Unbounded Dynamic Concave Utilities via BSDEs

TL;DR

This paper develops a framework for representing unbounded dynamic concave utilities as solutions to BSDEs with unbounded terminal values, extending existing theories to more general growth conditions.

Contribution

It introduces a novel representation of unbounded dynamic concave utilities via BSDEs with unbounded terminal values, utilizing recent existence and uniqueness results.

Findings

Representation of unbounded utility as BSDE solutions

Attainability of the infimum in the utility

Extension of BSDE theory to unbounded generators

Abstract

The dynamic concave utility (or the dynamic convex risk measure) of an unbounded endowment is studied and represented as the value process in the unique solution of a backward stochastic differential equation (BSDE) with an unbounded terminal value, with the help of our recent existence and uniqueness results on unbounded solutions of scalar BSDEs whose generators have a linear, super-linear, sub-quadratic or quadratic growth. Moreover, the infimum in the dynamic concave utility is proved to be attainable. The Fenchel-Legendre transform (dual representation) of convex functions, the de la Vall\'{e}e-Poussin theorem, and Young's and Gronwall's inequalities constitute the main ingredients of the dual representation.