Ren, Z., Tan, X., Touzi, N., & Yang, J. (2023). Entropic Optimal Planning for Path-Dependent Mean Field Games. SIAM Journal on Control and Optimization, 61(3), 1415–1437. https://doi.org/10.1137/22M1484444
E105 - Institut für Stochastik und Wirtschaftsmathematik
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Journal:
SIAM Journal on Control and Optimization
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ISSN:
0363-0129
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Date (published):
2023
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Number of Pages:
23
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Publisher:
SIAM PUBLICATIONS
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Peer reviewed:
Yes
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Keywords:
McKean-Vlasov dynamic; mean field games; optimal transport; planning problem
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Abstract:
In the context of mean field games, with possible control of the diffusion coefficient, we consider a path-dependent version of the planning problem introduced by P. L. Lions: given a pair of marginal distributions (\mu 0,\mu 1), find a specification of the game problem starting from the initial distribution \mu 0 and inducing the target distribution \mu 1 at the mean field game equilibrium. Our main result reduces the path-dependent planning problem to an embedding problem, that is, constructing a McKean-Vlasov dynamics with given marginals (\mu 0,\mu 1). Some sufficient conditions on (\mu 0,\mu 1) are provided to guarantee the existence of solutions. We also characterize, up to integrability, the minimum entropy solution of the planning problem. In particular, as uniqueness does not hold anymore in our path-dependent setting, one can naturally introduce an optimal planning problem which would be reduced to an optimal transport problem along controlled McKean-Vlasov dynamics.
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Research Areas:
Mathematical Methods in Economics: 50% Fundamental Mathematics Research: 50%