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We present an approach to describe the stock market crises based on Mean Field Games (MFGs) and Optimal Control theory with a turnpike effect. The impact of the large amount of high-frequency traders (HFTs) can be modeled via a mean field term. We introduce the turnpike as a function that relies on the changes of the asset share price. An MFG is a coupled system of PDEs: a Kolmogorov–Fokker–Planck equation, evolving forward in time, and a Hamilton–Jacobi–Bellman equation, evolving backwards in time. The ill-posedness of this system comes from a turnpike effect. The numerical solution of an extremal problem that is dual to a PDE system is presented. We apply this approach to describe the Chinese stock market crash in 2015 considering the representative stock of CITIC Securities (ticker 600030). We consider HFTs that form a dominating bull and bear market. As a result, the bull strategy imitators do not make any profit.

Numerical study of the stock market crises based on mean field games approach