Abstract

We present an approach to describe the stock market crises based on Mean Field Games and Optimal Control theory with a turnpike effect. The impact of the large amount of high-frequency traders (HFTs) can be modelled via mean field term. A Mean Field Game is a coupled system of PDEs: a Kolmogorov–Fokker–Planck equa-tion, evolving forward in time and describing evolution of the HFTs probability den-sity function spread by the amount of asset shares; and a Hamilton–Jacobi–Bellman equation, evolving backwards in time and defining the strategy of the HFTs. These equations form a boundary value problem