Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-07-06T05:33:50.316Z Has data issue: false hasContentIssue false

Dynamic models of long-memory processes driven by Lévy noise

Published online by Cambridge University Press:  14 July 2016

V. V. Anh*
Affiliation:
Queensland University of Technology
C. C. Heyde*
Affiliation:
Australian National University
N. N. Leonenko*
Affiliation:
Cardiff University
*
Postal address: School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane QLD 4001, Australia. Email address: v.anh@qut.edu.au
∗∗ Postal address: School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia.
∗∗∗ Postal address: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4YH, UK.

Abstract

A class of continuous-time models is developed for modelling data with heavy tails and long-range dependence. These models are based on the Green function solutions of fractional differential equations driven by Lévy noise. Some exact results on the second- and higher-order characteristics of the equations are obtained. Applications to stochastic volatility of asset prices and macroeconomics are provided.

MSC classification

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2002 

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

[1]. Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, New York.Google Scholar
[2]. Alós, E., Mazet, O., and Nualart, D. (2000). Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than ½. Stoch. Process. Appl. 86, 121139.Google Scholar
[3]. Andrews, G. E., Askey, R., and Roy, R. (1999). Special Functions. Cambridge University Press.Google Scholar
[4]. Anh, V. V., and Leonenko, N. N. (2000). Scaling laws for fractional diffusion-wave equation with singular initial data. Statist. Prob. Lett. 48, 239252.Google Scholar
[5]. Anh, V. V., and Nguyen, C. N. (2000). Stochastic analysis of fractional Riesz–Bessel motion. Random Operators Stoch. Equat. 8, 105126.Google Scholar
[6]. Anh, V. V., Angulo, J. M., and Ruiz-Medina, M. D. (1999). Possible long-range dependence in fractional random fields. J. Statist. Planning Infer. 80, 95110.CrossRefGoogle Scholar
[7]. Baillie, R. T. (1996). Long memory processes and fractional integration in econometrics. J. Econometrics 73, 559.CrossRefGoogle Scholar
[8]. Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 4168.Google Scholar
[9]. Barndorff-Nielsen, O. E. (2001). (1999). Superpositions of Ornstein–Uhlenbeck type processes. Res. Rep. 1999-2, MaPhySto, Aarhus University. Theory Prob. Appl. 45, 175194.Google Scholar
[10]. Barndorff-Nielsen, O. E. and Pérez-Abreu, V. (1999). Stationary and self-similar processes driven by Lévy processes. Stoch. Process. Appl. 84, 357369.Google Scholar
[11]. Barndorff-Nielsen, O. E., and Shephard, N. (1998). Aggregation and model construction for volatility models. Working paper No. 10, Center for Analytical Finance, University of Aarhus.Google Scholar
[12]. Barndorff-Nielsen, O. E., and Shephard, N. (1998). Incorporation of a leverage effect in a stochastic volatility model. Res. Rep. 1998-18, MaPhySto, University of Aarhus.Google Scholar
[13]. Barndorff-Nielsen, O. E., and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Statist. Soc. B 63, 167241.CrossRefGoogle Scholar
[14]. Bertoin, J. (1989). Sur une intégrale pour les processes à α variation bornée. Ann. Prob. 17, 15211535.Google Scholar
[15]. Bibby, M. and Sörensen, M. (1997). A hyperbolic diffusion model for stock prices. Finance Stoch. 1, 2441.Google Scholar
[16]. Bingham, N. H. (1972). A Tauberian theorem for integral transforms of Hankel type. J. London Math. Soc. 5, 493503.Google Scholar
[17]. Brockwell, P. J. (2001). Continuous-time ARMA processes. In Handbook of Statistics, Vol. 19, Stochastic Processes: Theory and Methods, eds Shanbhag, D. N. and Rao, C. R., Elsevier, Amsterdam, pp. 249275.Google Scholar
[18]. Brockwell, P. J. (2001). Lévy-driven CARMA processes. Ann. Inst. Statist. Math. 53, 113124.CrossRefGoogle Scholar
[19]. Carmona, P., Coutin, L., and Montseny, G. (1998). Applications of a representation of long memory Gaussian processes. Preprint. LAAS Rep. 97422, CNRS, Toulouse.Google Scholar
[20]. Chambers, M. J. (1996). The estimation of continuous parameter long-memory time series models. Econometric Theory 12, 374390.CrossRefGoogle Scholar
[21]. Comte, F. (1996). Simulation and estimation of long-memory continuous time models. J. Time Series Anal. 17, 1936.CrossRefGoogle Scholar
[22]. Comte, F., and Renault, E. (1996). Long memory continuous-time models. J. Econometrics 73, 101149.Google Scholar
[23]. Comte, F., and Renault, E. (1998). Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291323.Google Scholar
[24]. Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17, 17491766.Google Scholar
[25]. Dai, W., and Heyde, C. C. (1996). Itô's formula with respect to fractional Brownian motion and its application. J. Appl. Math. Stoch. Anal. 9, 439448.Google Scholar
[26]. Ding, Z., and Granger, C. W. J. (1996). Modelling volatility persistence of speculative returns: a new approach. J. Econometrics 73, 185215.CrossRefGoogle Scholar
[27]. Djrbashian, M. M. (1993). Harmonic Analysis and Boundary Value Problems in Complex Domain. Birkhäuser, Basel.CrossRefGoogle Scholar
