In this seminar, we discuss some aspects of mean-field type stochastic optimal control and mean field games (MFGs). The focus will be on both PDE and stochastic methods for solving MFGs.We meet Monday, 15:oo - 17:ooin RUD 25, Room 3.00.7. The first meeting will be on April 16th. Here is a tentative outline of the seminar.
- Review of stochastic optimal control (if necessaey)
- Analytic approaches to MFGs (J. Lasry and P. Lions, Mean Field Games, Jpn. J. Math., 2 (2007), 229-260)
- Probabilistic approaches to MFGs and forward-backward SDEs (R. Carmona, F. Delarue, Probabilistic Analysis of Mean-Field Games, SIAM J. Control Optim.,51 (2013), 2705-2734)
- Mean field type control: Mac-Kean-Vlaslov FBSDs (R. Carmona, F. Delarue, and A. Lachapelle, Control of McKeanVlasov Dynamics versus
Mean Field Games, Math. Financ. Econ., 7 (2013), 131-166.)
- Relaxed solutions in optomal control (U. Haussmann, W. Suo, Singular Optimal Stochastic Controls I: Existence, SIAM. J. ControlOptim. 33 (1995), 916-936.)
- Relaxed solutions to MFGs ( D. Lacker, Mean Field Games via Controlled Martingale Problems: Existence of Markovian Equilibria, Stoch. Proc. Appl. 125 (2015), 2856-2894.)
- MFGs with singular control (G. Fu, U. Horst, Mean-field games with singular controls; SIAM J. Control Optim., 55 (2017), 3833-3868)
- A MFG of optimal portfolio liquidation (G. Fu, P. Graewe, U. Horst, A. Popier, preprint)
Basic knowledge of stochastic calculus (Brownian motion, Ito integral, SDEs) is required; basic kowledge of optimal stochastic control is deirable. Here is a good reference for stochastic control (available as e-book):
- Huyên Pham: Continuous-time stochastic control and optimization with financial applications, Springer (2009)
These books provide a compresensive analysis of MFGs:
- R. Carmona, F. Delarue: Probabilistic Theory of Mean Field Games With Applications I, II; Springer (2018)
Some of the presentations will touch on our own research; a limited number of topics for a Master thesis on MFGs will be available. More information will b eposted here soon.