Applications of generalized statistics in critical phenomena and financial markets
Final Report Abstract
The present project applies methods and techniques known from nonequilibrium statistical physics and information theory to systems exhibiting generalized statistics. Broadly speaking, generalized statistics characterizes processes, which have broad (or heavy-tail) distributions. The difficulty in working with generalized statistics lies in the fact that the Central Limit Theorem alongside with the usual methods of statistical physics cannot be applied. Theoretical qualification for such “non-canonical” distributions is provided by means of the generalized Central Limit Theorem of P. Lévy. Phenomena obeying generalized statistics are very diverse and structurally rich including fractional diffusion processes, multifractals, volatility fluctuations, or certain polymer growth models. Primary focus of the project has been in presently intensely studied systems represented by distributions emerging either from the Rényi and Tsallis Maximum-Entropy prescription or from Superstatistics. Central applications have been in financial markets, polymer physics, and in the theory of critical phenomena in strongly-interacting many-particle systems.
Publications
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Green function of the double-fractional Fokker-Planck equation: Path integral and stochastic differential equations, Phys. Rev. E 88, 052106 (2013)
H. Kleinert and V. Zatloukal
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Local-time representation of path integrals, Phys. Rev. E 92, 062137 (2015)
P. Jizba and V. Zatloukal
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Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton-Jacobi theory, Int. J. Geom. Methods Mod. Phys. 13, 1650072 (2016)
V. Zatloukal
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Option pricing beyond Black–Scholes based on double-fractional diffusion, Physica A: Statistical Mechanics and its Applications 449, 200–214 (2016)
H. Kleinert and J. Korbel
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Tsallis thermostatics as a statistical physics of random chains
P. Jizba, J. Korbel and V. Zatloukal
