Although the concept of entropy is originated from thermodynamics, its concepts and relevant principles, especially the principles of maximum entropy and minimum cross-entropy, have been extensively applied in finance. In this paper, we review the concepts and principles of entropy, as well as their applications in the field of finance, especially in portfolio selection and asset pricing. Furthermore, we review the effects of the applications of entropy and compare them with other traditional and new methods.

Applications of Entropy in Finance: A Review

Arbitrary amplitude ion-acoustic solitary waves are addressed in a two-component plasma with a q-nonextensive electron velocity distribution. Our results show that in such a plasma solitary waves, the amplitude and nature of which depend sensitively on the q-nonextensive parameter can exist. Due to the electron nonextensivity, our plasma model can admit compressive as well as rarefactive ion-acoustic solitons. For q>1 (−1<q<1), the lower limit of the allowable Mach numbers is smaller (greater) than its Boltzmannian counterpart, allowing therefore the possibility of the existence of subsonic ion-acoustic solitons. As the nonextensive character of the plasma becomes important, the potential pulse amplitude increases while its width is narrowed. For −1<q<0, our plasma model can admit only rarefactive ion-acoustic solitary waves involving relatively high Mach numbers. Due to the flexibility provided by the nonextensive q-parameter, our results should help in providing a good fit between theoretical and experi...

Ion-acoustic solitary waves in a plasma with a q-nonextensive electron velocity distribution

A first theoretical work is presented to study variable charge dust acoustic solitons within the theoretical framework of the Tsallis statistical mechanics. Our results reveal that the spatial patterns of the variable charge solitary wave are significantly modified by electron nonextensive effects. In particular, it may be noted that for −1 1 provides qualitatively opposite results: electron nonextensivity makes the solitary structure more spiky. Our results should help in providing a go...

Variable charge dust acoustic solitary waves in a dusty plasma with a q-nonextensive electron velocity distribution

Electron-acoustic double-layers (EA-DLs) are addressed in a plasma with a q-nonextensive electron velocity distribution. The domain of their allowable Mach numbers depends drastically on the plasma parameters and, in particular, on the electron nonextensivity. As the electrons evolve far away from their thermodynamic equilibrium, the negative EA-DLs shrinks and may develop into compressive EA-DLs. Our results may be relevant to the double-layers observed both in the auroral region and the plasma sheet of Earth’s magnetosphere (during enhanced magnetic activity). These DLs associated parallel electric fields are thought to be responsible for particle (electrons and ions) acceleration. Furthermore, our theoretical analysis brings a possibility to develop more refined theories of nonlinear cosmic DLs that may occur in astrophysical plasmas.

Electron acoustic double layers in a plasma with a q -nonextensive electron velocity distribution

The modulational instability (MI) of ion-acoustic waves (IAWs) in a two-component plasma is investigated in the context of the nonextensive statistics proposed by Tsallis [J. Stat. Phys. 52, 479 (1988)]. Using the reductive perturbation method, the nonlinear Schrodinger equation (NLSE) which governs the MI of the IAWs is obtained. The presence of the nonextensive electron distribution is shown to influence the MI of the waves. Three different ranges of the nonextensive q-parameter are considered and in each case the MI sets in under different conditions. Furthermore, the effects of the q-parameter on the growth rate of MI are discussed in detail.

Modulational instability of ion-acoustic waves in a plasma with a q-nonextensive electron velocity distribution

In the present effort we consider the most general non linear particle kinetics within the framework of the Fokker-Planck picture. We show that the kinetics imposes the form of the generalized entropy and subsequently we demonstrate the H-theorem. The particle statistical distribution is obtained, both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy. The present approach allows to treat the statistical distributions already known in the literature in a unifying scheme. As a working example we consider the kinetics, constructed by using the $\kappa$-exponential $\exp_{_{\{\kappa\}}}(x)= (\sqrt{1+\kappa^2x^2}+\kappa x)^{1/\kappa}$ recently proposed which reduces to the standard exponential as the deformation parameter $\kappa$ approaches to zero and presents the relevant power law asymptotic behaviour $\exp_{_{\{\kappa\}}}(x){\atop\stackrel\sim x\to \pm \infty}|2\kappa x|^{\pm 1/|\kappa|}$. The $\kappa$-kinetics obeys the H-theorem and in the case of Brownian particles, admits as stationary state the distribution $f=Z^{-1}\exp_{_{\{\kappa\}}}[-(\beta mv^2/2-\mu)]$ which can be obtained also by maximizing the entropy $S_{\kappa}=\int d^n v [ c(\kappa)f^{1+\kappa}+c(-\kappa)f^{1-\kappa}]$ with $c(\kappa)=-Z^{\kappa}/ [2\kappa(1+\kappa)]$ after properly constrained.

G. Kaniadakis

Papers

H-Theorem and Generalized Entropies Within the Framework of Non Linear Kinetics