On backward stochastic differential equations and strict local martingales

TL;DR

This paper investigates backward stochastic differential equations driven by local martingales, revealing multiple solutions in the strict local martingale case and establishing conditions for uniqueness when a Lyapunov function exists.

Contribution

It demonstrates the existence of multiple solutions to BSDEs driven by strict local martingales and links these solutions to viscosity solutions of associated PDEs, depending on martingale properties.

Findings

Multiple solutions exist for BSDEs with strict local martingales.

Different solutions generate distinct viscosity solutions to PDEs.

Existence of a Lyapunov function ensures uniqueness of the viscosity solution.

Abstract

We study a backward stochastic differential equation whose terminal condition is an integrable function of a local martingale and generator has bounded growth in $z$. When the local martingale is a strict local martingale, the BSDE admits at least two different solutions. Other than a solution whose first component is of class D, there exists another solution whose first component is not of class D and strictly dominates the class D solution. Both solutions are $\mathbb{L}^p$ integrable for any $0<p<1$. These two different BSDE solutions generate different viscosity solutions to the associated quasi-linear partial differential equation. On the contrary, when a Lyapunov function exists, the local martingale is a martingale and the quasi-linear equation admits a unique viscosity solution of at most linear growth.