Lévy-based spatial-temporal modelling, with applications to turbulence

O. E. Barndorff-Nielsen   ; J. Schmiegel

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 1, pp. 65-90

Résumé

This paper involves certain types of spatial-temporal models constructed from Lévy bases. The dynamics is described by a field of stochastic processes $X=\{X_t(\sigma)\}$, on a set $\mathscr S$ of sites $\sigma$, defined as integrals $$ X_t(\sigma)=\int_{-\infty}^t\int_{\mathscr S}f_t(\rho,s;\sigma)\,Z(\mathrm d\rho\times\mathrm ds), $$ where $Z$ denotes a Lévy basis. The integrands $f$ are deterministic functions of the form $f_t(\rho,s;\sigma)=h_t(\rho,s;\sigma)\mathbf 1_{A_t(\sigma)}(\rho,\sigma)$, where $h_t(\rho,s;\sigma)$ has a special form and $A_t(\sigma)$ is a subset of $\mathscr S\times \mathbb R_{\leqslant t}$. The first topic is OU (Ornstein–Uhlenbeck) fields $X_t(\sigma)$, which represent certain extensions of the concept of OU processes (processes of Ornstein–Uhlenbeck type); the focus here is mainly on the potential of $X_t(\sigma)$ for dynamic modelling. Applications to dynamical spatial processes of Cox type are briefly indicated. The second part of the paper discusses modelling of spatial-temporal correlations of SI (stochastic intermittency) fields of the form $$ Y_t(\sigma)=\exp\{X_t(\sigma)\}. $$ This form is useful when explicitly computing expectations of the form $$ \mathsf E\{Y_{t_1}(\sigma_1)\cdots Y_{t_n}(\sigma_n)\}, $$ which are used to characterize correlations. The SI fields can be viewed as a dynamical, continuous, and homogeneous generalization of turbulent cascades. In this connection an SI field is constructed with spatial-temporal scaling behaviour that agrees with the energy dissipation observed in turbulent flows. Some parallels of this construction are also briefly sketched.

Bibliographie

[1] O. E. Barndorff-Nielsen, “Probability and statistics; selfdecomposability, finance and turbulence”, Proceedings of the Conference “Probability towards 2000” (New York, 1995), Lecture Notes in Statist., 128, eds. L. Accardi, C. C. Heyde, Springer-Verlag, New York, 1998, 47–57 | MR | Zbl

[2] O. E. Barndorff-Nielsen, “Processes of normal inverse Gaussian type”, Finance Stoch., 2:1 (1998), 41–68 | DOI | MR | Zbl

[3] O. E. Barndorff-Nielsen, “Superposition of Ornstein–Uhlenbeck type processes”, Teoriya veroyatn. i ee primen., 45:2 (2000), 289–311 | MR | Zbl

[4] O. E. Barndorff-Nielsen, “Modelling by Lévy processes”, Selected Proceedings of the Symposium on Inference for Stochastic Processes (Athens, 2000), IMS Lecture Notes Monogr. Ser., 37, eds. I. V. Basawa, C. C. Heyde, R. L. Taylor, Inst. Math. Statist., Beachwood, 2001, 25–31 | MR

[5] O. E. Barndorff-Nielsen, J. L. Jensen, M. Sørensen, “Parametric modelling of turbulence”, Philos. Trans. Roy. Soc. London A, 332 (1990), 439–455 | DOI

[6] O. E. Barndorff-Nielsen, J. L. Jensen, M. Sørensen, “A statistical model for the streamwise component of a turbulent velocity field”, Ann. Geophys., 11 (1993), 99–103

[7] O. E. Barndorff-Nielsen, J. Pedersen, K. Sato, “Multivariate subordination, self-decomposability and stability”, Adv. in Appl. Probab., 33:1 (2001), 160–187 | DOI | MR | Zbl

[8] O. E. Barndorff-Nielsen, J. Schmiegel, “Lévy-based spatial-temporal modelling”, In preparation, 2003

[9] O. E. Barndorff-Nielsen, N. Shephard, “Modelling by Lévy processes for financial econometrics”, Lévy Processes – Theory and Applications, eds. O. E. Barndorff-Nielsen, T. Mikosch, S. Resnick, Birkhäuser, Boston, 2001, 283–318 | MR | Zbl

