We develop an approach to analysis of stock market crises based on the generalized nonparametric method. The generalized nonparametric method is based on solvability and regularization of ill-posed inverse problem in Pareto's demand theory. Our approach allows one to select a few companies that may be considered as the main reason for the crisis. We apply this approach to study the Chinese stock market crash in 2015.
According to Pareto's theory of consumer demand a rational representative consumer should choose their consumption bundle as the solution of mathematical programming problem of maximization of utility function under their budget constraint. The inverse problem of demand analysis is to recover the utility function from the demand functions. The answer to the question of solvability of this problem is based on the revealed preference theory. If the problem is unsolvable, one should apply regularization procedure by introducing irrationality indices. When recovering the utility function one puts a priori requirements on it. In this paper, we suggest including positively-homogeneity property into these requirements. We compare the setups with and without this requirement both theoretically and empirically and provide evidence in favor of requiring this property when computing economic indices for consumer representing consumption behavior of large number of various households for large time intervals.
We consider Sturm–Liouville problems on the finite interval with non-Borg conditions. Using eigenvalues of four Sturm–Liouville problems, we construct the spectral data and show that the mapping from potential to spectral data is a bijection. Moreover, we obtain estimates of spectral data in terms of potentials.
Observed polar motion consists of uniform circular motions at both positive (prograde) and negative (retrograde) frequencies. Generalized Euler–Liouville equations of Bizouard, taking into account Earth's triaxiality and asymmetry of the ocean tide, show that the corresponding retrograde and prograde circular excitations are coupled at any frequency. In this work, we reconstructed the polar motion excitation in the Chandler band (prograde and retrograde). Then we compared it with geophysical excitation, filtered out in the same way from the series of the Oceanic Angular Momentum (OAM) and Atmospheric Angular Momentum (AAM) for the period 1960–2000. The agreement was found to be better in the prograde band than in the retrograde one.