Robust non-zero-sum investment–consumption games under multivariate stochastic covariance models

Abstract

This paper discusses a non-zero-sum stochastic differential game involving multiple players and model ambiguity. Each economic agent is concerned about the relative performance of other agents in terms of their average terminal wealth and intermediate consumption. The goal is to find the optimal investment–consumption strategy that is robust under the worst-case scenario of adverse probability measures. The competitive and ambiguity-averse agents can invest in an incomplete financial market composed of a risk-free asset, an equity index, and a single equity depicted by a class of generically non-Markovian multivariate stochastic covariance models, under which the market prices of risks hinge on a multivariate affine-diffusion factor process. The unified modeling framework includes some state-of-the-art stochastic covariance models as particular cases. A backward stochastic differential equation approach coupled with the martingale optimality principle addresses the robust non-Markovian stochastic differential game. Closed-form solutions are derived to the robust Nash equilibrium investment–consumption policies, the probability perturbation processes associated with the well-defined worst-case scenarios, and the corresponding value functions. Under certain technical conditions, we verify the admissibility of the solutions. Finally, we performed numerical experiments to showcase the impact of model parameters on robust investment–consumption strategies.