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Uniqueness of adapted solutions to scalar BSDEs with Peano-type generators

Shengjun Fan, Ying Hu, Shanjian Tang

TL;DR

The paper tackles the nontrivial issue of uniqueness for scalar BSDEs with Peano-type generators when the terminal value is strictly positive. It develops two complementary strategies: a verification-based duality that identifies the Y-component as the value process of a stochastic control problem, and an analytic route that transforms the BSDE to a convex quadratic form to apply a $\theta$-difference technique; these culminate in a dual representation $Y_t=\essinf_{q_\cdot\in\mathcal{H}} Y_t^{q_\cdot}$ and the identification of an optimal control $q^*$ achieving the infimum, ensuring uniqueness within the positive-Y class. The paper also provides a sharp result for a special Peano-type BSDE (including the Kreps–Porteus utility case) using a variable change and the $\theta$-difference method, highlighting conditions under which uniqueness holds without requiring integrability of $1/\xi$. Together, these results extend the understanding of existence and uniqueness for non-Lipschitz BSDEs and have implications for stochastic control and utility-based finance models.

Abstract

A Backward Stochastic Differential Equation (BSDE) with a Peano-type generator, is known to have infinitely many solutions when the terminal value is vanishing, and is shown to have possibly multiple solutions even when the terminal value is not vanishing but nonnegative. In this paper, we study the uniqueness of adapted solutions of such a BSDE when the terminal value is almost surely positive. Two methods are developed. The first one is to connect the BSDE to an optimal stochastic control problem: under suitable integrability of the terminal values, with a verification argument, we prove that the first component of the adapted solution pair is the value process for the optimal stochastic control problem. The second one appeals to a change of variables, and is more inclined to analysis: by a change of variables, the original BSDE is reduced to a convex quadratic BSDE, and then using the $θ$-difference method, we give a sharp result in some special case, which includes the BSDE governing the well-known Kreps-Porteus utility.

Uniqueness of adapted solutions to scalar BSDEs with Peano-type generators

TL;DR

The paper tackles the nontrivial issue of uniqueness for scalar BSDEs with Peano-type generators when the terminal value is strictly positive. It develops two complementary strategies: a verification-based duality that identifies the Y-component as the value process of a stochastic control problem, and an analytic route that transforms the BSDE to a convex quadratic form to apply a -difference technique; these culminate in a dual representation and the identification of an optimal control achieving the infimum, ensuring uniqueness within the positive-Y class. The paper also provides a sharp result for a special Peano-type BSDE (including the Kreps–Porteus utility case) using a variable change and the -difference method, highlighting conditions under which uniqueness holds without requiring integrability of . Together, these results extend the understanding of existence and uniqueness for non-Lipschitz BSDEs and have implications for stochastic control and utility-based finance models.

Abstract

A Backward Stochastic Differential Equation (BSDE) with a Peano-type generator, is known to have infinitely many solutions when the terminal value is vanishing, and is shown to have possibly multiple solutions even when the terminal value is not vanishing but nonnegative. In this paper, we study the uniqueness of adapted solutions of such a BSDE when the terminal value is almost surely positive. Two methods are developed. The first one is to connect the BSDE to an optimal stochastic control problem: under suitable integrability of the terminal values, with a verification argument, we prove that the first component of the adapted solution pair is the value process for the optimal stochastic control problem. The second one appeals to a change of variables, and is more inclined to analysis: by a change of variables, the original BSDE is reduced to a convex quadratic BSDE, and then using the -difference method, we give a sharp result in some special case, which includes the BSDE governing the well-known Kreps-Porteus utility.
Paper Structure (9 sections, 6 theorems, 159 equations)

This paper contains 9 sections, 6 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and spaces
  4. Two Lemmas
  5. Main result
  6. Statement of the main result
  7. Proof of the main result
  8. Sharp result on some special BSDE
  9. An a priori estimate

Key Result

Lemma 2.1

For $\rho(\cdot)\in \mathbf{S}$, let Then, $\rho^*(\cdot)$ is a nonincreasing convex function on ${\mathbb R}_+$ taking values in ${\mathbb R}_+\cup \{+\infty\}$, and Moreover, for any $x_0\in {\mathbb R}_{++}$, by picking $q=\rho'(x_0)\in{\mathbb R}_+$ we have $\rho^*(q)\in{\mathbb R}_+$ and

Theorems & Definitions (18)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Corollary 3.4
  • ...and 8 more