We construct an abstract framework in which the dynamic programming principle (DPP) can be readily proven. It encompasses a broad range of common stochastic control problems in the weak formulation, and deals with problems in the “martingale formulation” with particular ease. We give two illustrations; first, we establish the DPP for general controlled diffusions and show that their value functions are viscosity solutions of the associated Hamilton–Jacobi–Bellman equations under minimal conditions. After that, we show how to treat singular control on the example of the classical monotone-follower problem.
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Extended Mean Field Control Problems: stochastic maximum principle and transport perspective
We study Mean Field stochastic control problems where the cost function and the
state dynamics depend upon the joint distribution of the controlled state and the
control process. We prove suitable versions of the Pontryagin stochastic maximum
principle, both in necessary and in sufficient form, which extend the known conditions
to this general framework. Furthermore, we suggest a variational approach to study
a weak formulation of these control problems. We show a natural connection between
this weak formulation and optimal transport on path space, which inspires a novel
discretization scheme.
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- Award ID(s):
- 1716673
- NSF-PAR ID:
- 10074614
- Date Published:
- Journal Name:
- arXiv.org
- ISSN:
- 2331-8422
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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