Cemracs 2017: numerical probabilistic approach to MFG
ESAIM. Proceedings, Tome 65 (2019), pp. 84-113
Cet article a éte moissonné depuis la source EDP Sciences
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.
Affiliations des auteurs :
Andrea Angiuli 1 ; Christy V. Graves 2 ; Houzhi Li 3 ; Jean-François Chassagneux 3 ; François Delarue 4 ; René Carmona 5
@article{EP_2019_65_a5,
author = {Andrea Angiuli and Christy V. Graves and Houzhi Li and Jean-Fran\c{c}ois Chassagneux and Fran\c{c}ois Delarue and Ren\'e Carmona},
title = {Cemracs 2017: numerical probabilistic approach to {MFG}},
journal = {ESAIM. Proceedings},
pages = {84--113},
year = {2019},
volume = {65},
doi = {10.1051/proc/201965084},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201965084/}
}
TY - JOUR AU - Andrea Angiuli AU - Christy V. Graves AU - Houzhi Li AU - Jean-François Chassagneux AU - François Delarue AU - René Carmona TI - Cemracs 2017: numerical probabilistic approach to MFG JO - ESAIM. Proceedings PY - 2019 SP - 84 EP - 113 VL - 65 UR - http://geodesic.mathdoc.fr/articles/10.1051/proc/201965084/ DO - 10.1051/proc/201965084 LA - en ID - EP_2019_65_a5 ER -
%0 Journal Article %A Andrea Angiuli %A Christy V. Graves %A Houzhi Li %A Jean-François Chassagneux %A François Delarue %A René Carmona %T Cemracs 2017: numerical probabilistic approach to MFG %J ESAIM. Proceedings %D 2019 %P 84-113 %V 65 %U http://geodesic.mathdoc.fr/articles/10.1051/proc/201965084/ %R 10.1051/proc/201965084 %G en %F EP_2019_65_a5
Andrea Angiuli; Christy V. Graves; Houzhi Li; Jean-François Chassagneux; François Delarue; René Carmona. Cemracs 2017: numerical probabilistic approach to MFG. ESAIM. Proceedings, Tome 65 (2019), pp. 84-113. doi: 10.1051/proc/201965084
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