An Extended McKean–Vlasov Dynamic Programming Approach to Robust Equilibrium Controls Under Ambiguous Covariance Matrix

This paper studies a general class of time-inconsistent stochastic control problems under ambiguous covariance matrix. The time inconsistency is caused in various ways by a general objective functional and thus the associated control problem does not admit Bellman’s principle of optimality. Moreover...

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Bibliographic Details
Published in:Applied mathematics & optimization Vol. 88; no. 3; p. 91
Main Authors: Lei, Qian, Pun, Chi Seng
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2023
Springer Nature B.V
Subjects:
Calculus of Variations and Optimal Control; Optimization
Control
Covariance matrix
Dynamic programming
Equilibrium
Game theory
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical and Computational Physics
Optimal control
Partial differential equations
Principles
Robust control
Simulation
Stochastic processes
Systems Theory
Theoretical
Dynamic programming/optimal control
35B37
Extended dynamic programming
Time inconsistency
McKean–Vlasov dynamics
Bellman–Isaacs PDE system
91B28
34H05
Ambiguous covariance matrix
91A40
ISSN:0095-4616, 1432-0606
Online Access:Get full text
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Summary:This paper studies a general class of time-inconsistent stochastic control problems under ambiguous covariance matrix. The time inconsistency is caused in various ways by a general objective functional and thus the associated control problem does not admit Bellman’s principle of optimality. Moreover, we model the state by a McKean–Vlasov dynamics under a set of non-dominated probability measures induced by the ambiguous covariance matrix of the noises. We apply a game-theoretic concept of subgame perfect Nash equilibrium to develop a robust equilibrium control approach, which can yield robust time-consistent decisions. We characterize the robust equilibrium control and equilibrium value function by an extended optimality principle and then we further deduce a system of Bellman–Isaacs equations to determine the equilibrium solution on the Wasserstein space of probability measures. The proposed analytical framework is illustrated with its applications to robust continuous-time mean-variance portfolio selection problems with risk aversion coefficient being constant or state-dependent, under the ambiguity stemming from ambiguous volatilities of multiple assets or ambiguous correlation between two risky assets. The explicit equilibrium portfolio solutions are represented in terms of the probability law.
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ISSN:0095-4616
1432-0606
DOI:10.1007/s00245-023-10069-3