Backward SDEs with superquadratic growth

TL;DR

This paper investigates the solvability of backward stochastic differential equations with superquadratic growth, revealing conditions under which solutions exist or are non-unique, especially in Markovian settings.

Contribution

It establishes the non-existence of bounded solutions for certain superquadratic BSDEs and demonstrates the existence of multiple solutions when solutions do exist, advancing understanding of their behavior.

Findings

Superquadratic BSDEs may lack bounded solutions for some terminal conditions.

If a bounded solution exists, then infinitely many solutions are possible.

Existence of solutions is proven for Markovian BSDEs with bounded continuous terminal values.

Abstract

In this paper, we discuss the solvability of backward stochastic differential equations (BSDEs) with superquadratic generators. We first prove that given a superquadratic generator, there exists a bounded terminal value, such that the associated BSDE does not admit any bounded solution. On the other hand, we prove that if the superquadratic BSDE admits a bounded solution, then there exist infinitely many bounded solutions for this BSDE. Finally, we prove the existence of a solution for Markovian BSDEs where the terminal value is a bounded continuous function of a forward stochastic differential equation.