Portfolio optimization under nonlinear utility

This paper studies the utility maximization problem of an agent with nontrivial endowment, and whose preferences are modeled by the maximal subsolution of a backward stochastic differential equation (BSDE). We prove existence of an optimal trading strategy and relate our existence result to the existence of a maximal subsolution to a controlled decoupled forward-BSDE (FBSDE). Using BSDE duality, we show that the utility maximization problem can be seen as a robust control problem admitting a saddle point if the generator of the BSDE additionally satisfies a specific growth condition. We show by convex duality that any saddle point of the robust control problem agrees with a primal and a dual optimizer of the utility maximization problem, and can be characterized in terms of a BSDE solution.