On measure solutions of backward stochastic differential equations

TL;DR

This paper explores measure solutions for backward stochastic differential equations with quadratic growth generators, showing their relation to classical solutions and discussing cases of non-uniqueness with unbounded terminal conditions.

Contribution

It introduces the concept of measure solutions for BSDEs with quadratic growth and analyzes their relation to classical solutions under various terminal conditions.

Findings

Measure solutions coincide with classical solutions when terminal conditions are bounded.

Non-uniqueness of solutions can occur with unbounded terminal conditions, with measure solutions providing alternative solutions.

Examples demonstrate coexistence of classical and measure solutions in certain cases.

Abstract

We consider backward stochastic differential equations (BSDE) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. In case the terminal condition is bounded and the generator fulfills the usual continuity and boundedness conditions, we show that measure solutions with equivalent measures just reinterpret classical ones. In case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in case of non-uniqueness classical solutions that fail to be measure solutions can coexists with different measure solutions.