Continuous-Time Random Walk with multi-step memory: an application to market dynamics

• Published in: The European physical journal. B, Condensed matter physics Vol. 90; no. 11; pp. 1 - 15
• Main Authors: Gubiec, Tomasz, Kutner, Ryszard
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2017; Springer; Springer Nature B.V
• Subjects: Autocorrelation functions; Comparative analysis; Complex Systems; Computer memory; Condensed Matter Physics; Current State and Outlook; Empirical analysis; Fluid- and Aerodynamics; Histograms; Markets; Physics; Physics and Astronomy; Random variables; Random walk; Regular Article; Solid State Physics; Stochastic processes; Topical issue: Continuous Time Random Walk Still Trendy: Fifty-year History
• ISSN: 1434-6028; 1434-6036
• DOI: 10.1140/epjb/e2017-80216-3

An extended version of the Continuous-Time Random Walk (CTRW) model with memory is herein developed. This memory involves the dependence between arbitrary number of successive jumps of the process while waiting times between jumps are considered as i.i.d. random variables. This dependence was established analyzing empirical histograms for the stochastic process of a single share price on a market within the high frequency time scale. Then, it was justified theoretically by considering bid-ask bounce mechanism containing some delay characteristic for any double-auction market. Our model appeared exactly analytically solvable. Therefore, it enables a direct comparison of its predictions with their empirical counterparts, for instance, with empirical velocity autocorrelation function. Thus, the present research significantly extends capabilities of the CTRW formalism.