An Extended McKean — Vlasov Dynamic Programming Approach to Robust Equilibrium Controls under Ambiguous Covariance Matrix

34 Pages Posted: 16 May 2020

See all articles by Qian Lei

Qian Lei

Nanyang Technological University (NTU) - School of Physical and Mathematical Sciences

Chi Seng Pun

Nanyang Technological University (NTU) - School of Physical and Mathematical Sciences

Date Written: April 20, 2020

Abstract

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

Lei, Qian and Pun, Chi Seng, An Extended McKean — Vlasov Dynamic Programming Approach to Robust Equilibrium Controls under Ambiguous Covariance Matrix (April 20, 2020). Available at SSRN: https://ssrn.com/abstract=3581429 or http://dx.doi.org/10.2139/ssrn.3581429

Qian Lei

Nanyang Technological University (NTU) - School of Physical and Mathematical Sciences ( email )

S3 B2-A28 Nanyang Avenue
Singapore, 639798
Singapore

Chi Seng Pun (Contact Author)

Nanyang Technological University (NTU) - School of Physical and Mathematical Sciences ( email )

SPMS-MAS-05-22
21 Nanyang Link
Singapore, 637371
Singapore
(+65) 6513 7468 (Phone)

HOME PAGE: http://personal.ntu.edu.sg/cspun/

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