On backward stochastic differential equations and strict local martingales

We study a backward stochastic differential equation (BSDE) 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 Lp 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.