Utility Maximization Under Endogenous Pricing

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 .