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October 2017 A mean-field stochastic control problem with partial observations
Rainer Buckdahn, Juan Li, Jin Ma
Ann. Appl. Probab. 27(5): 3201-3245 (October 2017). DOI: 10.1214/17-AAP1280

Abstract

In this paper, we are interested in a new type of mean-field, non-Markovian stochastic control problems with partial observations. More precisely, we assume that the coefficients of the controlled dynamics depend not only on the paths of the state, but also on the conditional law of the state, given the observation to date. Our problem is strongly motivated by the recent study of the mean field games and the related McKean–Vlasov stochastic control problem, but with added aspects of path-dependence and partial observation. We shall first investigate the well-posedness of the state-observation dynamics, with combined reference probability measure arguments in nonlinear filtering theory and the Schauder fixed-point theorem. We then study the stochastic control problem with a partially observable system in which the conditional law appears nonlinearly in both the coefficients of the system and cost function. As a consequence, the control problem is intrinsically “time-inconsistent”, and we prove that the Pontryagin stochastic maximum principle holds in this case and characterize the adjoint equations, which turn out to be a new form of mean-field type BSDEs.

Citation

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Rainer Buckdahn. Juan Li. Jin Ma. "A mean-field stochastic control problem with partial observations." Ann. Appl. Probab. 27 (5) 3201 - 3245, October 2017. https://doi.org/10.1214/17-AAP1280

Information

Received: 1 August 2015; Revised: 1 January 2017; Published: October 2017
First available in Project Euclid: 3 November 2017

zbMATH: 1380.93282
MathSciNet: MR3719957
Digital Object Identifier: 10.1214/17-AAP1280

Subjects:
Primary: 60H10, 60H30, 93E03, 93E11, 93E20

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 5 • October 2017
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