Benchmarks in the utility function have various interpretations, including performance guarantees and risk constraints in fund contracts and reference levels in cumulative prospect theory. In most literature, benchmarks are a deterministic constant or a fraction of the underlying wealth; as such, the utility is still a univariate function of the wealth. In this paper, we propose a framework of multivariate utility optimization with general benchmark variables, which include stochastic reference levels as typical examples. The utility is state-dependent and the objective is no longer distribution-invariant. We provide the optimal solution(s) and fully investigate the issues of well-posedness, feasibility, finiteness and attainability. The discussion does not require many classic conditions and assumptions, e.g., the Lagrange multiplier always exists. Moreover, several surprising phenomena and technical difficulties may appear: (i) non-uniqueness of the optimal solutions, (ii) various reasons for non-existence of the Lagrangian multiplier and corresponding results on the optimal solution, (iii) measurability issues of the concavification of a multivariate utility and the selection of the optimal solutions, and (iv) existence of an optimal solution not decreasing with respect to the pricing kernel. These issues are thoroughly addressed, rigorously proved, completely summarized and insightfully visualized. As an application, the framework is adopted to model and solve a constraint utility optimization problem with state-dependent performance and risk benchmarks.
Download