Fractional econophysics: Market price dynamics with memory effects

In recent years, a new branch of the econophysics has appeared and began to actively develop, which can be called fractional econophysics. We can define fractional econophysics as a new direction of research applying methods developed in physical sciences, to describe processes in economics and finance, basically those including power-law memory and spatial nonlocality. The mathematical tool of this branch of econophysics is the fractional calculus. The birth of the fractional econophysics can be dated 2000 and it can be primarily associated with the works of a group, which includes E. Scalas, F. Mainardi, R. Gorenflo, M. Raberto, in the field of the continuous-time finance. The fractional econophysics was born on the border of the centuries: the first paper was submitted to Physica A on 10 December 1999. Then a lot of work was done on the adaptation and application of fractional dynamics methods, physical model and equations, previously used in the physical sciences, to the description of processes in economics and finance. In fact, at the end of 2019 and at the beginning of 2020 there will be a twenty-year anniversary of fractional econophysics. In this paper, using the fractional econophysics approach, we consider dynamics of market prices, in which power-law memory effects are taken into account. We propose new economic model of the price dynamics in the market for a single product. In this model we assume that economic entities (merchants, buyers, suppliers) can remember how stocks of goods and their prices have changed over time.