[10] O. E. Barndorff-Nielsen, N. Shephard, “Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics”, J. R. Stat. Soc. Ser. B Stat. Methodol., 63:2 (2001), 167–241 | DOI | MR | Zbl

[11] O. E. Barndorff-Nielsen, N. Shephard, “Normal modified stable processes”, Theory Probab. Math. Statist., 2001, no. 65, 1–20 | MR | Zbl

[12] O. E. Barndorff-Nielsen, N. Shephard, “Integrated OU processes and non-Gaussian OU-based stochastic volatility models”, Scand. J. Statist., 30:2 (2003), 277–295 | DOI | MR

[13] O. E. Barndorff-Nielsen, N. Shephard, Financial Volatility: Stochastic Volatility and Lévy Based Models, Cambridge Univ. Press, Cambridge, 2004 (to appear)

[14] O. E. Barndorff-Nielsen, S. Thorbjørnsen, “A connection between classical and free infinite divisibility”, Research Report 2003-7, Univ. of Aarhus, MaPhySto, Aarhus, 2003

[15] T. Bohr, M. H. Jensen, G. Paladin, A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge Nonlinear Science Series, 8, Cambridge Univ. Press, Cambridge, 1998 | MR | Zbl

[16] A. Brix, Spatial and Spatio-Temporal Models for Weed Abundance, Ph.D. Thesis, Royal Veterinary and Agricultural University, Copenhagen, 1998

[17] A. Brix, “Generalized gamma measures and shot-noise Cox processes”, Adv. in Appl. Probab., 31:4 (1999), 929–953 | DOI | MR | Zbl

[18] A. Brix, J. Chadœuf, “Spatio-temporal modelling of weeds by shot-noise $G$ Cox processes”, Biom. J., 44:1 (2002), 83–99 | 3.0.CO;2-W class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[19] A. Brix, P. J. Diggle, “Spatiotemporal prediction for log-Gaussian Cox processes”, J. R. Stat. Soc. Ser. B Stat. Methodol., 63:4 (2001), 823–841 | DOI | MR | Zbl

[20] M. Çag̃lar, E. Çinlar, “Lyapunov exponents of Poisson shot-noise velocity fields”, Stochastic Process. Appl., 94:1 (2001), 29–49 | DOI | MR

[21] P. Carmona, F. Petit, M. Yor, “Exponential functionals of Lévy processes”, Lévy Processes – Theory and Applications, eds. O. E. Barndorff-Nielsen, T. Mikosch, S. Resnick, Birkhäuser, Boston, 2001, 41–55 | MR

[22] J. Cleve, M. Greiner, “The Markovian metamorphosis of a simple turbulent cascade model”, Phys. Lett. A, 273:1–2 (2000), 104–108 | DOI | MR | Zbl

[23] D. Duffie, D. Filipović, W. Schachermayer, “Affine processes and applications in finance”, Ann. Appl. Probab., 13:3 (2001), 984–1053 | MR

[24] H. C. Eggers, T. Dziekan, M. Greiner, “Translationally invariant cumulants in energy cascade models of turbulence”, Phys. Lett. A, 281:4 (2001), 249–255 | DOI | MR | Zbl

[25] U. Frisch, Turbulence. The Legacy of A. N. Kolmogorov, Cambridge Univ. Press, Cambridge, 1995 | MR

[26] S. Ghashgaie, W. Breymann, J. Peinke, P. Talkner, Y. Dodge, “Turbulent cascades in foreign exchange markets”, Nature, 381 (1996), 767–770 | DOI

[27] M. Greiner, “Turbulent random multiplicative branching processes”, Lectures on Multiscale and Multiplicative Processes in Fluid Flows, MaPhySto Lecture Notes, 11, eds. E. Waymire et al., MaPhySto, Univ. of Aarhus, Aarhus, 2002, 81–88

[28] B. Jørgensen, The Theory of Dispersion Models, Chapman Hall, London, 1997 | MR

[29] B. Jouault, M. Greiner, P. Lipa, “Fix-point multiplier distributions in discrete turbulent cascade models”, Phys. D, 136 (2000), 125–144 | DOI | Zbl

