Abstract
Stochastic optimal control problems with constraints on the probability distribution of the final output are considered. Necessary conditions for optimality in the form of a coupled system of partial differential equations involving a forward Fokker–Planck equation and a backward Hamilton–Jacobi–Bellman equation are proved using convex duality techniques.
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Acknowledgements
The author thanks the anonymous referees for their comments and careful proofreading of the paper. The author was partially supported by the ANR (Agence Nationale de la Recherche) project ANR-16-CE40-0015-01 on Mean Field Games. Part of this research was performed, while the author was visiting the Institute for Mathematical and Statistical Innovation (IMSI), which is supported by the National Science Foundation (Grant No. DMS-1929348). The author wishes to thank Professor Pierre Cardaliaguet (Paris Dauphine) for fruitful discussions all along this work.
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Appendix
Appendix
Since it appears twice in our article and in particular in the proof of Theorem 3.2, we recall the statement of the von Neumann theorem we are using. The statement and proof can be found in Appendix of [36] and in a slightly different setting, in [40].
Theorem A.1
(Von Neumann) Let \({\mathbb {A}}\) and \({\mathbb {B}}\) be convex sets of some vector spaces and suppose that \({\mathbb {B}}\) is endowed with some Hausdorff topology. Let \({\mathcal {L}}\) be a function satisfying:
Suppose also that there exist \(a_* \in {\mathbb {A}}\) and \(C_* > \sup _{a \in {\mathbb {A}} } \inf _{b \in {\mathbb {B}}} {\mathcal {L}}(a,b)\) such that:
Then,
Remark A.1
The fact that the infimum in the “\(\inf \sup \)” problem is in fact a minimum is part of the theorem.
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Daudin, S. Optimal Control of Diffusion Processes with Terminal Constraint in Law. J Optim Theory Appl 195, 1–41 (2022). https://doi.org/10.1007/s10957-022-02053-8
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DOI: https://doi.org/10.1007/s10957-022-02053-8