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This paper introduces one unique price motion process with fractional Brownian motion. We introduce the imaginary number into the agent’s subjective probability for the reason of convergence; further, the result similar to Ito Lemma is proved. As an application, this result is applied to Merton’s dynamic asset pricing framework. We find that the four order moment of fractional Brownian motion is entered into the agent’s decision-making. The decomposition of variance of economic indexes supports the possibility of the complex number in price movement.

This work is devoted to development of instruments of the statistical mechanics as well as to their application in studying microstructural behavior of the metal matrix composites reinforced with randomly positioned silicon carbide (SiC) particles. The micro-scale representative volume element (RVE) models of the materials were studied. Their microstructure was described with the correlation functions of different orders. Analytical expressions for the local stress and strain fields’ statistics were obtained using the solution of the boundary value problem in statistical formulation. The developed analytical model allows to take into account both geometrical parameters of microstructure and physical–mechanical properties of constituents. The case studies of multiphase Ti + SiC and Al + SiC metal matrix composites were investigated. Analysis of influence of the microstructural parameters on behavior of each of the phases was performed with obtaining of the numerical results for the statistics of stress and strain fields. [ABSTRACT FROM AUTHOR]

This paper introduces one unique price motion process with fractional Brownian motion. We introduce the imaginary number into the agent's subjective probability for the reason of convergence; further, the result similar to Ito Lemma is proved. As an application, this result is applied to Merton's dynamic asset pricing framework. We find that the four order moment of fractional Brownian motion is entered into the agent's decision-making. The decomposition of variance of economic indexes supports the possibility of the complex number in price movement. [ABSTRACT FROM AUTHOR]