Multifractality of products of geometric Ornstein-Uhlenbeck type processes

 

Host Institution:

La Trobe University

Title of Seminar:

Multifractality of products of geometric Ornstein-Uhlenbeck type processes

Speaker's Name:

Prof. N.N. Leonenko

Speaker's Institution:

Cardiff University, UK

Time and Date:

Friday 14 August 2009 at 11:00 am

Seminar Abstract:

This is joint work with V.V. Anh (Queensland University of Technology, Brisbane) and N.-R. Shieh (National Taiwan University, Taipei).

 

We consider multifractal products of stochastic processes as de…ned in Mannersalo et al. (2002), but we provide a new interpretation of the conditions on the mean, variance and covariance functions of the resulting cumulative processes in terms of the moment generating functions. We show that the

logarithms of the corresponding limiting processes have an in…nitely divisible distribution such as the gamma and variance gamma distributions (resulting in the log-gamma and log-variance gamma scenarios respectively), the inverse Gaussian and normal inverse Gaussian distributions (yielding the log-inverse

Gaussian and log-normal inverse Gaussian scenarios respectively). We describe the behavior of their q-th order moments and Rényi functions, which are non-linear, hence displaying their multifractality. A property on the dependence structure of the limiting processes, leading to their possible long-range dependence, is also obtained.

We consider very different scenarios such as the log-gamma and log-inverse Gaussian scenarios as typical examples of our approach. We should also note some related results by Barndorff-Nielsen and Schmiegel (2004) who introduced some Lévy-based spatiotemporial models for parametric modelling of turbulence. Log-in…finitely divisible scenarios related to independently scattered random measures were introduced in Bacry and Muzy (2003) and others. We should note that Chris Heyde (1999) proposed to use a multifractality into risky asset model with fractal activity time (see also Heyde and Leonenko (2005)).

Similar results can be obtained for the multifractal products of stationary diffusion processes (Anh, Leonenko and N.-R. Shieh (2009b)) and birth-death processes processes (Anh, Leonenko and N.-R. Shieh (2009a)).

REFERENCES

[1] V.V. Anh, N.N. Leonenko and N.-R. Shieh (2008) Multifractality of products of geomertic Ornstein-Uhlenbeck type processes, Adv. Appl. Prob., 40, 1129-1156.

[2] V.V. Anh, N.N. Leonenko and N.-R. Shieh (2009a) Multifractal scaling of products of birth-death processes, Bernoulli, 15 (2), 508-531.

[3] V.V. Anh, N.N. Leonenko and N.-R. Shieh (2009b) Multifractal products of stationary diffusion processes, Stochastic Analysis and Applications, 27, 475-499.

[4] E. Bacry and J.F. Muzy (2003) Log-in…nitely divisible multifractal processes, Comm. Math. Phys. 236, 449-475.

[5] O.E. Barndorf-Nilsen and Yu. Shmigel (2004) Spatio-temporal modeling based on Lévy processes, and its applications to turbulence, (Russian) Uspekhi Mat. Nauk 59 (2004), 63–90; translation in Russian Math. Surveys 59, 65-90.

[6] C.C. Heyde, (1999). A risky asset model with strong dependence through fractal activity time. J. Appl. Prob., 36, 1234-1239.

[7] C.C. Heyde and N.N. Leonenko (2005) Student processes, Adv. Appl. Prob., 37, 342-365.

[8] P. Mannersalo, I. Norris and R. Riedi(2002), Multifractal products of stochastic processes: construction and some basic properties, Adv. Appl. Prob., 34, 888-–903.

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