Risky asset models with dependence and heavy-tailed distributions

   Cardiff School of Mathematics

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  Prof N Leonenko, Dr A Pepelyshev  Applications accepted all year round  Self-Funded PhD Students Only

About the Project

It is now generally accepted that heavy-tailed distributions occur commonly in practice. Their use is now widespread in communication network, risky asset and insurance modelling. However, the study of stationary processes having these heavy-tailed distributions as their one-dimensional distributions and also having a full range of possible dependence structures has received rather little attention. We focus on such processes with Student and related tempered stable marginals. The Student family with ν degrees of freedom covers the range of power tail possibilities, the spectrum including the Cauchy distribution and ranging through to the Gaussian distribution.

   In the field of finance, distributions of logarithmic asset returns can often be fitted extremely well by Student t-distribution or Variance Gamma distribution. In particularly, a number of authors have advocated using a t-distribution with ν degrees of freedom, typically 3≤ν≤5. This implies infinite kth moments for k≥ν.

   Another issue in modelling economic and financial time series is that their sample autocorrelation functions (acf) may decay quickly, but their absolute or squared increments may have acfs with non-negligible values for large lags. These ubiquitous phenomena call for an effort to develop reasonable models which can be integrated into the economic and financial theory. This approach has a long history, certainly dating back to Mandelbrot's work in the 1960s, in which use of heavy-tailed distributions (stable or Pareto type) was advocated. Tempered stable processes are also can be used in financial econometrics.

   A particular motivation is the modelling of risky asset time series related to a fractal activity time geometrical Brownian motion (FATGBM), which have a heavy-tailed or semi heavy–tailed log returns and significant dependence in squares including long-range dependence. 

  One of the main aims of the project is to develop a stochastic analysis (including pricing formulae, hedging procedures), the analysis higher frequency data and the statistical inference theory for estimation of unknown parameters of such models, including the so-called Hurst parameter or parameter of long-range dependence.

The project will also consider some further extension these models for vector-valued stochastic processes for modelling of management portfolios.

A particular motivation is the modelling of risky asset time series related to a fractal activity time geometrical Brownian motion (FATGBM), which have a heavy-tailed or semi heavy–tailed log returns and significant dependence in squares including long-range dependence. 

The project considers also  a statistical methods for analysis of log-retunes of such a risky assets and its counterpart related to the analysis of higher frequency data.

For further details please contact Prof. Nikolai Leonenko

Mathematics (25)

Funding Notes

We are interested in pursuing this project and welcome applications if you are self-funded or have funding from other sources, including government sponsorships or your employer


[1] Heyde, C. C.; Leonenko, N. N. (2005) Student processes. Adv. in Appl. Probab. 37, no. 2, 342-365
[2] Leonenko N.N, Petherick, S. and A. Sikorskii (2011) The Student subordinated models with dependence for risky asset returns, Communications in Statistics- Theory and Methods, 40, 3509-3523
[3] Leonenko, N.N, Petherick, S. and Sikorskii, A (2012) Fractal activity time models for risky asset with dependence and generalized hyperbolic distributions, Stochastic Analysis and Applications, 30, N3, 476-492
[4] Leonenko N.N, Petherick, S. and Sikorskii A. (2012) Normal Inverse Gaussian model for risky asset with dependence, Statistic and Probability Letters, 82, 109-115
[5] Leonenko, N.N, Petherick, S. and Taufer, E. (2013) Multifractal scaling for risky asset modelling, Physica A: Statistical Mechanics and its Applications, Volume 392, issue 1, 7-16
[6] Kerss, A.D.J., Leonenko, N.N. and and Sikorskii, A. (2014) Risky asset models with tempered stable fractal activity time, Stochastic Analysis and Applications, 32, 642-663
[7] Kerss, A.D.J., Leonenko, N.N. and and Sikorskii, A. (2014) Fractional Skellam processes with applications to finance, Fractional Calculus and Applied Analysis, 17. No.2, pp 532-551
[8] Grahovac, D. and Leonenko, N (2014) Detecting multifractal stochastic processes under heavy-tailed effects, Chaos, Solitons, Fractals, 65, 78-89
[9] Grahovac, D. , Leonenko, N and Taqqu, M. S. (2015) Scaling properties of the structure function of linear fractional stable motion and estimation of its parameters, Journal of Statistical Physics, 158, 105-119
[10] Grahovac, D., Jia. M., Leonenko, N. and Taufer, E. (2015) Asymptotic properties of the partition function and applications in tail index inference of heavy-tailed data, Statistics, DOI: 10.1080/02331888.2014.969267

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