Robust asset-liability management games in a stochastic market with stochastic cash flows under HARA utility

Abstract: This paper investigates an optimal asset-liability management problem involving two strategically interactive managers with ambiguity aversion under a multivariate stochastic covariance model characterized by hybrid stochastic volatility and stochastic interest rates. Two ambiguity-averse managers participate in a financial market comprising a money market account, a market index, a stock, and zero-coupon bonds to enhance profits, where interest rates are determined via an affine model, which includes both the Cox–Ingersoll–Ross model and the Vasicek model as specific instances, while the market index and stock price are driven by a general class of non-Markovian multivariate stochastic covariance models. Moreover, the two competitive managers, subject to idiosyncratic liability commitments and influenced by the random nature of cash inflow or outflow in their investment decision making, have varying risk preferences described by the hyperbolic absolute risk aversion (HARA) utility function, with the power utility function as a special case. Each manager aims to develop a robust investment strategy to outperform their competitors by maximizing the expected terminal utility of the relative surplus in worst-case scenarios. A backward stochastic differential equation method coupled with the martingale optimality principle is used to solve this robust non-Markovian stochastic differential game, resulting in closed-form expressions for robust Nash equilibrium investment strategies, the density generator processes under worst-case probability measures, and the corresponding value functions. Finally, numerical examples are provided to illustrate their financial implications.