Statistical mechanics and Bayesian Inference addressed to the Osborne Paradox

One of the greatest contributors of the 20th century among all academician in the field of statistical finance, M. F. M. Osborne published in 1956 [6] an essential paper and proposed to treat the question of stock market motion through the prism of both the Law of Weber-Fechner [1, 4] and the branch of physics developed by James Clerk Maxwell, Ludwig Boltzmann and Josiah Willard Gibbs [3, 5] namely the statistical mechanics. He proposed an improvement of the known research made by his predecessor Louis Jean-Baptiste Alphonse Bachelier, by not considering the arithmetic changes of stock prices as means of statistical measurement, but by drawing on the Weber-Fechner Law, to treat the changes of prices. Osborne emphasized that as in statistical mechanics, the probability distribution of the steady-state of subjective change in prices is determined by the condition of maximum probability, a statement close to the Gibbs distribution conditions. However, Osborne also admitted that the empirical observation of the probability distribution of logarithmic changes of stock prices was emphasizing obvious asymmetries and consequently could not perfectly confirm his prior theory. The purpose of this paper is to propose an explanation to what we could call the Osborne paradox and then address an alternative approach via Bayesian inference regarding the description of the probability distribution of changes in logarithms of prices that was thenceforth under the prism of frequentist inference. We show that the stock market returns are locally described by equilibrium statistical mechanics with conserved statistics variables, whereas globally there is yet other statistics with persistent flowing variables that can be effectively described by a superposition of several statistics on different time scales, namely, a superstatistics.