Quadratic Mean-Field BSDEs and Exponential Utility Maximization

In this paper, we study a class of real-valued mean-field backward stochastic differential equations (BSDEs) with generators of quadratic growth in the control variable and the mean-field term. Under this assumption, together with a bounded terminal condition, we establish the existence and uniqueness of solutions. Our approach departs from classical fixed-point arguments and instead combines Malliavin calculus with refined BMO and stability estimates. The result bridges the gap between the quadratic BSDE results of [Ann. Probab. 45 (2017), pp.~3795--3828] and Hao et al. [Ann. Appl. Probab. 35 (2025), pp.~2128--2174]. Moreover, motivated by the structure of the mean-field exponential utility maximization problem introduced in our paper, we extend our framework to terminal conditions without continuity or the Markovian assumption. We establish the existence and uniqueness of solutions under a smallness terminla value on the terminal conditions. We then apply this extended theory to solve a mean-field exponential utility maximization problem, which developing the classical framework of Hu et al. [Ann. Appl. Probab. 15 (2005), pp.~1691--1712] to a fully coupled quadratic mean-field setting.