A Framework of State-dependent Utility Optimization with General Benchmarks

TL;DR

This paper introduces a comprehensive framework for utility optimization involving stochastic benchmarks, addressing key mathematical challenges and demonstrating applications to constrained problems with state-dependent performance and risk levels.

Contribution

It develops a general theoretical framework for state-dependent utility optimization with stochastic benchmarks, including solutions and feasibility analysis.

Findings

Provides optimal solutions for stochastic benchmark utility problems.

Addresses measurability and feasibility issues in utility optimization.

Demonstrates applications to constrained optimization with state-dependent benchmarks.

Abstract

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 variable; thus, the utility is also a function of the wealth. In this paper, we propose a general framework of state-dependent utility optimization with stochastic benchmark variables, which includes stochastic reference levels as typical examples. We provide the optimal solution(s) and investigate the issues of well-definedness, feasibility, finiteness, and attainability. The major difficulties include: (i) various reasons for the non-existence of the Lagrange multiplier and corresponding results on the optimal solution; (ii) measurability issues of the concavification of a state-dependent utility and the selection of the optimal solutions. Finally, we show how to apply the framework to solve some constrained utility optimization problems with state-dependent performance and risk benchmarks as some nontrivial examples.