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This paper is concerned with the coupling of the inverse problem theory with the thermostatted kinetic theory. Specifically an inverse problem is proposed where the data vector consists of m known measures, the data kernel is a m×n matrix which depends on the distribution function vector that is solution of the thermostatted kinetic theory model, and the unknown source or signal consists of a n-dimensional vector. In particular the paper focuses on the under-determined inverse problem, namely m < n, and the solution is obtained by employing the principle of maximum Shannon entropy of the information theory. Applications refer to the financial market and specifically to the derivation of the information which triggers the evolution of global stock market indexes. Future research directions are also discussed into the last section of the paper.

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