Recursive utility optimization with concave coefficients

TL;DR

This paper develops a duality-based approach to recursive utility maximization with concave coefficients, deriving explicit solutions and optimal terminal wealth for investors with ambiguity aversion.

Contribution

It introduces a novel convex duality framework for recursive utility problems with nonlinear, nonsmooth coefficients, providing explicit solutions and applications.

Findings

Derived a variational formulation using Fenchel-Legendre transform

Translated the primal problem into a dual minimization problem

Explicitly solved for optimal terminal wealth in three investor cases

Abstract

This paper concerns the recursive utility maximization problem. We assume that the coefficients of the wealth equation and the recursive utility are concave. Then some interesting and important cases with nonlinear and nonsmooth coefficients satisfy our assumption. After given an equivalent backward formulation of our problem, we employ the Fenchel-Legendre transform and derive the corresponding variational formulation. By the convex duality method, the primal "sup-inf" problem is translated to a dual minimization problem and the saddle point of our problem is derived. Finally, we obtain the optimal terminal wealth. To illustrate our results, three cases for investors with ambiguity aversion are explicitly worked out under some special assumptions.