scholarly journals Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations

2019 ◽  
Author(s):  
Masaaki Fujii
Keyword(s):  
Probabilistic Approach ◽  
Mean Field ◽  
Mean Field Games ◽  
Control Problems ◽  
Multiple Populations ◽  
Field Type ◽  
Mean Field Type Control

scholarly journals Control problem on space of random variables and master equation

2019 ◽  
Vol 25 ◽  
pp. 10
Author(s):  
Alain Bensoussan ◽  
Sheung Chi Phillip Yam
Keyword(s):  
Hilbert Space ◽  
Control Problem ◽  
Master Equation ◽  
Mean Field ◽  
Random Variables ◽  
Probability Measures ◽  
Control Problems ◽  
Field Type ◽  
Collège De France ◽  
Mean Field Type Control

In this article, we study a control problem in an appropriate space of random variables; in fact, in our set up, we can consider an arbitrary Hilbert space, yet we specialize only to a Hilbert space of square-integrable random variables. We see that the control problem can then be related to a mean field type control problem. We explore here a suggestion of Lions in (Lectures at College de France, http://www.college-de-france.fr) and (Seminar at College de France). Mean field type control problems are control problems in which functionals depend on probability measures of the underlying controlled process. Gangbo and Święch [J. Differ. Equ. 259 (2015) 6573–6643] considered this type of problem in the space of probability measures equipped with the Wasserstein metric and use the concept of Wasserstein gradient; their work provides a completely rigorous treatment, but it is quite intricate, because metric spaces are not vector spaces. The approach suggested by Lions overcomes this difficulty. Nevertheless, our present proposed approach also benefits from the useful concept of L-derivatives as introduced in a recent interesting treatise of Carmona and Delarue [Probabilistic Theory of Mean Field Games with Applications. Springer Verlag (2017)]. We also consider Bellman equation and the Master equation of mean field type control. We provide also some extension of the results of Gangbo and Święch [J. Differ. Equ. 259 (2015) 6573–6643].

scholarly journals Cemracs 2017: numerical probabilistic approach to MFG

2019 ◽  
Vol 65 ◽  
pp. 84-113
Author(s):  
Andrea Angiuli ◽  
Christy V. Graves ◽  
Houzhi Li ◽  
Jean-François Chassagneux ◽  
François Delarue ◽  
...  
Keyword(s):  
Optimization Problems ◽  
Continuation Method ◽  
Large Population ◽  
Probabilistic Approach ◽  
Mean Field ◽  
Picard Iteration ◽  
Mean Field Games ◽  
Benchmark Problems ◽  
Control Problems ◽  
Great Success

This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see [29], and by Huang, Caines, and Malhamé, see [26]. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme; importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges.

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