[28]. Eberlein, E., and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli 1, 281299.Google Scholar
[29]. Eberlein, E., and Raible, S. (1999). Term structure models driven by general Lévy processes. Math. Finance 9, 3153.CrossRefGoogle Scholar
[30]. Fox, R., and Taqqu, M. S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14, 517532.Google Scholar
[31]. Gay, R., and Heyde, C. C. (1990). On a class of random field models which allows long range dependence. Biometrika 77, 401403.CrossRefGoogle Scholar
[32]. Granger, C. W. J. (1980). Long memory relationships and the aggregation of dynamic models. J. Econometrics 14, 227238.Google Scholar
[33]. Granger, C. W. J., and Ding, Z. (1996). Varieties of long-memory models. J. Econometrics 73, 6172.Google Scholar
[34]. Helson, S. L. (1993). A closed solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327343.Google Scholar
[35]. Heyde, C. C. (1999). A risky asset model with strong dependence. J. Appl. Prob. 36, 12341239.CrossRefGoogle Scholar
[36]. Heyde, C. C., and Liu, S. (2001). Empirical realities for a minimum description risky asset model. The need for fractal features. J. Korean Math. Soc. 38, 10471059.Google Scholar
[37]. Hu, Y. and Øksendal, B. (1999). Fractional white noise calculus and applications to finance. Preprint, University of Kansas.Google Scholar
[38]. Hull, J., and White, A. (1987). The pricing of options of assets with stochastic volatilities. J. Finance 42, 281300.Google Scholar
[39]. Iglói, E., and Terdik, G. (1999). Bilinear stochastic systems with fractional Brownian motion input. Ann. Appl. Prob. 9, 4677.CrossRefGoogle Scholar
[40]. Iglói, E., and Terdik, G. (1999). Long-range dependence through gamma-mixed Ornstein–Uhlenbeck process. Electron. J. Prob. 4, 133.Google Scholar
[41]. Inoue, A. (1993). On the equations of stationary processes with divergent diffusion coefficients. J. Fac. Sci. Univ. Tokyo, Sec. IA Math. 40, 307336.Google Scholar
[42]. Kramkov, D. O. (1996). Optimal decomposition of supermartingales and hedging contingent in incomplete security markets. Prob. Theory Relat. Fields 105, 459479.Google Scholar
[43]. Leonenko, N. N., and Woyczynski, W. A. (1999). Parameter identification for singular random fields arising in Burgers’ turbulence. J. Statist. Planning Infer. 80, 113.CrossRefGoogle Scholar
[44]. Marinucci, D., and Robinson, P. M. (1999). Alternative forms of fractional Brownian motion. J. Statist. Planning Infer. 80, 111122.Google Scholar
[45]. Marinucci, D., and Robinson, P. M. (2000). Weak convergence of multivariate fractional processes. Stoch. Process. Appl. 86, 103120.CrossRefGoogle Scholar
[46]. Mikosch, T. and Norvaiŝa, R. (2000). (1997). Stochastic integral equations without probability. Preprint, University of Groningen. Bernoulli 6, 401434.Google Scholar
[47]. Mueller, C. (1998). The heat equation with Lévy noise. Stoch. Process. Appl. 74, 6782.Google Scholar
[48]. Norros, I., Valkeila, E., and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motion. Bernoulli 5, 571587.Google Scholar
[49]. Oppenheim, G., and Viano, M.-C. (1999). Obtaining long-memory by aggregating random coefficients of discrete and continuous time simple short memory processes. Pub. IRMA 49, 116.Google Scholar
[50]. Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego.Google Scholar
[51]. Protter, P. (1992). Stochastic Integration and Differential Equations. Springer, New York.Google Scholar
[52]. Robinson, P. (1995). Gaussian semiparametric estimation of long-range dependence. Ann. Statist. 23, 16301661.Google Scholar
[53]. Robinson, P. (1995). Log-periodogram regression in time series with long-range dependence. Ann. Statist. 23, 10481072.CrossRefGoogle Scholar
[54]. Rogers, L. C. C. (1997). Arbitrage with fractional Brownian motion. Math. Finance 7, 95105.Google Scholar
[55]. Rydberg, T. H. (1999). Generalized hyperbolic diffusion process with applications in finance. Math. Finance 9, 183201.Google Scholar
[56]. Samko, S. G., Kilbas, A. A., and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach, Yverdon.Google Scholar
[57]. Samorodnitsky, G., and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes . Chapman and Hall, New York.Google Scholar
[58]. Schoutens, W. (2000). Stochastic Processes and Orthogonal Polynomials (Lecture Notes Statist. 146). Springer, New York.Google Scholar
[59]. Shiryaev, A. N. Some problems in the spectral theory of higher-order moments. I. Theory Prob. Appl. 5, 265284.Google Scholar
[60]. Surgailis, D. (1979). On the Markov property of a class of linear infinity divisible fields. Z. Wahrscheinlichkeitsth. 49, 293311.Google Scholar
[61]. Viano, M.-C., Deniau, C., and Oppenheim, G. (1994). Continuous-time fractional ARMA processes. Statist. Prob. Lett. 21, 323336.Google Scholar
[62]. Willinger, W., Taqqu, M. S., and Teverovsky, V. (1999). Stock market prices and long-range dependence. Finance Stoch. 3, 113.CrossRefGoogle Scholar
[63]. Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251282.Google Scholar
[64]. Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. J. Prob. Theory Relat. Fields 111, 333374.Google Scholar