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Unbounded Dynamic Concave Utilities via BSDEs

Shengjun Fan, Ying Hu, Shanjian Tang

TL;DR

This paper addresses representing dynamic concave utilities for unbounded endowments via BSDEs with unbounded terminal values. It develops a comprehensive dual representation framework using the Fenchel-Legendre transform and de la Vallée-Poussin theorem, handling generators with linear, super-linear, sub-quadratic, and quadratic growth. For endowments with finite exponential moments, it proves the well-posedness of the dual representation: the dynamic utility $U_t(\xi)$ coincides with the value process $Y_t$ of a BSDE driven by the conjugate generator $g$, and the infimum is attained at a minimizer in the subdifferential $ ext{∂}g(Z)$. The results extend prior bounded-endowment theory to unbounded settings, provide precise tail-integrability regimes across four growth scenarios, and enable practical computation of unbounded dynamic utilities via numerical schemes such as Monte Carlo.

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.

Unbounded Dynamic Concave Utilities via BSDEs

TL;DR

This paper addresses representing dynamic concave utilities for unbounded endowments via BSDEs with unbounded terminal values. It develops a comprehensive dual representation framework using the Fenchel-Legendre transform and de la Vallée-Poussin theorem, handling generators with linear, super-linear, sub-quadratic, and quadratic growth. For endowments with finite exponential moments, it proves the well-posedness of the dual representation: the dynamic utility coincides with the value process of a BSDE driven by the conjugate generator , and the infimum is attained at a minimizer in the subdifferential . The results extend prior bounded-endowment theory to unbounded settings, provide precise tail-integrability regimes across four growth scenarios, and enable practical computation of unbounded dynamic utilities via numerical schemes such as Monte Carlo.

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.
Paper Structure (9 sections, 4 theorems, 156 equations)

This paper contains 9 sections, 4 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Statement of the main result
  4. Proof of \ref{['thm:3.1']}
  5. Two auxiliary propositions
  6. Proof of Assertion (i) of \ref{['thm:3.1']}
  7. Proof of Assertion (ii) of \ref{['thm:3.1']}
  8. Proof of Assertion (iii) of \ref{['thm:3.1']}
  9. Proof of Assertion (iv) of \ref{['thm:3.1']}

Key Result

Proposition 2.3

Denote $\bar{\xi}:=|\xi|+\int_0^T \bar{h}_t{\rm d}t$ and $Y^*:=\sup_{t\in [0,T]}|Y_t|$. We have the following assertions.

Theorems & Definitions (12)

  • Example 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • Example 3.2
  • Remark 3.3
  • Proposition 4.1
  • proof
  • Remark 4.2
  • ...and 2 more