Utility Maximization Under Endogenous Pricing
研究大投资者在不完全市场中面临内生永久市场冲击时的期望效用最大化问题，通过广义次梯度将最优性刻画为耦合正倒向随机微分方程，并证明解的存在性与光滑性。
We study the expected utility maximization problem of a large investor who is allowed to make transactions on tradable assets in an incomplete financial market with endogenous permanent market impacts. The asset prices are assumed to follow a nonlinear price curve quoted in the market as the utility indifference curve of a representative liquidity supplier. Using generalized subgradients, we show that optimality can be fully characterized via a system of coupled forward-backward stochastic differential equations (FBSDEs) that corresponds to a nonlinear backward stochastic partial differential equation (BSPDE). We show existence of solutions to the optimal investment problem and the FBSDEs in the case in which the driver function of the representative market maker grows at least quadratically or the utility function of the large investor grows at least quadratically or is exponential. Furthermore, we derive smoothness results for the existence of solutions of BSPDEs. Examples are provided when the market is complete, the driver is positively homogeneous or the utility function is exponential.grows atleast Funding: T. Nguyen acknowledges support from the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2021-02594]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0376 .