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We study a stochastic control/stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We demonstrate that the stochastic control/stopping problem with expectation constraints (CSEC) is independent of a specific probability setting and is equivalent to the constrained stochastic control/stopping problem in weak formulation (an optimization over joint laws of Brownian motion, state dynamics, diffusion controls and stopping rules on an enlarged canonical space). Using a martingale-problem formulation of controlled SDEs in spirit of Stroock and Varadhan (2006), we characterize the probability classes in weak formulation by countably many actions of canonical processes, and thus obtain the upper semi-analyticity of the CSEC value function. Then we employ a measurable selection argument to establish a dynamic programming principle (DPP) in weak formulation for the CSEC value function, in which the conditional expected costs act as additional states for constraint levels at the intermediate horizon. This article extends (El Karoui and Tan, 2013) to the expectation-constraint case. We extend our previous work (Bayraktar and Yao, 2024) to the more complicated setting where the diffusion is controlled. Compared to that paper the topological properties of diffusion-control spaces and the corresponding measurability are more technically involved which complicate the arguments especially for the measurable selection for the super-solution side of DPP in the weak formulation.
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A Framework for the Dynamic Programming Principle and Martingale-Generated Control Correspondences
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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- Award ID(s):
- 1815017
- PAR ID:
- 10108978
- Date Published:
- Journal Name:
- Applied Mathematics & Optimization
- ISSN:
- 0095-4616
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00245-019-09589-8
