A framework of state-dependent utility optimisation with general benchmarks

Benchmarks in utility functions have various interpretations, including performance guarantees and risk constraints in fund contracts and reference levels in cumulative prospect theory. In most of the existing literature, benchmarks are either a deterministic constant or a fraction of the underlying wealth; thus the utility is still a function of the wealth. In this paper, we propose a general framework of state-dependent utility optimisation with stochastic benchmark variables, which includes stochastic reference levels as typical examples. We provide the solution(s) and investigate the issues of well-definedness, feasibility, finiteness and attainability. The major challenges include: (i) various reasons for the non-existence of Lagrange multipliers and the corresponding results of the solution; (ii) measurability issues related to the concavification of a state-dependent utility function and the selection of solutions. Finally, we apply the framework to solve some constrained utility optimisation problems with state-dependent performance and risk benchmarks serving as non-trivial examples.