The paper considers the problem of optimizing the terminal wealth of a portfolio of assets from the perspective of the fund’s manager. The manager has his utility function \(U\), which fulfills standard assumptions (i.a. is concave) but is paid according to an incentive scheme \(g(W_T)\), which is a convex function of the terminal wealth. Consequently, the manager seeks to maximize the expected value of the function \(\bar{U}=U \circ g\), which is neither concave, nor convex.
The authors consider three problems with the following value functions: the initial problem \(u(x) = \sup E[\bar{U}(W_T)]\), the concavified problem \(w(x) = \sup E[\bar{U}^{**}(W_T)]\) and the dual problem \(v(x) = \inf E[\bar{U}^{*}(Y_T)]\), where \(\bar{U}^{*}\) is the convex conjugate of \(\bar{U}\), \(\bar{U}^{**}\) is the biconjugate (and concavification) and \(Y\) belongs to the set of dual processes to the processes of admissible wealth in the initial problem.
The main theorems of the article concern the existence and uniqueness of the solution to these three problems. The authors prove the existence of the dual problem and give a characterization of it using the set of local martingale measures. Moreover, they show that if the terminal value of the optimal solution to the dual problem, \(\hat{Y}_T\), has a continuous distribution, then there exists a unique solution to the concavified problem, and it coincides with the solution to the initial problem.
The authors consider the problem of optimizing terminal wealth in the standard one-dimensional Black-Scholes model and in several incomplete models: lognormal mixture model and models of stochastic volatility. They give conditions under which the assumption about the continuity of \(\hat{Y}_T\) is fulfilled.