Unbounded Dynamic Concave Utilities via BSDEs

2024/04/22 by Shengjun Fan, Ying Hu, Fan, Shengjun +3
Economics, Econometrics and Finance · #Economic theories and models #FOS: Mathematics #Monetary Policy and Economic Impact #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2404.14059

openalex created_date 2024/04/17 · openalex publication_date 2024/04/22 · openalex updated_date 2026/08/03

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 Vallée-Poussin theorem, and Young's and Gronwall's inequalities constitute the main ingredients of the dual representation.

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