We propose a time-delayed Cucker-Smale type model(CS model), which can be applied to modeling (1) collective dynamics of self-propelling agents and (2) the dynamical system of stock return volatility in a financial market. For both models, we assume that it takes a certain amount of time to collect/process information about the current position/return configuration until velocity/volatility adjustment is made. We provide a sufficient condition under which flocking phenomena occur. We also identify the initial configu- ration for a two-agent case, in which collective behaviors are accelerated by changes in the delay parameter. Numerical illustrations and financial simula- tions are carried out to verify the validity of the model.
2020 Mathematics Subject Classification. Primary: 82C22, 91B70, 91G80. Key words and phrases. Cucker\u2013Smale model, flocking, volatility, time delay. Bae was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2021R1A2C1093383), Cho by PRIN Project 2017 (No. 2017KKJP4X entitled \u201cInnovative numerical methods for evolutionary partial differential equations and applications\u201d) funded by Italian Ministry of Instruction, University and Research (MIUR), Yoo by Ajou University Research Fund, and Yun by Samsung Science and Technology Foundation under Project Number SSTF-BA1801. \u2217Corresponding author: Jane Yoo.
