An Extended McKean — Vlasov Dynamic Programming Approach to Robust Equilibrium Controls under Ambiguous Covariance Matrix
34 Pages Posted: 16 May 2020
Date Written: April 20, 2020
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.
Keywords: Time-Inconsistency, Ambiguous Covariance Matrix, McKean — Vlasov Dynamics, Extended Dynamic Programming, Bellman — Isaacs PDE System, Portfolio Selection
JEL Classification: C72, D81, G11
Suggested Citation: Suggested Citation