Jarosław Klamut, Tomasz Gubiec

Continuous-time random walk (CTRW) is a stochastic process with continuous and fluctuating waiting (interevent) time. It was firstly introduced to physics by Montroll and Weiss [1]. Since then it has been used to model anomalous transport and diffusion, hydrogen diffusion in nanostructure compounds, electron transfer, aging of glasses, transport in porous media, diffusion of epicenters of earthquakes aftershocks, cardiological rhythms, human travel and many more [2]. CTRW is also successfully applied in econophysics [3], for example its version with the one-step memory was used to describe stock price dynamics [4], especially autocorrelation function (ACF) of price changes.

However, empirical ACF of absolute values of price changes decays much slower than ACF of price changes. Is the one-step jump memory able to describe this phenomena? In order to answer this question, we introduce a directed CTRW with one-step jump memory, which turned out to be analytically solvable.

[1] E. W. Montroll, G. H. Weiss, J. Math. Phys, 6(2):167181, 1965.
[2] R. Kutner, J. Masoliver, arXiv:1612.02221v1 (2016)
[3] E. Scalas, Complex Networks of Economic Interactions, pp 3-16.
Springer, 2006.
[4] T. Gubiec, R. Kutner, Phys. Rev. E 82, 046119 (2010)