[30] O. Kallenberg, Random Measures, Akademie-Verlag, Berlin, 1986 | MR

[31] A. N. Kolmogorov, “Lokalnaya struktura turbulentnosti v neszhimaemoi vyazkoi zhidkosti pri ochen bolshikh chislakh Reinoldsa”, Dokl. AN SSSR, 30:4 (1941), 299–303

[32] A. N. Kolmogorov, “Rasseyanie energii pri lokalno izotropnoi turbulentnosti”, Dokl. AN SSSR, 32:1 (1941), 19–21 | MR

[33] A. N. Kolmogorov, “Utochnenie predstavlenii o lokalnoi strukture turbulentnosti v neszhimaemoi vyazkoi zhidkosti pri bolshikh chislakh Reinoldsa”, Mécanique de la turbulence: Actes du Colloque International du CNRS, 1962, 447–451 ; Рђ. Рќ. Колмогоров, Р�збранныРμ труды. МатРμматика Рё РјРμС...аника, Наука, Рњ., 1985, 348–352 | MR | Zbl | MR

[34] S. Kwapień, W. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992 | MR | Zbl

[35] C. Meneveau, K. R. Sreenivasan, “The multifractal nature of turbulent energy dissipation”, J. Fluid Mech., 224 (1991), 429–484 | DOI | MR | Zbl

[36] J. Møller, Shot noise Cox processes, Research Report 2002-18, MaPhySto, Univ. of Aarhus, Aarhus, 2002

[37] J. Møller, A. R. Syversveen, R. P. Waagepetersen, “Log Gaussian Cox processes”, Scand. J. Statist., 25:3 (1998), 451–482 | DOI | MR

[38] A. S. Monin, A. M. Yaglom, Statisticheskaya gidromekhanika, t. 1, 2, Nauka, M., 1965, 1967

[39] E. Nicolato, E. Venardos, “Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type”, Math. Finance, 13:4 (2003), 445–466 | DOI | MR | Zbl

[40] A. M. Obukhov, “Some specific features of atmospheric turbulence”, J. Fluid Mech., 13 (1962), 77–81 | DOI | MR

[41] J. Pedersen, The Lévy–Ito decomposition of an independently scattered random measure, Research Report 2003-2, MaPhySto, Univ. of Aarhus, Aarhus, 2003

[42] B. Rüdiger, Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces, Research Report, SFB 611, Universität Bonn, Bonn, 2003

[43] G. Samorodnitsky, M. Taqqu, Stable non-Gaussian Random Processes, Chapman Hall, New York, 1994 | MR

[44] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[45] W. Schoutens, Lévy Processes in Finance, Wiley, New York, 2003 | Zbl

[46] J. Schmiegel, Ein dynamischer Prozess für die statistische Beschreibung der Energiedissipation in der vollentwickelten Turbulenz, Dissertation TU Dresden, Germany, 2002

[47] J. Schmiegel, Some remarks on Taylor's Frozen Flow Hypothesis, Research Report, MaPhySto, Univ. of Aarhus, Aarhus, 2003

[48] J. Schmiegel, T. Dziekan, J. Cleve, B. Jouault, M. Greiner, “Scaling functions in a random multiplicative energy-cascade model of turbulence”, Phys. Rev. E, 2002 (to appear)

[49] J. Schmiegel, H. C. Eggers, M. Greiner, Dynamical random multiplicative cascade model in 1+1 dimensions, arXiv: cond-mat/0106347

[50] J. Schmiegel, H. C. Eggers, M. Greiner, “A spatio-temporal and causal multifractal stochastic process”, Phys. A, 2003 (to appear)

[51] A. N. Shiryaev, Kolmogorov and the Turbulence, Miscellanea, 12, MaPhySto, Univ. of Aarhus., Aarhus, 1999

[52] K. R. Sreenivasan, R. A. Antonia, “The phenomenology of small-scale turbulence”, Annu. Rev. Fluid Mech., 29:2 (1997), 435–472 | DOI | MR

[53] R. L. Wolpert, K. Ickstadt, “Poisson/gamma random field models for spatial statistics”, Biometrika, 85:2 (1998), 251–267 | DOI | MR | Zbl