd edn. Cambridge University Press.Google Scholar
Klüppelberg, C., Lindner, A. and Maller, R. (2004). A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour. J. Appl. Prob. 41, 601–622.Google Scholar
Klüppelberg, C., Lindner, A., and Maller, R. A. (2006). Continuous-time volatility modelling: COGARCH versus Ornstein-Uhlenbeck models. In From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, eds Kabanov, Y. et al., Springer, Berlin, pp. 393–419.Google Scholar
Kolmogorov, A. N. (1941). Stationary sequences in Hilbert space. Bull. Moskov. Gos. Univ. Mat. 2, 1–40 (in Russian).Google Scholar
Kolmogorov, A. N. (1986). Selected Works of A. N. Kolmogorov, Vol. 2. Nauka, Moskva (in Russian).Google Scholar
Kramer, H. P. (1959). A generalised sampling theorem. J. Math. Phys. 38, 68–72.Google Scholar
Levinson, N., and McKean, H. P. Jr. (1964). Weighted trigonometrical approximation on R1 with application to the germ field of a stationary Gaussian noise. Acta Math. 112, 99–143.Google Scholar
Lévy, P. (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris.Google Scholar
Lloyd, S. P. (1959). A sampling theorem for stationary (wide sense) stochastic processes. Trans. Amer. Math. Soc. 92, 1–12.Google Scholar
Loève, M. (1948). Fonctions aléatoires du second ordre. Supplement to Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris, pp. 228–352.Google Scholar
Loève, M. (1978). Probability Theory. II (Graduate Texts Math. 46), 4th edn. Springer, New York.Google Scholar
Mandrekar, V., and Salehi, H. (1972). The square-integrability of operator-valued functions with respect to a non-negative operator-valued measure and the Kolmogorov isomorphism theorem. Indiana Univ. Math. J. 20, 545–563.Google Scholar
Mandrekar, V., and Salehi, H. (eds) (1983). Prediction Theory and Harmonic Analysis: The Pesi Masani Volume. North-Holland, Amsterdam.Google Scholar
Markowitz, H. (1952). Portfolio selection. J. Finance 7, 77–91.Google Scholar
Markowitz, H. M. (1959). Portfolio Selection: Efficient Diversification of Investments. John Wiley, New York.Google Scholar
Martin, R. T. W. (2010). Symmetric operators and reproducing kernel Hilbert spaces. Complex Anal. Operat. Theory 4, 845–880.Google Scholar
Masani, P. (1968). Orthogonally scattered measures. Adv. Math. 2, 61–117.Google Scholar
Mikosch, T. (2000). Modelling dependence and tails of financial time series. In Symposium in Honour of Ole E. Barndorff-Nielsen, Aarhus University, pp. 61–73.Google Scholar
Nelson, D. B. (1990). Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.Google Scholar
Niemi, H. (1975). Stochastic processes as Fourier transforms of stochastic measures. Ann. Acad. Sci. Fenn. Ser. A I 591, 47 pp.Google Scholar
Nikol'skivi, N. K. (1986). Treatise on the Shift Operator. Spectral Function Theory (Grundl. Math. Wiss. 273). Springer, Berlin.Google Scholar
Nikolski, N. K. (2002). Operators, Functions and Systems: An Easy Reading, Vol. 1. American Mathematical Society, Providence, RI.Google Scholar
Paley, R. E. A. C., and Wiener, N. (1934). Fourier Transforms in the Complex Domain. American Mathematical Society, New York.Google Scholar
Partington, J. R. (1997). Interpolation, Identification, and Sampling. Oxford University Press.Google Scholar
Payen, R. (1967). Fonctions aléatoires du second ordre à valeurs dans un espace de Hilbert. Ann. Inst. H. Poincaré B (N.S.) 3, 323–396.Google Scholar
Ramsay, J. O., and Silverman, B. W. (1997). Functional Data Analysis. Springer, New York.Google Scholar
Ramsay, J. O., and Silverman, B. W. (2002). Applied Functional Data Analysis: Methods and Case Studies. Springer, New York.Google Scholar
Rao, M. M. (1985). Harmonizable, Cramér, and Karhunen classes of processes. In Time Series in the Time Domain (Handbook Statist. 5), North-Holland, Amsterdam, pp. 279–310.Google Scholar
Richard, P. H. (1992). Harmonizability, V-boundedness and stationary dilation for Banach-valued processes. In Probability in Banach Spaces, eds Dudley, R. M., Hahn, M. G. and Kuelbs, J., Birkhäuser, Basel, pp. 189–205.Google Scholar
Riedle, M. (2011). Cylindrical Wiener processes. In Séminaire de Probabilités XLIII (Lecture Notes Math. 2006), Springer, Berlin, pp. 191–214.Google Scholar
Riedle, M. (2011). Infinitely divisible cylindricalmeasures on Banach spaces. Studia Math. 207, 235–256.Google Scholar
Robert, C. P., and Casella, G. (2004). Monte Carlo Statistical Methods. Springer, New York.Google Scholar
Schmidt, F. (1978). Banach-space-valued stationary processes with absolutely continuous spectral function. In Probability Theory on Vector Spaces (Lecture Notes Math. 656), Springer, Berlin, pp. 237–244.Google Scholar
Schwartz, L. (1973). Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford University Press.Google Scholar
Shiryaev, A. N. (1989). Kolmogorov: life and creative activities. Ann. Prob. 17, 866–944.Google Scholar
Simon, B. (2005). Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory. American Mathematical Society, Providence, RI.Google Scholar
Simon, B. (2005). Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory. American Mathematical Society, Providence, RI.Google Scholar
Simon, B. (2011). Szeg Ho's Theorem and its Descendants. Spectral Theory for L2 Perturbations of Orthogonal Polynomials. Princeton University Press.Google Scholar
Szegö, G. (1939). Orthogonal Polynomials. American Mathematical Society, New York.Google Scholar
Vakhania, N. N., Tarieladze, V. I., and Chobanyan, S. A. (1987). Probability Distributions on Banach Spaces. Reidel, Dordrecht.Google Scholar
Weston, J. D. (1949). The cardinal series in Hilbert space. Proc. Camb. Phil. Soc. 45, 335–341.Google Scholar
Whittaker, J. M. (1935). Interpolatory Function Theory (Camb. Tracts Math. 33). Cambridge University Press.Google Scholar
Wiener, N. (1949). Extrapolation, Interpolation and Smoothing of Stationary Time Series. With Engineering Applications. John Wiley, New York.Google Scholar
Wiener, N. (1981). Collected Works With Commentaries: The Hopf-Wiener Integral Equation; Prediction and Filtering; Quantum Mechanics and Relativity; Miscellaneous Mathematical Papers (ed. Masani, P. R.), Vol. III. MIT Press, Cambridge, MA.Google Scholar