Collective behaviors of stochastic agent-based models and applications to finance and optimization

In this paper, we present a survey of recent progress on the emergent behaviors of stochastic particle models which arise from the modeling of collective dynamics. Collective dynamics of interacting autonomous agents is ubiquitous in nature, and it can be understood as a formation of concentration in a state space. The jargons such as aggregation, herding, flocking and synchronization describe such concentration phenomena. Recently it became one of the emerging topics in the applied mathematics community due to possible engineering applications and close relation with nonlocal partial differential equations. When an autonomous agent system interacts with unknown environment as an open system, the effects of hidden and unidentified interactions between the environment and the autonomous system are often realized by stochastic noises in agent dynamics, and the temporal evolution of the autonomous system results in stochastic collective models. From the viewpoint of dynamical systems theory, it is very interesting how collective dynamics emerges from initial state. As concrete examples, we consider four specific stochastic collective models (stochastic Winfree and Kuramoto models for synchronization, the stochastic Cucker-Smale model for flocking, and a first-order stochastic nonlinear consensus model), and we also briefly review the state-of-The-Art results for these models on the emergence of collective dynamics and discuss their applications in finance and optimization.