Closed-loop Equilibrium for Time-Inconsistent McKean-Vlasov Controlled Problem

Mei, Hongwei; Zhu, Chao

doi:10.48550/arxiv.2002.06952

Cited by 1 publication

(2 citation statements)

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“…Following the same arguments as in Theorem 3.2, we can profit from the resent results available in Djete, Possamaï, and Tan [22] to obtain the next dynamic programming principle. We also remark time-inconsistent McKean-Vlasov problems have been recently studied in [59].…”

Section: Assumption E Assumption A(ii) and A(iii) Together Withmentioning

confidence: 89%

“…Han and Wong [37] study the case where the state variable follows a Volterra process and, by associating an extended path-dependent Hamilton-Jacobi-Bellman equation system, obtains a verification theorem. Finally, Mei and Zhu [59] deals with a class of time-inconsistent control problems for McKean-Vlasov dynamics which are, for example, a natural framework to study mean-variance problems.…”

Section: Introductionmentioning

confidence: 99%

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Me, myself and I: a general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents

Hernández¹,

Possamaï²

2020

Preprint

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We develop a theory for continuous-time non-Markovian stochastic control problems which are inherently time-inconsistent. Their distinguishing feature is that the classical Bellman optimality principle no longer holds. Our formulation is cast within the framework of a controlled non-Markovian forward stochastic differential equation, and a general objective functional setting. We adopt a game-theoretic approach to study such problems, meaning that we seek for sub-game perfect Nash equilibrium points. As a first novelty of this work, we introduce and motivate a new definition of equilibrium that allows us to establish rigorously an extended dynamic programming principle, in the same spirit as in the classical theory. This in turn allows us to introduce a system of backward stochastic differential equations analogous to the classical HJB equation. We prove that this system is fundamental, in the sense that its well-posedness is both necessary and sufficient to characterise the value function and equilibria. As a final step we provide an existence and uniqueness result. Some examples and extensions of our results are also presented.

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Section: Assumption E Assumption A(ii) and A(iii) Together Withmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%