Robust Nash equilibrium for defined contribution pension games with delay under multivariate stochastic covariance models

Abstract

This paper explores a stochastic differential investment game problem with delay among n defined contribution pension fund managers. These managers are concerned with relative performance and model ambiguity and participate in an incomplete financial market comprising a risk-free asset, a market index, and a stock. The market index and stock are described by a class of potentially non-Markovian multivariate stochastic covariance models, with the market prices of risks dependent on a multivariate affine-diffusion factor process. Managers' wealth processes are modeled by stochastic differential delay equations, considering performance-related capital inflow and outflow. Each manager aims to maximize the expected exponential utility of his terminal wealth with delay relative to the averages among his competitors under the worst-case scenario of the alternative measures and seek a robust investment strategy. By employing a backward stochastic differential equation approach to address this robust non-Markovian control problem, we derive, in closed form, the robust Nash equilibrium investment strategies, the probability perturbation processes under the well-defined worst-case scenarios, and the corresponding value functions. The admissibility of robust equilibrium policies is confirmed under specific technical conditions. Finally, we conduct numerical examples to demonstrate the impact of model parameters on robust investment policies and derive economic interpretations from the results.