From optimal execution in front of a background noise to mean field games

A large number of mathematical frameworks are available to control optimally of the execution of a large order (see for instance "Optimal control of trading algorithms: a general impulse control approach" SIAM J. Financial... [ view full abstract ]

A large number of mathematical frameworks are available to control optimally of the execution of a large order (see for instance "Optimal control of trading algorithms: a general impulse control approach" SIAM J. Financial Mathematics, 2:1, 404-438 by Bouchard, Dang, Lehalle in 2011 or "General intensity shapes in optimal liquidation" Mathematical Finance, 25:3, 457-495. Guéant and Lehalle, in 2015), and some frameworks are emerging to manage the life cycle of small orders in an orderbook (like in "Optimal liquidity-based trading tactics", by Lehalle, Mounjid, and Rosenbaum, arxiv 2018). In all these framework an isolated investor faces a background noise coming from the aggregation of other market participants' behaviours. With recent progresses in Mean Field Games (MFG), it is now possible to propose analyses of the same problems in a closed loop, going further than current isolated views. I will expose proposed approaches for both cases (see "Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis" Mathematics and Financial Economics,10:3, 223-262, by Lachapelle, Lasry, Lehalle, Lions, 2016 for small orders and "Mean field game of controls and an application to trade crowding" Mathematics and Financial Economics, by Cardaliaguet and Lehalle, 2017 for large orders) and explain how MFG can answer to a lot of needs in modelling liquidity on financial markets.