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

TL;DR

This paper investigates the conditions under which solutions to scalar BSDEs with Peano-type generators are unique when the terminal value is positive, employing control and analytical methods to establish uniqueness results.

Contribution

It introduces two novel methods—control-based and analytical—to prove the uniqueness of adapted solutions for a class of BSDEs with Peano-type generators.

Findings

Uniqueness of solutions is established when the terminal value is almost surely positive.

The control method links the BSDE to an optimal stochastic control problem.

The analytical method reduces the problem to a convex quadratic BSDE, enabling sharp results.

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 $\theta$-difference method, we give a sharp result in some special case, which includes the BSDE governing the well-known Kreps-Porteus utility.